Probabilistic Methods in Fluids
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editors:
I M Davies N Jacob
A Truman Department of Mathematics University of Wales Swansea UK
0 Hassan K Morgan
N P Weatherill School of Engineering University of Wales Swansea
Proceedings of the Swansea 2002 Workshop
Probabilistic Methods in Fluids Wales, UK
ye L
14 - 19 April 2002
World Scientific NewJersey London Singapore Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.
PROBABILISTIC METHODS IN FLUIDS Proceedings of the Swansea 2002 Workshop Copyright 0 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, includingphotocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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Contents Preface
. . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
IRIMA
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
. . . . . . . . . . . . . . . . . . . . . . . . .
xi
Participants
Sergio Albeverio and Yana Belopolskaya . . . . . . . . . . . . . Probabilistic Approach to Hydrodynamic Equations
1
Hakima Bessaih and Franco Flandoli . . . . . . . . . . . . . . 22 A Mean Field Result for 3D Vortex Filaments Bjorn Bottcher and Niels Jacob . . . . . . . . . . . . . . . . . 35 Remarks on Meixner-type Processes Zdzistaw Brzeinaak . . . . . . . . . . . . . . . . . . . . . . Some Remarks on It6 and Stratonovich Integration in 2-smooth Banach Spaces
48
Tomas Caraballo . . . . . . . . . . . . . . . . . . . . . . . The Long-time Behaviour of Stochastic 2D-Navier-Stokes Equations
70
Pao-Liu Chow . . . . . . . . . . . . . . . . . . . . . . . . Semilinear Stochastic Wave Equations
84
Nigel J. Cutland . . . . . . . . . . . . . . . . . . . . . . . Stochastic Navier-Stokes Equations: Loeb Space Techniques & Attractors
97
Arnaud Debussche . . . . . . . . . . . . . . . . . . . . . The 2D-Navier-Stokes Equations Perturbed by a Delta Correlated Noise
115
Sergio Albeverio and Benedetta Ferrario . . . . . . . . . . . . Invariant Measures of Lkvy-Khinchine Type for 2D Fluids
130
Franco Flandoli . . . . . . . . . . . . . . . . . . . . . . . Some Remarks on a Statistical Theory of Turbulent Flows
144
Christophe Giraud . . . . . . . . . . . . . . . . . . . . . Some Properties of Burgers Turbulence with White Noise Initial Conditions
161
V
vi
Yuri E. Gliklikh . . . . . . . . . . . . . . . . . . . . . . . Deterministic Viscous Hydrodynamics via Stochastic Processes on Groups of Diffeomorphisms
179
Niels Jacob and Aubrey Truman . . . . . . . . . . . . . . . Further Classes of Pseudo-differential Operators Applicable to Modelling in Finance and Turbulence
191
Benjamin Jourdain and Tony Lelihre . . . . . . . . . . . . . Mathematical Analysis of a Stochastic Differential Equation Arising in the Micro-Macro Modelling of Polymeric Fluids
205
Hannelore Lisei and Michael Scheutzow . . . . . . . . On the Dispersion of Sets under the Action of an Isotropic Brownian Flow
. . . .
224
Aubrey Truman, Chris N . Reynolds and David Williams . . . . . Stochastic Burgers Equation in d-dimensions - A One-dimensional Analysis: Hot and Cool Caustics and Intermittence of Stochastic Turbulence
239
A m e n Shirikyan . . . . . . . . . . . . . . . . . . . . . . A Version of the Law of Large Numbers and Applications
263
Maricin SlodiEka . . . . . . . . . . . . . . . . . . . . . . Comprehensive Models for Wells
272
Enrique Thomann and Mina Ossiander . . . . . . . . . . . . Stochastic Cascades Applied to the Navier-Stokes Equations
287
Aubrey Truman and Jiang-Lun Wu . . . . . . . . . . . . . . 298 Stochastic Burgers Equation with Lkvy Space-Time White Noise TushengZhang . . . . . . . . . . . . . . . . . . . . . . . A Comparison Theorem for Solutions of Backward Stochastic Differential Equations with Two Reflecting Barriers and Its Applications
324
Aubrey Truman and Huaizhong Zhao . . . . . . . . . . . . . Burgers Equation and the WKB-Langer Asymptotic L2 Approximation of Eigenfunctions and Their Derivatives
332
Preface This volume contains papers presented at the “Probabilistic Methods in Fluids Workshop” which was hosted by the Department of Mathematics, University of Wales Swansea between the 14th and l g t h of April 2002. The aim of the meeting, the first IRIMA workshop, was to bring together internationally reknowned researchers from the areas of Pure Mathematics, Applied Mathematics and Engineering t o participate in a workshop, on probabilistic methods for fluids, and through collaboration further the mathematical understanding of the fundamental problems in this field. This international workshop successfully allowed leading researchers to present, reflect upon and discuss their recent work in the probabilistic modelling of fluids. This field stretches across Pure Mathematics, Applied Mathematics and Engineering and consequently is ideally placed to benefit from regularly arranged workshops for collaborative purposes. The Workshop mainly concentrated on the understanding of turbulence in stochastic fluid dynamics, a problem which has numerous applications in science and engineering and has defied many attempts to success full^ model it. The workshop bridged a gap between the recent year of activity at the University of Warwick and the year of emphasis at Princeton, which started in Autumn 2002. As such the workshop ensured that the research momentum in Britain, in this subject, was maintained. In this volume probabilistic approaches to hydrodynamic equations are reviewed and deterministic viscous hydrodynamics is discussed in terms of stochastic processes on groups of diffeomorphisms. At the Workshop significant progress was made in understanding the intermittence of stochastic turbulence for Burgers equation and the application of L6vy processes t o the Mathematics of Finance, both of which are represented in the proceedings. Other noteworthy developments concerned the Strong Law of Large Numbers and ergodicity of the Gaussian invariant measures for 2dimensional Navier-Stokes equations with space-time white noise and periodic boundary conditions and mean field results for 3-dimensional vortex filaments. Also, new results are presented on Loeb space techniques and attractors for stochastic Navier-Stokes equations. The long time behaviour of stochastic 2-dimensional Navier-Stokes equations is investigated as are vii
...
Vlll
perturbations by delta correlated noise. Burgers turbulence for white noise initial conditions is discussed in detail and the Cauchy problem for stochastic Burgers equation with L6vy space-time white noise is also examined. A complete mathematical analysis of stochastic differential equations arising in micro-macro modelling of polymeric fluids is given.
Scientific Organising Committee S. Albeverio, Y.I. Belapolskaya, Z. Brzezniak, A. Chorin, F. Flandoli, B. Rozovski, A. Truman We are especially grateful to Zdzislaw Brzezniak for his contribution to the organisation and success of the workshop, and to Roger Tribe for his guidance.
finding The workshop was supported by EPSRC grant GR/96545/01 “Probabilisitic Methods for Fluids - IRIMA” and we are indebted to EPSRC for their financial support and advice. Local Organisation We wish to thank Jane Barham and Janice Lewis for their forebearance before, during and after the workshop in providing secretarial and administrative support. We must thank also Bjorn Boettcher, Victoriya Knopova, Scott Reasons and Chris Reynolds for their contribution towards the successful running of the workshop. Finally, we thank the referees for their important but anonymous contribution in helping us to finish this volume on time. I M Davies N Jacob A Truman University of Wales Swansea, December 2002
0 Hassan K Morgan N P Weatherill
International Research Institute in Mathematics and its Applications Patron: Sir Michael Atiyah OM, FRS There is no such thing as Applied Science only the Applications of Science, Henri Poincark
We have established the International Research Institute in Mathematics and Its Applications, IRIMA, (Sefydliad Ymchwil Rhyngwladol i Fathemateg a’i Chymwysiadau, SYRIFAC) with the aim of conducting a series of research programmes in Mathematics and its applications t o Engineering and Science. In so doing we aim to accelerate the transfer of modern Mathematics to Engineering and the Sciences. These programmes should be seen to be interdisciplinary, with the express intention of providing a forum for interaction between groups of mathematicians, engineers and scientists, while at the same time preserving the integrity of the Mathematics being utilised. The Institute will be based in Swansea and will draw on existing strengths in Stochastic Processes, Physical Mathematics, Finite Element Methods and Theoretical Computer Science. Swansea (in the person of Oleg Zienkiewicz FRS) pioneered the use of Finite Element Methods in Engineering. More recently, his research group, which includes Profs. Nigel Weatherill, Ken Morgan FREng and Roger Owen FREng, has done vitally important research work in a number of different application areas, including work on the European Airbus and Thrust SSC, the supersonic car. Swansea also has international centres of research excellence in Probability Theory and ix
X
Applications as represented by the presence of Profs. David Williams FRS, Leonid Pastur and Aubrey Truman, in Theoretical Computer Science in Professor John Tucker’s research group and in Theoretical Particle Physics in the research team of Professor David Olive FRS.
Scientific Advisory Panel Prof. A. Truman (Chair), Prof. S. Albeverio (Bonn), Prof. C. Dafermos (Brown, Providence RI), Prof. D. Elworthy (Warwick), Dr. N. Jacob, Prof. R. Mackay FRS (Warwick), Prof. K. Morgan FREng, Prof. D. Olive FRS, Dr. M. Overhaus (Deutsche Bank AG London), Dr. D.P. Rowse (BAE Systems), Dr. D. Burridge (Meteorological Office), Prof. R. Owen FREng., Prof. E. Rees (Edinburgh), Dr. C. Sparrow (IN1 Cambridge and Warwick), Prof. J. Tucker, Prof. N. Weatherill, Prof. D. Williams FRS and Prof. 0. Zienkiewicz FRS FREng.
Workshop Participants Yana Belopolskaya, Department of Mathematics, St. Petersburg University for Architecture and Civil Engineering Hakima Bessaih, Dipartimento di Matematica Applicata, UniversitA di Pisa Bjoern Boettcher, Department of Mathematics, University of Wales Swansea Zdzislaw Brzezniak, Department of Mathematics, The University of Hull Tomiis Caraballo, Dpto. Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matemiiticas, Sevilla Pao-Liu (Paul) Chow, Dept. of Mathematics, Wayne State University Nigel J. Cutland, Department of Mathematics, The University of Hull Constantine Dafermos, Division of Applied Mathematics, Brown University Ian M Davies, Department of Mathematics, University of Wales Swansea Arnaud Debussche, ENS de Cachan, Bruz Karl Doppel, Fachbereich Mathematik und Informatik, FU Berlin Benedetta Ferrario, Institut fur Angewandte Mathematik, Bonn Universitat Franco Flandoli, Dipartimento di Matematica, Universitg di Pisa Mark Freidlin, Dept. of Mathematics, University of Maryland Christophe Giraud, Laboratoire J.A. Dieudonne, Universite de Nice SophiaAnti polis Yuri E. Gliklikh, Mathematics Faculty, Voronezh State University Oleg Gulinskii, Moscow Institute of Information Transmission Problems, Moscow Oubay Hassan, School of Engineering, University of Wales Swansea Niels Jacob, Department of Mathematics, University of Wales Swansea Mark Kelbert, European Business Management School, University of Wales Swansea Viktoriya Knopova, Department of Mathematics, University of Wales Swansea
xi
xii
Vassili Kolokoltsov, Department of Computing and Mathematics, Nottingham Trent University Markus Kraft, Department of Chemical Engineering, University of Cambridge Sergei Kuksin, Department of Mathematics, Heriot-Watt University Jose A. Langa-Rosado, Dpto. Ecuaciones Diferenciales y Analisis Numerico, Facultad de Matemhticas, Sevilla Tony Lelievre, CERMICS ENPC, Champs sur Marne Nikolai Leonenko, School of Mathematics, Cardiff University Yuhong Li, Department of Mathematics, The University of Hull Hannelore Lisei, Institut fur Mathematik, Technische Universitat Berlin Terry Lyons, The Mathematical Institute, University of Oxford Salah Mohammed, Department of Mathematics, SIU-C Carbondale Ken Morgan, School of Engineering, University of Wales Swansea Szymon Peszat, Institute of Mathematics, Polish Academy of Sciences, Krakow Scott Reasons, Department of Mathematics, University of Wales Swansea Chris Reynolds, Department of Mathematics, University of Wales Swansea James Robinson, Mathematics Institute, University of Warwick Institut Galilee, Mathematiques, Universite Paris 13 Francesco RUSSO, Michael Scheutzow, Institut fur Mathematik, T U Berlin RenQ Schilling, Department of Mathematics, University of Sussex Armen Shirikyan, Department of Mathematics, Heriot-Watt University Maridn SlodiEka, Department of Mathematical Analysis, Faculty of Engineering, Ghent University Andrew Stuart, Mathematics Institute, University of Warwick Enrique Thomann, Department of Mathematics, Oregon State University Alexander Tokarev, Department of Mathematics, University of Wales Swansea
xiii
Michael Tretyakov, Department of Mathematics and Computer Science, University of Leicester Aubrey Truman, Department of Mathematics, University of Wales Swansea Alexei Tyukov, School of Mathematical Sciences, University of Sussex Nigel Weatherill, School of Engineering, University of Wales Swansea David Williams, Department of Mathematics, University of Wales Swansea Wojbor A. Woyczynski, Department of Statistics, Case Western Reserve University Jiang-Lun Wu, Department of Mathematics, University of Wales Swansea Oleg Zaboronski, Mathematics Institute, University of Warwick Tusheng Zhang, Department of Mathematics, University of Manchester Huaizhong Zhao, Department of Mathematical Sciences, Loughborough University
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PROBABILISTIC APPROACH TO HYDRODYNAMIC EQUATIONS
S. ALBEVERIO Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D-53115 Bonn, Germany SFB 256, Bonn, BiBoS, Bielefeld - Bonn, CERFIM, Locarno and USI (Switzerland) YA. BELOPOLSKAYA St. Petersburg State University for Architecture and Civil Engineering, Russia, 198005, St. Petersburg, 2-ja Krasnoarmejskaja 4 We construct diffusion processes associated with the Navier-Stokes system in R3 and use them t o prove the existence and uniquenes of local solution of the Cauchy problem for this system in some functional space. AMS Subject classification: 60 H 15, 35 Q 30 Key words: Stochastic processes, Navier-Stokes system, probabilistic representations
1. Diffusion process and the Navier-Stokes equations
Among tremendous number of papers and books devoted to the investigation of the Navier-Stokes system there is relatively small number of works with probabilistic background. The very idea to consider the stochastic process with the drift velocity field subjected to the Navier-Stokes equation belongs to Nelson l and was presented in his functional analysis of the finite energy Navier-Stokes flow. To construct a diffusion process such that the Navier-Stokes equation can be treated as the backward Kolmogorov equation (BKE) for this process we consider the stochastic differential equation similar to one studied in '. The difference is in the relation used to define the drift coefficient. We apply here the probabilistic approach to study the Cauchy problem for nonlinear PDEs by reducing it to the investigation of stochastic differential equations with coefficients functionally depending on the distribution of the SDE solution developed in papers by Dalecky and Belopolskaya 2-3. 1
2
Notice that among recent works the papers by Busnello and Flandoli, Busnello, are mostly close to our approach. In these papers the authors deal with the equation for the circulation of the velocity field and the BiotSavart law (instead of the original Navier-Stokes system) and study it by probabilistic methods. In the present paper we construct diffusion processes that can be used for the probabilistic representation of the velocity field and the pressure itself. To this end we consider the Cauchy problem for the Navier-Stokes system
divu
=0
(2)
where u(t,z) E R3,z E R3,t E [0, co),P is a positive constant and p ( t , x) E R1 and change ( 2 ) for the Poisson equation
- A p ( t , x) = y ( t ,x), y = Tr[VuI2 (3) connecting the velocity field and the pressure. The topic of main interest for us in the present paper is the construction of diffusion processes in R3 associated with (1),(3). Let (0,F,P ) be a probability space and w ( t ) E R3,B ( t ) E R3 be a couple of independent Wiener processes defined on it. Denote by E the expectation with respect to P and by EE the conditional expectation with respect to a stochastic process [ ( t ) .Consider the Cauchy problem for the stochastic differential equation d<(T) = -U(t
- 7 ,< ( T ) ) d 7+ fJdw('T),
<(o) = 5
(4)
and assume the unknown drift coefficient u to be determined by
Since p is an unknown function as well we close (4),(5) by the relation
div u(t,z)= 0.
(6)
System (4)-(6) was considered previously in 6, 7. In this paper we study a system consisting of (4), ( 5 ) and
instead of (6). It is easy to check using Ito's formula that if u is smooth and together with p satisfy (1),(3)then u , pcan be represented in the form
(5), (7).
3
The inverse statement is valid as well. Namely, given a solution to (4), (5), (7) such that u , p are smooth enough we can state that (5), (7) determine classical ( or respectively C') solution to (l), (3) and hence to (1)0). By general results of diffusion process theory and in particular the Bismut-Elworthy formula we show that heuristic differentiation of (4), (5), (7) leads to
and
where 6 i k is the Kronecker symbol and the usual convention of summation over repeated indices is made. In addition we notice that Bismut-Elworthy's formula allows to derive from (7) the representation for V p
V d t ,). = -
m l
-SE [ y ( t ,
+ B(s))B(s)lds
(10)
and use i t t o eliminate the dependence on p from (5). After that we obtain a nonlinear integral equation involving u, Vu,show that it gives rise to a contractive mapping in a certain functional space and find the fixed point of this mapping by a successive approximation procedure. To realize this program we need some auxiliary results about the behavior of the solution to the Poisson equation which can be proved using standard techniques of integral estimates based on the Holder inequality. Let us recall some results concerning the probabilistic representation of the solution to the Poisson equation
where y ( t ,.) : R3 -+ R3 is a measurable function depending on the parameter t E [0, m). In the next two lemmas we recall the integral estimates for the solution of the Poisson equation which we need below. Let LY = L Y ( R ~~ ,n =) {j(.) E ~n : where 11 . 11 is a Euclidian norm in Rn.
{sR3~lf(.)llydx}+
=
Ilfll,
4
Lemma 1.1. Let y ( t ) E Lq2 n L 4 ( R 3 )for some 1 5 q 2 < 3 < q < 00 and arbitrary t from a compact [0,TI. Then for each x E R3 the integral
h"
+[y(t,
%
+ B(S))Bk(s)lds
converges. Moreover the function
belongs to Cb(R3)and (13) IPk(t, .)I100 5 Ilr(t,.)l142,4. If in addition q 2 > $, then Fk(t,x)= 2&p(t,z) . Lemma 1.2. Let y ( t ) E Lq2 nL4(R3)for some 1 < q 2 < 2 < 3 < q , t E [0,TI. Then for each r > 2 the functions x H Fk(t,x) belong to L' and
(14) IIFk(t)llr I Kllr(t)1142 , 4 ( 1 + Ilr(t)1142 A). Let us construct the solution for (4),(5), (7) by a successive approximation method. To this end we consider (15)
uO(t,x) = uo(x), p ( t ) = x, w
p k ( t ,x) = J
+
ETr [Vu0l2(t, x B(s))ds,
po(t,x) =
ETr [Vukl2(t, x f B(s))ds
0
where y k ( t , x )=
auk $$,
auk
(Ic
=
1 , 2 , . . . and q , j = 1,2,3) and
uk+l(t,x) = E[UO(<"(t))-
/
t
v p y t - 7,<'"(7))d7].
0
(18)
To prove the convergence of the successive approximations we have to add to (15)-(18) the successive approximations for the derivatives q(t) = at -&(t) and V u of the solutions to (4),(5), (7) and hence we need a probabilistic representation for the gradient V u . We recall the Bismut- Elworthy formula for a diffusion process satisfying a stochastic differential equation. Let v ( t , x ) , ( x E R 3 , t E [O,T]),be a differentiable vector field of sublinear growth (in x) and let fo E C2(R3).Consider the Cauchy problem for the stochastic differential equation d
=
+
-v(t, &(t))dt adw,
&(O) = z
(19)
5
and put
u(t,).
= E[uo(Jz(t))].
(20)
Then Bismut- Elworthy’s formula states that if ‘u has bounded spatial derivatives then u is a smooth function and Vu admits the representation
Here $ ( t ) satisfies the linear equation drlji = Vrn’ui(t,,$Z(t))rlrnj(t)dt, V k i ( 0 ) = Sji
(22)
and S j i is the Kronecker symbol. (We use the notation A = u @ g for the matrix A = ( a i k ) with matrix elements aik = u i g k and assume the summation in the repeating indices. ) It results from (10) and BismutElworthy’s formula applied t o f(t, z) = EVp(t,& ( t ) ) that the heuristic expressions for derivatives of (4),(5), (7) have the form d V j i ( t ) = -Vmui(t - 7, J(T))Vrnj(t)dt, rlji(0) = S j i
(23)
and
where
We set 0 = {O(t,z) = ( u i ( t , z ) , V j u i ( t , z ) ){}e, ( t , z ) : Ile(t)IIL. < m} < r < 2, and by 0 2 for r > 3. We then have is denoted by 01 for 0 = 01 n 0 2 . Let M = C([O,TI, 0 ) be the linear space of continuous functions defined on [O,T]and valued in 6 with the norm
[IP(t)llo + [V4t)lal.
l l ~ l I ~ , r= x S~PtG[O,T]
We prove that (un,Vun)converges in the norm of the space M for some fixed interval [0,TI. In fact we consider the space G = M U S where S is the space of vector and matrix valued processes with the norm determined by llE112 = sup&11,$(t)l12. In section 2 we determine the interval [O,T]and prove the convergence of (15)-(18), (23)-(26) in 6. This leads us to the following main results.
6
Theorem 1.1.Assume that (210, V U OE) 0. Then there exists a bounded interval [0,TI and a unique solution ( t ( t )u, ( t ,z), p ( t , z), V ( t ) , V u ( t ,z))to (4), (5), (7)-(9) in for each r E [0,TI. Theorem l.2.Assume that the conditions of theorem 1.1 hold. Then there exists an interval [0,TI such that for all t E [0,TI there exists a unique solution t o (1), (2) in M . The assertion of theorem 1.2 is a consequence of theorem 1.1 and the results of the theory of diffusion processes according to which the function u ( t ,z) given by (4),(5) satisfies the backward Kolmogorov equation of the form (1). Notice that the solution constructed in this way is a generalized solution to (1) since we can prove that u has a Holder continuous gradient but the existence of the second spatial derivative for the function u is not claimed. 2. Convergence of successive approximations To check the convergence of the successive approximations determined by (15)-(18) we need some auxiliary results about the solution [,"(t)to the equation rt
(27) with the smooth drift coefficient v ( t , x ) E R3,x E R 3 , t E [0, co) and its derivatives with respect t o the initial data V"(t)given by
V v ( t - 7, &(.))
0
VZ(T)dT,
where I is the identity matrix and Vv o denotes the matrix product. Lemma 2.1. Let v ( t ,z) E R3,(t E [0,co),z E R3) satisfy the estimates
I K,(t), s~P"Ilv(t,z)1I2
s~PZIIVv(t,s)1I2 5 K,(t),
and IIW1x)
-
Vv(t,Y)1l2I L:(t)Ilz - YII
2
where K, ( t ) ,K t ( t ) LA , ( t ) are positive time dependent scalar functions bounded on bounded interval [0,TI. Then
7
and 21 21 S O t [ K t ( t - - 7 ) + L t ( t - r ) ] d ~
EIlr12(t)- rlY(t)1121 I 1 1 2 - YII e
(31)
hold for t E [O, TI. Proof. I t results from Ito's formula and Gronwall's lemma that
Elltdt) - t Y ( t ) l 1 2 1 t
5 1l2-YIl2'+211
Ilrz(.)
El(4-7&(7))
- EY(7))/12(1-1)d7i
-"(t-7ltY(7)),&(4
1 1 5 - YII
21 21
e
-Jy(7))1
lot K,'(t-r)dr
To prove (30) we notice that for 2 = 1 we have
Ellrl(t)l12
< 1-+ 2
-
it
E(Vv(t - 7 ,t(7)>17(.>1rl(T))d7
1 t
I 1t 2
K t ( t - 7)E11rl(7)/)2d7
where K i ( t ) = supZ((Vv(t, .)I/. Then for arbitrary I the required estimates are derived in a standard way. Let us give some details on the proof of (31). To check (31) we consider the case 1 = 1 and derive the estimate for EllrlZ(t)- 17Y(t)ll2. E11r15(t)- rlY(t)ll2 5
8
The solution &(t)of (27) gives rise to the stochastic flow x addition
Vd&(T)
H
& ( t ) . In
d,V&(T) = -V[v(t - T , < ~ ( T ) ] ~ T .
For given $ ( t ) = V & ( t )we shall denote by J ( t , x) = det $ ( t ) the Jacobian of the random map x -+ &(t). Lemma 2.2. The Jacobian J ( t , x) satisfies the equation
dt J ( t , x) = J ( t , x)div v(& ( t ) ) . Proof. The determinant of a matrix is a mi tilinear function in the columns (or rows). Hence, fixing x,we have
dtVlE' V1t2 V1J3 d t V 2 t 1 V2t2V2E3 d t V 3 t 1 V3t2V3t3
V1C1d t U 1 t 2 V1J3
V2t' d t V 2 t 2
V2t3
V3J1d t V 3 t 2 V3J3
By (27) we have d t V j [ l ( t ) = V j [ v i ( & ( t ) ) ] d t Substituting . this relation along with v i [ v j ( < 2 ( t )= )] V k d V i t k into the above expressions for dt J we get
xi=,
JVlv'
+ J V 2 v 2 + J V 3 v 3 = (div
U ) J.
0
Remark If the drift vector field v(t, x) in (27) possesses the property d i v v = 0 , we deduce d t J = 0 and J ( t ) = J ( 0 ) = I . 0 Assume that v is a smooth divergence free vector field and consider functions p , and u given by
In what follows we denote by 11 . 11 either the Euclidian norm of a vector or the norm of a matrix respectively. As a rule for matrices we choose IlAll = rnaxjklajkl or the equivalent norm IJAIJ= TrA. Lemma 2.3. Let v(t),uo E C2 n L' and assume the estimates
II~oII, < tor, I I v w I I I ~
< c&,
SUP,
JJv~o(z)IJ
I K,,
SUP,
Il~o(x)IlI KO,
9
Il~(t)llTI CUT(t)I llVw(t)IIp< c:T(t)lS u P z I I W , .)I1
5 K,'(t)
hold. Then there exists an interval [0,T I ] (with T I depending on the functions U O , w )and functions ,B(t),y ( t ) bounded on this interval, such that the inequalities supzIIV4t,z)ll 5 P(t)
(34)
llvu(t)llT< y ( t )
(35)
and
9
hold f o r the function u ( t , x )given by (32), (33), 0 I t < T I and < r < 2 or r > 3. Proof. From the heuristic expressions (23)-(26) rewritten for the functions determined by (32), (33) by Jensen and Holder inequalities we deduce that ( ( V u ( t - s , x ) ( Il m 1 ( t - s , x ) + m z ( t - s 1 s )
(36)
where
m ( t - 3 , ~ )=
6"
0
-$(EIIVp(t - ~,
Ellr1z(7)112d~)~]de. (38)
To derive the estimate for ml(t - s, x) we notice that by (30) K: (t--7)dT
m i ( t - s , z ) I EllVuo(tS,,(t))/leLt
and derive the estimate for n1(t - s,z) = E / I V ~ O ( E , , ~ ( ~ ) ) / ( . Changing variables under the integral sign we deduce
Here J l ( t ) = d e t [ ~ ( t ) - lis] the Jacobian of the random transformation inverse to the transformation L H z e~ & ( t ) determined by (27). By Lemma 2.2 we conclude that
To derive the estimate for mz we apply the Holder inequality to the right hand side of (37). Taking into account (30) this yields -
~ , < z ( e ) ) I I z ~ l I r 1 z ( ~ ) 1 1 2 )5~ d e
10
To estimate respect to
\lm211rwe
e t o get
+
c6
&
s)llr IeJ3t
llm2(t -
I = 1. Choose for 1 43 44
where that
apply the Holder inequality to the integral with
43
K t ( t - ~ )v d( t~,s) where
< 2 and q 4 > 2 to obtain
t
do]& depends on t , s. If r > 4 and 2 1 then we can apply Jensen’s inequality to get =
where C7 depends on t , s. If 1
the estimate a: that
q4
are chosen so
& 5 1 a similar inequality can be derived by
< a valid for q , a > 1 assuming without loss of generality
[(qvp(t
- e,
> 1.
Finally, we derive the required estimate for v in terms of the Lr- norm of V p , using the properties of the Jacobian J proved in Lemma 2.2. In this way we obtain
44 s ) Ic7
I’ s,,
IlVp(t - 0, z)llrEIJl(t
-
8, z)lTdzdQ
L k3 t
I c7
IIVP(t - 8, z)IITdzdQ.
+
Recall that by (14) we have IIVp(t)ll, I K ~ ~ T r [ V ~ ] ~ ( t ) ~ l ~ ~ , < q1 < 2 and 3 < q 2 . Thus we obl l T ~ [ V u ] ~ ( t ) /for j ~r~ > , ~ 2~ and ] tain
11
where Mt = tQCt and finally choosing t large enough we get
Using the notation Cur@)= IIVu(t)ll. we deduce from the above estimates that
where K = Ka-,T 2Tl. Later we choose either 1 < r < 2 or r > 3. Let us derive next the estimate for K t ( t ) = s ~ p , I l V u ( t , x ) l 1 ~Using . the relation of the type ( 1 3 ) inspite of (14) and above considerations we obtain
Denote by
P(t - s) and y(t - s) functions that satisfy the relations
P(t - s ) = K o1 eJ,"~
t
?2(t
~ ( t - 7+~ 7
Finally, we notice that the functions y(t the system of ODE d y = Py
ds
+ KY2[1+ y2],
dP = p2 ds
+
Ky2,
-
-
e),J8' P ( ~ - - T ) ~dB1 T
(42)
s) and P ( t - s) are governed by
y(0) =
c;,,
P(0) = KO.
(44)
(45)
By the general theory of ODE systems we know that there exists a unique bounded solution to this system over an interval [O,Tl] depending on KJ,CJr.Finally we notice that if K,(t) 5 p(t) and C&(t) 5 y(t) then IIVull, 5 y ( t ) and sup,IIVu(t,x)II5 p(t) on the interval [O,Tl]as well. 0 Lemma 2.4 Under the conditions of Lemmas 2.1 - 2.3 there exist functions M i @ ) ,M,(t), K u ( t )and Z,(t) (bounded o n [O,Tl)for above 7'1 ) such
12
that the vector fields u given by (33) and V p for p given b y (32) obey the estimates II~(t)llql,r,qz
5 n(t)<
ll4t,x)ll 5 Pl(t),
0 0 1
supzIlVp(t.x)II 5 Z,(t) and ll~P(t)llql,T,q2
-5
where 1 < r < $ or r = 4 and < q 1 < 2 , q 2 > 3. Proof. The proof of these estimates can be derived from (32), (33) using the results of Lemma 1.2 and Lemmas 2.1 - 2.3. 0 Lemma 2.5.Assume that conditions of Lemma 2.1 - Lemma 2.3 hold. Let in addition u0 E C 2 ( R 3 )and llu0llcz 5 Kz. Then there exists an interval [0,T I ]with TI < T depending on KZ, KOand the function Lk(t - s ) bounded on this interval such that the estimate
I1Vu(t - s , x ) - V u ( t - s , y ) l l 2 < LL(t - s)11x -
(46)
holds. Proof. It is easy to check that
$(t,x,Y) = 11v4t,).
-
V u ( t ,Y)1I2
I 2Cl(t, 5 , Y) + 2 C 2 ( t , Zl Y)
where
C l ( 4 Z,Y)
= E / l [ ~ ~ o ( € z ( t ) ) 7 7 z ( t) ~~0(Jy(t))rly(t)1I2,
By the assumptions about no and
we derive form (23)-(26) that
Substituing (29)-(31) in (460 WE DEDUCE
13
To derive the estimate for satisfies the estimate
(2(t)
we notice first that y(t, x) = Tr[V2w(t, x)]
pUTTING
WE CAN WRITE n9T) IN THE FORM
and perform some computations based on Holder inequality and Fubini theorem. For N l ( t ) we derive using (46)
where C is a positive constant depending on LA and K;. To estimate N2(t) we choose Ic, I, m such that = 1, use (45) and the Holder inequality to derive
i+k+:
N2(t) 5
JT"
+
:2K;(t)(EI(V.u(t,z B ( s ) )- Vu(t, Y + B(s))Ilm)&
5.
for q' = Finally, choosing q < 2 and T < z q we prove that the integrals in the latter expression converge that leads t o the estimate
N ( t ) L CIK,(t)L:(t)Jlz
-
Yll.
14
Now we check that
where C is a positive constant depending on LA and KA. It remains to apply the Holder inequality to derive the estimate
€or
2+
= 1 and
a < 2. We use (30) to derive
st
$(t,5 , Y) I IIx - YII’[C~ + u - ~ I C0 ~ ~
C4[KLv(t-T)+K]dT
with positive constants Cs, C4 depending only on t and /3 and by Lemma 2.4 K = Supo
IIVu(k2) - V 4 t , Y ) / I 25 L:(t)llx - YII 2
‘
From the above estimates we get that there exist absolute positive constants
C3, C4 such that L i ( t )5
[c3+ u-1]c4eJot C~[KL:(~-T)+KI~T
Let us construct a majorizing function governed by the equation K(t
- s)
= MeL’
K(t)
for Lh(t) as the function to be
c 4 [ K K ( f, -7 )+K]d7
15
+
where A4 = C4[C3 a-1]. Choose It - sI i TI to ensure that Ki(i-1) < K < 03. As a result we deduce that n(t - s) solves the Cauchy problem
dK(t - S ) ds
+
)M = C ~ [ KK ] K , ~ ( 0=
and can be explicitly represented in the form
M eC4 (t-s 1 K ( t - s, =
where C
=
1
+
c
-
eC4(t-s)
Hence, if 0 < t < T3 where
c41c3+c-11. K
T3 = min(Tl,T2)
(53)
and
then K ( t - s) is bounded and V u ( t ,z) possesses the required property. 0 Lemma 2.6. Let the assumptions of Lemma 2.1 -2.4 hold. Then there exists a positive function L p ( t )bounded o n the interval [0,T3] given by (53) such t h d the fu,nction. V p ( t ,z) = E [ J r~ - ~ T r [ V v ] ' z( t+ , B(s))B(s)ds] satisfies the estimate
IlVP(tl.>
-
VP(t,Y>ll I LP(t)llZ - YII.
The assertion of this lemma can be deduced from the estimates derived in section 1, Lemma 1.2 and Lemma 2.5. 0 Lemma 2.7. Under the conditions of Lemma 2.1 the estimates
E l l l 2 ( t ) - E,""t)ll2
(54)
(55)
[Zk
hold f o r the solutions ( t ) ,~ x ~ (" tk) to (27), (28) for 5 = I , 2. Proof. We deduce from (27) that
16
and Gronwall's lemma yields (54). We prove the second estimate applying Gronwall's lemma t o (28) that yields q r l z v ' ( t ) - 77""2(t)1I2
I
Before coming back to successive approximation system (15)- (18) and its derivatives in x variable
where
we need one important remark. Since the gradient Vuk was proved to be uniformly (in k) bounded in L' norm for t E [O,T3)and
vipyt, x) =
.I
m l
-E[y'"(t,x S
+ B(S))Bi(S)]dS
3 auk auf where y'"(t, x) = we deduce that V p k ( t ,x) is uniformly (in k) bounded in L' norm as well. Moreover given rk(t)E Lq n C1icy(R3), Q E (0,1),1 5 q 5 we know by Schauder's theorem that IIV2p(t)Ilc; 5
dF
4
KIIYII Lqncg. Let u k , Vuk be successive approximations of tensor fields u,Vu defined by (15)-(18) and (55)-(58). Now we can prove our main result stated in Theorem 1.1. Proof of Theorem 1.1 Let us prove that u k ( t ) ,Vuk(t) given by (15) - (18), (55) - (58) converge in L' norm. Set
17
We start our considerations with general remark that to estimate all above functions we apply the Fatou-Fubini theorem to change the order of integration in t and z variables as well as Holder's and Jensen's inequalities. We denote further by K,, i = 1 ' 2 , . . . , constants which depend only on t and r and assume that s is choosen so that f = 1. Many of our computations use the Fubini theorem and the Holder inequality and the properties of the Jacobian J that allows to change the variables under the integral sign. Since we have used already this reasoning before in proving previous Lemmas we do not give below the detailed description of them. To estimate aL(t) we apply (34)' the Gronwall lemma and the Holder inequality to deduce
+
I K l [ es,"
1 t
KZPT(t-T)dT
+
,qt
-
T)dT]
(60)
To derive the estimate for ,Li(t) by the Lipschitz property of uo(z)and Holder inequality we deduce
By Lemma 2.4 and 2.5 we derive
In addition due to (7)
18
To derive the estimate for Z k r we rewrite it in the form 5
where
El/VUo(E,k-l ( t ) I/)'11 p k( t )- T
1 ; ' ( t )= L
p -
( t )11 ' d z
3
and the last three terms are derived by Bismut-Elworthy's formula that along with Fubini's theorem and the Holder inequality gives
To estimate Zkl, we use the Lipschitz property of Vuo(z) and estimates from Lemma 2.7 to obtain
I,&) 5 (L$aL(t)e-l;: /3 ( t-
T )d~
(64)
For 2zT(t)we derive from Lemma 2.7 estimates and properties of VUOthat
19
In addition
Ak(t) 5
t
L K5[/
r;(t - T ) ( Q ~ ( T ) ) ~ ~ T ] .
0
By the Holder inequality we have
and by estimates from Lemma 2.1 and the Holder inequality applied in the 0 variable we deduce for ml = r m < 2
with positive constant K6 depending on t and account (67) we get
ml.
Finally taking into
tO DERIVE THE ESTIMATE FOR iS(T) WE RECALL THAT WE HAVE
DUE TO THE ESTIMATES GIVEN IN SECTION 1. tHIS ALLOWS TO DEDUCE THAT
tO DERIVE THE ESTIMATE FOR
(T0 WE USE THE ESTIMATES FROM lEMMA
20
By the properties of stochastic integrals we have
where and
-rl1
Let us combine the above estimates (58)- (70) t o derive the following inequality t IIGk(t)lIT
I
M ( t - ~)IIGk-l(~)IITd~.
Here Gk(t,z) = ( a k ( t ) ,A k ( t ) ,L k ( t ) , Z k ( t ) )and M ( t ) is the corresponding positive scalar bounded function that can be read out of (58)-(70). By the above arguments we deduce that QnTn vn = SuPO
limn-mVn
= 0.
To prove that the solution constructed in this way is unique in 0 1 , suppose on the contrary that there exist two solutions u ( t , z ) ,<,"(t)and v ( t ,z),<,"(t)to (4), (5), (7) satisfying the same initial condition u(0,z) = v(0,x) = ug(z). Using the estimates of Lemma 2.5 we derive (71)
t
Ilu(t - T ) - w ( t - T)IITdT
1
+ c1
Ilu(t - ).
- v ( t - T)II.dTde
where the positive constants C , C1 depend on the interval [0,T ) and estimates for functions L h ( t ) , Kk(t) derived in the above Lemmas. Finally (71) yields that IJu(t)- v(t)lJ. = 0.
21
Acknowledgements We are very grateful1 to Professor Aubrey Truman for the kind invitation to an interesting and stimulating conference and to the University of Wales for the hospitality. The financial support by DFG Project 436 RUS 113/593 and by Grant RFBR 02-01-00483 are gratefully acknowledged.
References 1. Nelson E. Les e‘coulements incompressibles d’energie finie. Colloques intern. d u CNRS” 117,159, (1962). 2. Belopolskaya Ya., Dalecky Yu. Investigation of the Cauchy problem for quasilinear parabolic systems with the help of Markov random processes. Izu, VUZ. Matematika, N 12, 5 (1978). 3. Belopolskaya Ya. I., Dalecky, Yu. L. Stochastic equations and differential geometry, Kluwer Acad. Publ., (1990). 4. Busnello B. A probabilistic approach t o the two-dimensional Nauier-Stokes equations. The Annals of Prob. 27, N 4, 1750,(1999). 5. Busnello B., Flandoli F., Romito M. A probabilistic representation f o r the vorticity of a 3D viscous fluid and f o r general systems of parabolic equataons. Preprint (2002). 6. Belopolskaya Ya. Probabilistic representation of solutions to boundary-value problems for hydrodynamic equations Zap. nauchn.sem. P O M I , 249, 77, (1997). 7. Belopolskaya Ya. Burgers equation o n a Hilbert manifold and the motion of incompressible fluid,Methods of Functional Analysis and Topology, 5 , N4, 15 (1999). 8. Elworthy K.D., X-M.Li. Formulae for the derivatives of heat semigroup. JFA 125,252, (1994).
A MEAN F I E L D R E S U L T F O R 3D VORTEX F I L A M E N T S
H. BESSAIH AND F . FLANDOLI Dipartimento di matematica applicata U.D i n i , V i a B o n a n n o 25/B 56126 Pisa, IT E-mail: bessaihodma. unipi. it, jlandoli@dma. unipi. it
A mean field result is proved for a n abstract model, under a class of conditions on t h e rescaling of the energy. Propagation of chaos, variational characterization of t h e limit Gibbs density h and a n equation for h are proved. The general results are applied t o a model of 3D vortex filaments described by stochastic processes, including Brownian motion and Brownian Bridge, other semimartingales, and fractional Brownian Motion.
1. I n t r o d u c t i o n
The importance of thin vortex structures in 3D turbulence has been discussed intensively in the last ten years, see 4 , Some mathematical models of vortex filaments, based on stochastic processes, have been proposed by Chorin 4, Gallavotti Lions-Majda 14, Flandoli 5, Flandoli-Gubinelli and Flandoli-Minelli The importance of these models for the statistics of turbulence or for the understanding of 3D Euler equations is under investigation. The limit properties (mean field) of a collection of many interacting vortices has been investigated by P. L. Lions and A. Majda l 4 for a particular model of “nearly parallel” vortices. The aim of our work is to investigate a similar limit for the model introduced in 5 , ‘. Here the expression for the kinetic energy is not approximated and filaments may fold, so some features are more realistic. However, the filament structures have a fractal cross section (as observed numerically) to eliminate a divergence in the energy. The structure of the paper is the following one. In the next section we present the abstract frameworks and state a mean field result for them. Section 3 is devoted to the proofs. Then in the final section we apply the general result to some models of vortex filament.
‘
22
23
2. Abstract mean field result
Let us define the abstract framework. Let ( R , d ) be a complete separable metric space (it will be the space of vortex structures) and let B be its Bore1 cT-algebra. Let po be a probability measure on (R, B).Let H s and H I be two random variables
H s : R -+ R,
HI :R x R
+R
with the meaning of self and interaction energy. Assume
N
i=l
i#j
(here and below, the second summation is extended from 1 to N ) with the understanding that for N = 1 it reduces to
H s ( w ) 2 0 po-a.s. (5) (hence also Sn2 HId& < a). Let us assume also the following conditions on H I : H I ( W , w’)f(w)f(W’)&;
2 0,
f E Lrn(R,Po),
(6)
and that
HI(u,u’) is symmetric in w and w‘, Let H N : ON defined as
4
2
po -a.s.
(7)
R, for any positive integer N , be the random variable
The variable H N has the meaning of a rescaled energy, where only the interaction energy is reduced as N grows. A physical motivation for this rescaling has to be found in each particular case. Denote the product measure of N copies of po by p t . Let
hN :RN + R be the probability density defined as
hN =
(N)-’ e - p H N ,
Z ( N )=
lN
ePPHNdpf
24
where ,8 > 0 is a given parameter, with the meaning of inverse temperature. Denote by p N the measure (it is not a product measure)
dpN
= hNdp,N.
Finally, denote by p N , k the k-marginals of p N on RN (by symmetry, the choice of the k-components is irrelevant). We have
dpNh = hN,k where hNik : Rk
---f
k
PO
R are given by
hN’k(W1l...lwk)=
s,r
hN(W1,...,WN)dpo(#k+1) ...dPo(Wiy)
Under the assumptions (2), (5), (3), (6) and (7), we have the following result. Theorem 2.1. For each k 2 1, pNik converges weakly as N --+ the sense of probability measures on R k ) to a product measure @)zk=l addition
dp
(in p. I n
00
= hdpo
where h E L““ ( R ,P O ) satisfies the equation h ( w ) = -e1 Z’
-P(Hs(w)+J,
HI (w+J’)h(w’)&o ( w ’ ) )
with
Moreover, h as the unique minimum of the following free energy functional
over the set { f L 0 pa
-
a s . , f E Loo (0,P O )
J, f (w)dpo = I}.
Remark 2.1. The theorem states that, in the limit N + 00, the filaments behave independently (the so called propagation of chaos). Moreover, the limit Gibbs density h of each filament is associated to an energy given by the sum of the self-energy of the filament plus the interaction term J , H I ( w ,.)hdpo. The latter describes the interaction between the filament and the mean field associated to h itself.
25
3. Proofs We introduce new notations to shorten the formulas. We set
H ( q J l , . . . , U p J ) = HS(W,), and H("qU1,.
. ., U N )
= HI(Wn,Um).
In the sequel, we simply write H(") and H("1") without their arguments. 3.1. Uniform bound on the marginals densities
Lemma 3.1. For every given k such that
2 1, there exists a positive constant C ( k )
hN)k5 C ( k ) a.s. on Rk for all N 2 k .
(16)
Proof. The proof of this lemma will be done in three steps. Let us define
In particular Z ( N ,N , 1) = ZN. Step 1 Given k 2 1, there exists a constant No(k) such that for all
N 2 No(k) 2k hNYk 5 ( Z ~ J ) -Z' ( N ,N - k , 1 - -) N Step 2 For every p
> 0 and for every N 2 k 2 1,
Z", N,P ) 2 (CZ(IL)YZ ( N ,N
- k, P
k
+ ;v)
where 0
Step 3 Let k 2 1 be given and let c k = 3k. Then, there exist constants C3(k) and Nl(Ic) such that c k
Z ( N ,N , 1 - -) N
I C 3 ( k ) Z ( NN, , l),
ViV
2 Nl(k).
To conclude the proof of the lemma, we collect the estimates of the three steps and have (for sufficiently large N )
26
hN'k 5 (ZN)-' Z ( N ,N
-
2k k , 1 - -) N
= (ZN)-' Z ( N , N
-
k , 1 - - $- -) N N
c k
5 ( Z N ) - ' ( C Z ) -Z~( N ,N , 1 -
z) c k
5 ( Z N ) - ' c 3 ( k ) ( C Z ) -Z~( N ,N , 1) = C3(k)(Cz)-"
The proof is complete. 3.2. Variational characterization of hN and known results
Let us introduce the following free energy functional
over the set
{
P= f 2o
p t
-
as.,
fEL~(oN;~:),
Lemma 3.2. The density hN is the unique minimum of F N .
Proof. The proof is classical, see
0
17.
3.3. Weak limit of hNyk From the uniform bound (16), by a diagonal procedure, we can extract a subsequence Nj of N , independent of k , still denoted by N in the sequel, such that for all given k and for N + DC, hNIk
-
-
h",k, weakly * in LO",
hNVk hmik, weakly in Lp for all p
21
for some hoo%k E L" ( O k ,p.,"). We easily have h"ik 2 0, Soh h">'dp.,"= 1, and h">k is symmetric (from the analogous properties of h N ) k ) . From the symmetry and Hewitt-Savage theorem l o we deduce that there exists a measure if on P such that
27
3.4. Convergence of the variational problems
Let us denote by II the set of all probability measures the following functional on
7r
on P. We define
II,
3 . 5 . Properties of the limit variational problem
Up to now we have proved that the Gibbs densities hNik have a subsequence converging t o some density hmikl with symmetry properties] and such that the associated measure i f minimizes the functional F ,
F(4 =
s,
F(f)dT(f),
where F is the functional given in theorem 2. Let us prove the following basic fact: Lemma 3.4. That there exists h E P such that ?i = 6h
(28)
and h is the unique minimum of F over P . Due to (28), this implies that hm3k factorizes] i.e. the associated measure is a product measure. This proves the first claim of Theorem 2. Moreover, i t proves that the functional F has a unique minimum. So, let us prove the lemma.
Proof. By definition] we have F(T) = S , F ( f ) d ~ ( f )for all 7r E II. Let us show that the set S of minimum points of F is non-empty and f is
28
concentrated on this set S. Since F is strictly convex, S reduces to a single point h, and therefore the claims of the lemma are proved. Let F be the infimum of F on P . For every 7r E II we have
F(%)=
s,
F ( f ) % ( d f )2
s,
r;’?i(df) = F
and on the other side, if h, is a minimizing sequence for F ,
F(%)5 F ( S h h ) = J’, F ( f ) S h , ( d f ) = F(h,)
-+
F
so
F ( % )= F . It follows that
Since F ( f ) - F 2 0, this implies F ( f ) - F = 0 %-as. This proves at the same time that 5’ is non-empty and % is concentrated on 5’.The proof is complete. Kl 4. Application to vortex filaments
First we describe in detail the application of the mean field theory to vortex filaments modelled by Brownian trajectories. At the end of the section we shortly describe a generalization to certain semimartingales (including Brownian bridge and models of vortices a t a solid boundary) and t o processes with finite p variation, p < 2 (including fractional Brownian motion with Hurst parameter H > and related nongaussian models).
i,
4.1. Brownian vortex filaments 4.1.1. Introduction
We consider a fluid in EX3. The kinetic energy
of a velocity field u(x) can be written in terms of the vorticity field [(z) = curlu(s) as
29
In the case of a vorticity field ideally concentrated along a curve y(t), t E [0,1], the vorticity field is formally defined as
1
1
E(.) where the parameter
=
r
b(x
-
y(t))?(t)dt,
r is the circulation and the energy takes the form
In the case of N curves y' , ...,y N , we have
For regular curves, as well as for many examples of curves given by paths of stochastic processes, this expression (with suitable interpretation for processes) is divergent. Physical vort,ex structures, although very thin, have a cross section. Re-introducing the cross section increases the degrees of freedom and makes the model less intrinsic, but helps t o eliminate the divergence of the energy. To keep a closer relation with the vortex structures observed in fluids, it is better to consider fractal cross sections instead of simply a tubular mollification. In the previous papers ', vorticity fields of such kind with finite energy have been constructed. They are formally expressed as
where p is a probability measure describing the cross section and subject to the assumptions given below, and (Wt)tE[o,llis a Brownian motion in EX3. The corresponding energy takes the form
and in i t is proved to be meaningful and finite, with probability one. In the case of N copies ( W / ) t E [ o , kl l ,= 1, ..., N , of 3-D Brownian motions, and N probability measures p', . . . , p N on EX3, the energy takes the form
In this sum the terms of the form
30
represent the self energies of the single filaments] while the terms
with n # m give us the interaction energy between the filaments n and m. By easy manipulations with Fourier transform (see 6 , we rewrite the self energy in the form
and similarly we rewrite the interaction energy in the form
(44) The presentation until now have been rather informal] but we give below rigorous definitions.
4.1.2. Space of configurations Following the previous description, a single vortex filament is defined to be an element of the product space
R
=
c x M-1,
where C = C ([0, TI;R3)and M-1 is the space of probability measures p in R3 defined below. The interpretation is that the filament has a core and a cross-section. The core is a 3-D curve, i.e. an element of C([O,TI;R3). The cross-section is a probability measure p, the support of the measure represents the geometric cross-section, while the measure weights the intensities of the different lines of vorticity. Thus R is the space of configurations of a fluid when the vorticity field is made of a single vortex filament with cross-section. The space of configurations of a collection of N vortex filaments is O N l the product of N copies of R. 4.1.3. Cross-section and its random selection
The cross-section of the vortex structures considered here will be described by the probability measures p of the following form. For any probability measure p on the Bore1 sets of R3,let us set
31
where b ( k ) = Jw3 ei"'"p(dz) and let us denote by M-1 the set of all p such that 11 p llT1< 00. We recall from classical potential theory, see l 2 that a probability measure p E M-1 is called a measure with finite energy. Given a set A in R3, there exists a probability measure p supported by A with finite energy if and only if the capacity of A is strictly positive. Finally, by Theorem 3.13 of 12, every compact set with Hausdorff dimension d > 1 has positive capacity. Therefore, it supports a probability measure p satisfying (46). On the space M of all probability measures p on (R3, 2 3 (R')) there is a metric d such that the convergence with respect to d is the weak convergence of probability measures:
for all bounded continuous functions f . We endow the subset M-1 with the metric d . Notice that ( M , d ) is complete, while ( M - 1 , d ) is not, but this fact has no importance in the sequel. Let us denote by BM the Borel c-algebra of (M-1, d ) . Let PM be a probability measure on ( M - 1 , B M ) . Our vortex structures will have a cross-section measure p choosen a t random with probability law PM. In the sequel, we denote by B the product a-algebra Bc @ BM on R
a=&@BM where BC is the Borel a-algebra on C = C ( [ 0 , T ] ; R 3 )Moreover, . if PC denotes the Wiener measure on C, we set PO = PC @ P M .
4.1.4. Reference measure The statistics of vortex filaments are given by probability laws on the configuration space defined by a Gibbs weigth with respect t o a reference measure. In the case of a single vortex filament, we choose as reference measure the product measure po on (R, B). In the case of N copies of vortex filaments the reference measure is the product measure p r on ( O N ,B N ) ,product of N copies of the measure po. Therefore we also have p r = P? @ P c , This
32
choice of the reference measure is very natural from a probabilistic point of view, but rather arbitrary from the fluid dynamic viewpoint. However, we do not have at the moment more accurate physical prescriptions.
4.1.5. The energy With the motivations given above, we define the self-energy H S : 0 -+ a vortex structure as the random variable
R of
where (Wt)t,[o,ll is the canonical process on the Wiener space C. In the case when p E M-1 is given, it is proved in that the random variable H S is well defined and it has finite expectation. As a joint function also of p, we prove a similar result. For the measurability of Hs, notice that it is defined in terms of integrals in Ic of the product of measurable functions of ( p , k), namely ,6 (k) (see a previous subsection) and measurable functions of k and the Wiener path. In particular if we assume that
then, we have
4.2. Other models
4.2.1. Brownian semimartingales This section is based on the paper ‘. The results described above for the Brownian motion extend with the same proofs to the case when the self energy is defined as
where (Xt)tE[o,ll is a Brownian semimartingale, i.e. a process of the form
X t = Wt
+
1 t
b,ds
33
with (Wt)t,[o,ll a 3D Brownian motion and (bt)tEjo,ll a progressively measurable process. We need the condition
t o have that H s is integrable, and therefore to apply the abstract result. This model with a Brownian semimartingale is quite flexible. It covers the Brownian bridge (hence closed filaments) and non Gaussian examples (based on Bessel processes) like processes living in a half space, with endpoints on the boundary of the halfspace (modelling filaments on a solid boundary). See for more details. Without any change, the same fact holds true for the model based on
where p k is the projection on the plane orthogonal to k . For open filaments there is an argument in showing that this expression is preferable from the fluid dynamic point of view. 4.2.2. Processes with finite p variation
In the self energy (53) has been defined in the case when X is a process with finite a variation, for a E (1,2), and p fulfills the stronger condition
The assumption on the process ( X t ) t c [ o , lisl that for every p exists a constant C, > 0 such that
IE [IXt - X,(P] I C,lt
-
q'a,
2
1 there
v s , t E [O, 11.
In this way we cover the fractional Brownian motion with Hurst parameter H E 1) and non Gaussian variants of it, like solutions t o nonlinear stochastic equations driven by the fractional Brownian motion, or modifications of the fractional Brownian motion conditioned to live in a half space. Restricted to the gaussian case, the same problem has been solved by different techniques by l6 under less restrictive conditions on p. Under the condition
(i,
the inequality (52) holds.
34
References 1. P. Blanchard., E. Briining. (1992). Variational Methods in Mathematical Physics. A unified Approach. Springer-Verlag. 2. E. Caglioti., P. L. Lions., C. Marchioro., M. Pulvirenti. (1992). A Special Class of Stationary Flows for Two-Dimensional Euler Equations: A Statistical Mechanics Description. Comm. Math. Phys 143,no 3, 501-525. 3. E. Caglioti., P. L. Lions., C. Marchioro., M. Pulvirenti. (1995). A Special Class of Stationary Flows for Two-Dimensional Euler Equations: A Statistical Mechanics Description 11. Comm. Math. Phys 174,no 2, 229-260. 4. A. Chorin. (1994). Vorticity and Turbulence. Springer-Verlag, New York. 5. F. Flandoli. A probabilistic description of small scale structures in 3D fluids. To appear on Annales Inst. Henri Poincark, Probab. & Stat. 6. F. Flandoli, M. Gubinelli. Gibbs ensembles of Vortex filaments. To appear on Prob. Theory and Related Fields. 7. F. Flandoli, I. Minelli. Probabilistics models of vortex filaments. To appear on Czechoslovak Mathematical Journal. 8. U . Frisch. (1998). Turbulence, Cambridge Univ. Press, Cambridge. 9. G. Gallavotti. (1996). Meccanica dei jluidi. Quaderni CNR- GNAFA n. 52, Roma. 10. E. Hewitt, L. J. Savage. (1955). Symmetric measures on Cartesian products. Trans. A m e r . Math. SOC40, pp470-501. 11. H. Kunita. (1984). Stochastic Differential Equations and Stochastic Flows of Diffeomorphisms, Ecole d’6t6 de Saint-Flour XII, 1982, LNM 1097, P.L. Hennequin Ed., Springer-Verlag, Berlin. 12. N. S. Landkof. (1972) Foundations of Modern Potential Theory, SpringerVerlag, New York. 13. P.L. Lions. (1997). O n Euler Equations and Statistical Physics, Scuola Normale Superiore. 14. P.L. Lions, A . Majda. (2000). Equilibrium Statistical Theory for Nearly Parallel Vortex Filaments. C. P. A. M, Vol. LIII, pp 0076-0142. 15. C. Marchioro, M. Pulvirenti. (1994) Mathematical Theory of Incompressible Noviscous Fluids, Springer- Verlag, Berlin. 16. D. Nualart, C. Rovira, S. Tindel. Probabilstic models for vortex filaments based on fractional Brownian motion. In preparation 17. D. Ruelle. (1969). Statistical mechanics: rigorous results, W. A. Benjamin, New York- Amsterdam.
REMARKS ON MEIXNER-TYPE PROCESSES
BJORN BOTTCHER AND NIELS JACOB Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales, UK E-mail: mabb @swan.ac.uk, n.jaco
[email protected]. uk We construct Feller processes called Meixner-type processes by making the parameters of the characteristic exponent of a Meixner process state space dependent. Our main tool is the theory of pseudo-differential operators. A further aim of this paper is t o popularize these methods among probabilists. Key words: Meixner process, Meixner-type process, pseudo-differential operators, L6vy-type processes. MSC-classification: 60J75, 60J35, 35899
1. Introduction L6vy processes are becoming more and more important in modeling, compare the collection of surveys recently edited by 0. Barndorff-Nielsen, Th. Mikosch and S.I. Resnick '. Several applications are related to the mathematics of finance, see in particular the contributions of 0. Barndorff-Nielsen and N. Shephard 4 , and E. Eberlein 7 , in '. By definition a L6vy process has stationary and independent increments. This is reflected by the fact that its transition probabilities form a convolution semigroup and that the generator A of the corresponding Feller semigroup is translation invariant. On smooth functions it is given by
where $ is the characteristic exponent of the LQvyprocess under consideration and 6,is the Fourier transform of u.Often $ depends on parameters, i.e.
$(c>
= +a,b,c,...
(6).
When we model with a specific LQvyprocess having the characteristic ex) happen that at certain threshold values the paponent $ u ~ b ~ c ~ ~ . it. ( <may rameters a , b, c, . . . change. Thus for fixed zo in the state space we have a Levy process with characteristic exponent $,a(.o)ib(.o),c(.o),... (<), but in general z ++ $~(.)~'(.)~'(.)~...(~) is not constant, i.e. the process modeling 35
36
such a situation will not any longer be a Levy process, but a Lkvy-type process. To our knowledge it had been 0. Barndorff-Nielsen and S. Z. Levendorski; who first handled such a situation in case of normal inverse Gaussian (type) processes. Their work had been stimulated by problems of modeling in finance. Their approach via pseudo-differential operators had partly been influenced by the survey 1 3 . In this paper we take up the ideas from of treating L&y(-type) processes with state space dependent parameters by using the theory of pseudodifferential operators. The family under consideration consists of Meixnertype processes, i.e. we start with the characteristic exponent (1) of a Meixner process and make the parameters state space dependent. The choice of these processes is due to more recent work of W. Schoutens 21 and W. Schoutens and J. Teugels 22 on modelling in finance with Meixner processes. In a first section we recall basic facts of Meixner processes and in the following section we introduce symbols of Meixner-type, i.e. symbols q(z,<) where for fixed 20 the function H q ( q , <) is a characteristic exponent of a Meixner process. It is proved that under reasonable restrictions on the 2-dependence of the parameter functions these symbols are elliptic symbols in the classical class S1(R).Finally in Section 3 we prove that to every Meixner-type process corresponds a unique Feller process, and some short time asymptotics of the corresponding transition functions is discussed. An asymptotic of the transition function with respect to the state space variable will go along the lines of the considerations in and is not discussed here. A remark t o the style of the paper: We aim to give more, and in a certain sense new tools into the hands of those who are modeling with LBvy processes or more general jump processes. More details (on a technical level) of the impact of pseudo-differential operators in the theory of Markov processes are given in 13, the more recent survey and in the monographs l4 and 15.
<
''
Acknowledgement: The second author would like to thank Ole Barndorff-Nielsen and Sergei Levendorski for stimulating discussions about their paper. The first named author acknowledges financial support from the EPSRC-Doctorial Training Grant of the Mathematics Department of the University of Wales Swansea.
37
2. Meixner Processes In this section we summarize various results on Meixner processes, i.e. realvalued LBvy processes whose characteristic exponent is a continuous negative definite function of type $ m , 6 , a , b ( < ) := -Zm<
+ 26
a< - ib 2
where m E R, 6 > 0, a > 0 and -7r < b < 7r. For the corresponding process (Xy'6'"'b)t20we find
According to W. Schoutens and J. Teugels 22 and B. Grigelionis lo the transition density for the Meixner process with characteristic exponent '$'m,6,a,b is given by
For the L6vy characteristic of $m,b,a,b (or ( X r 7 6 9 a 9 b ) t we > o find ) the drift
and the LBvy measure
where
is the one-dimensional Lebesgue measure. Clearly, there is i.e. we have $ m , 6 , a , b ( < ) = ir< JR,(,,) 1 - eixt The expectation of X y ' * l a l is b. given by
and its variance by
A straightforward calculation yields
+
+
38
and Im
~ L + 5 , ~ , b (= < ) -m<
+ 26tan-1
(9)
Moreover we find Re for all
‘$m,6,a,b(<)
2
a6
yl<1
< E R, 161 2 y ,as well as with some co > 0 IIm
‘$m,fi,a,b(()I
5 CO(1 + Re ‘$m,6,a,b(<)).
(11)
In addition it follows that I‘$m,fi,a,b(<)I
5 c l ( l + 151)
(12)
<
holds for all E R. From the considerations of Chr. Berg and G. Forst 5 , see also ample 4.7.32, and those in Z.-M. Ma and M. Rockner l9 we derive
14, Ex-
Proposition 2.1. The Meixner process ( X r ’ 6 ’ a ’ b ) t 2 0 is associated with the non-symmetric Dirichlet form given on S(R)by
E(u,
s,
‘&n,6,a,b(<)a(<).;(E)
d<.
(13)
Its domain D ( E ) is the classical Sobolev space H i ( R ) , hence ( I , D ( € ) )is regular, and on H1(R) we have b y (4) and (5) E(u,w ) = y
s,
u’(z)v(x)d x
Since for the study of symmetric Dirichlet forms much more (analytic) techniques are available, compare the monograph of M. F’ukushima, Y. Oshima and M. Takeda let us have a short look at the symmetric part of the Meixner process ( X r 1 6 1 a 1 b ) t 2 0 i.e. , the Lkvy process ( q s 1 a 3 b ) t 2 0 with characteristic exponent being the continuous negative definite function Re ‘$m,fi,a,b. w e find now
as well as
39
and
For b = 0 (3) yields the transition density of (yt6'a9b)t20
For the general case a longer calculation, see B. Bottcher following series representation of pf'a'b :
6,
leads to the
where
To proceed further we need some estimates for the derivative of Details of these calculations are given in B. Bottcher '.
$m,6,a,b.
Theorem 2.2. For all a E No there exists c, > 0 such that (a)
l$m,6,a,b(()l
5
+ 1c12)9
(20)
holds for all ( E R. This theorem tells us that in the sense of Definition (3.1) the continuous negative definite function ?/),$,a$ is a symbol in the class s1(& NoteIthat ). also Re $,,6,a,b belongs to S1(R). 3. Symbols of Meixner-type
As mentioned in the introduction we want to make the parameters m, 6, a and b in ( 1 ) state space, i.e. x-dependent and then identify (under some conditions) this function of x and ( as a symbol of a pseudo-differential operator generating a Markov process. In the following we denote by S(IR) the Schwartz space of rapidly decreasing functions.
Definition 3.1. A. An arbitrary often differentiable function q : IR x R 4 C is said to belong t o the symbol class Sk(IR),k E IR, if for all a,P E No there are constants cap 2 0 such that la,"a,Pq(T,r)l I cap(1
+ IEI2)+
(21)
40
holds for all x E R and 6 E R. B. A pseudo-differential operator with symbol q E Sk(R) is any extension of an operator of the form 4 ( z , w 4 x ) = (2T)
S,
eiZcq(x, c)ii(t)d t ,
u E s(R).
(22)
The class of all pseudo-differential operators with a symbol in Sk(IR) is denoted by Xb (R). Let us introduce a class of symbols which we would like t o call (smooth) Meixner symbols.
Definition 3.2. A function qm,&,a,b : R x &! Meixner symbol if it has the representation
-+
@. is called a (smooth)
with m, 6, a , b E C-(R) satisfying for all k E No and x E R 0 < a; 2 a(k)(x)I a; < 00
(24)
< b, I b(’)(z)5 b; < 7r 0 < 6, 5 S(”)(x)5 6; < 00
(25)
Im(”(x)I 5 m
(27)
-7r
(26)
k
where a t , b t , 62 and m k are real constants. The class of all Meixner symbols is denoted by MS(R).
Remark 3.3. By definition every Meixner symbol q E MS(R) is a negative definite symbol in the sense that for all x E R the function ++ q(x,<) is a continuous negative definite function.
<
Theorem 3.4. The class MS(R) is a subset of S’(R). Moreover q E MS(R) is elliptic in the sense that Re
holds for all x E R and
d x , €) 2 Yo(1 + l€I2)+
(28)
< E R,151 large.
Proof. Our proof follows the dissertation of B. Bottcher where more details are given. Instead of using a formula for higher order derivatives for composite functions, see L.E. F’raenkel 8, we use the special structure of symbols in MS(R) and some elementary results:
I tanh(x + iy) I I1 -&J , sin2 y
x
E
R and y
E
(--
7r7r
-),
2’ 2
(29)
41
1 sech2(z+ iy)l 5 cy(l + Jz12)-f, z E R , y E (--,72 r -)T2
and r
2 0, (30)
and (sech’ z)’ = -(sech2 z ) tanh z.
A.
(31)
5
Recall that sech z = Note that for IyI 5 C < (where C is a constant) we get the right hand side of (30) independent of y. Now, for q = qm,s,a,b E MS(R) we find the estimates
by observing the structure of the derivatives. Since
+ 26’(5) tan( -)b’(x) b (x) 2 a(.)(
ib(z) a’(z)<- ib’(z) + 26’(5) tanh( ) 2 )( 2 b”(z) b(z) b’’(z) + %(z) tan(-)--b(z) + 6(z)sec2(-)2 2 2 2 + 2S(z) tanh( 1 2 2
+ 2S(z) sech2(a(..)[
-
-
2
ib(x) a’(.)[ )(
(33)
- ib‘(x)
1
2
e),
we may use (31) to reduce the estimates for @q(z, p 2 3, t o the estimate for a,”q(z,[) and the estimates for z H (sech’ z ) z k ,and then (29) and (30) give the result. Next observe that
L$q(x, <) = -im(z)
a ( z ) <- ib(z) + 6 ( z ) a ( ztank( ) ) 2
which implies the desired estimates (21) for Q a:a;q(z, <) = -zm’(z)
= 1,,B = 0.
+ 6’(z)a(z)tanh( a(.)<
+ 6(z)a’(z)tanh( a(.)<
(34)
Further we have
- ib(z)
2
1
- ib(z)
2
1
a(z)<- ib(z) a’(z)E - ib’(z) + S ( z ) a ( zsech’( ) 1 2 )( 2 and again (29) and (30) yields the estimate (21). For o = 2 we find 1 a(.)< - ib(z) a,”q(x,[) = -a’(z)S(z) sech2( ) 2 2
(35)
42
which again by (29) and (30) leads to (21) for a = 2, /3 = 0. But now (32)( 3 5 ) together with (31) as well as (29)-(30) imply q E S'(R). The ellipticity condition (28) follows from the restrictions for the parameters and (10).
Corollary 3.5. Let q E M S ( R ) . Then Re q E S1(R) and Re q is elliptic in the sense of (28). Thus we may apply the theory of "classical" elliptic pseudo differential operators to pseudo-differential operators with symbols in M S ( R ) . This will be done in the next section. 4. Meixner-type Processes The aim of this section is to show that every pseudo-differential operator -q(x, D ) with q being a Meixner Symbol has an extension, in fact a unique extension, generating a Feller semigroup, hence gives rise to a Feller process, or equivalently, for every q E M S ( R ) there is a stochastic process (Xt)t>o with state space R such that
and (Tt)t?o,where
Ttu(z)= E " ( u ( X t ) ) ,
(37)
is a Feller semigroup on Cm(R), compare l 3 and R. Schilling 20. Since in principle the desired result is by now easily quotable, compare W. Hoh l1 and 1 2 , or the more comprehensive treatment in 1 5 , Chapter 2, we just outline the arguments and ideas to obtain the result, but we do not repeat longer calculations leading to the estimates needed. The construction of the Feller semigroup is based on the following variant of the Hille-Yosida-Ray theorem: If a linear operator ( A , D ( A ) )on Cm(R) is densely defined and satisfies the positive maximum principle, i.e. Vu E D(A) s.t. u(z)= supyEau(y) 2 0 implies ( A v ) ( z )5 0, and if for some X > 0 the range of X - A is dense in Cm(R), then ( A , D ( A ) )is closable and its closure generates a Feller semigroup. (For a proof we refer to l4 and the references given there on the origin of this result.) Since q is a negative definite symbol, it is clear that ( - q ( z , D ) , C ~ ( R ) ) satisfies the positive maximum principle and if we extend -q(z, D ) to some Sobolev space H t ( R ) such that q(z,D ) ( H t ( R ) )is continuously embedded into C,(R), then also (-q(z, D ) ,Ht(IR)) satisfies the positive maximum
43
principle, compare Theorem 2.6.1 in 15. The serious problem is the solvability of the equation Xu q ( x , D)u = f . This problem is overcome in several steps:
+
1. Show that for every f E L2(R) and X 2 0 sufficiently large there is a weak solution u E H i , i.e. u satisfies
+
~ x ( u $1 , := ~ ( u4 ), ~ ( u 4 1, 0 = (f, $10
for all
4EH~(R),
where B(.,.) is the continuous extension of (u, u) I+ ( q ( x ,D)u,u)o from H y R ) to Hi(R).
2. Show that for HS+’(R).
f
E H S ( R ) ,s
2 0,
every weak solution belongs to
-
3 . Finally, starting with ( - q ( x , D ) ,H 3 ( R ) ) ,note that 4x1 D ) ( H 3 ( R )c ) H2(R)
cco(R)l
and apply the Hille-Yosida-Ray theorem. Note that the fact M S ( R ) c S’(R) allows in fact an application of classical pseudo-differential operator theory as discussed for example by H. Kumano-go in 18, whereas in W. Hoh or in l5 larger classes of negative definite symbols which are not classical symbols are treated. Thus we arrive a t
Theorem 4.1. Let qm,6ia)bE M S ( R ) be given by a(.)<
-
2
ib(x)
I-
where for m, S,a , b the restrictions (24)-(27) do apply. T h e n ( - q m ~ 6 ~ a ~Db )( ,xH, 3 ( R ) ) extends uniquely t o a generator of a Feller semigroup (T,(oo))t20 = (Tp’6’a’b)t20 on Cco(R),and in addition (36) and (37) do hold. Corollary 4.2. If q = Re q m , 6 , a , b l q m ~ 6 ~Ea ~Mb S ( R ) i s as in Theorem (4.l), t h e n - q ( x , D ) extends t o a generator of a Feller semigroup too. The proof of Theorem (4.1), more precisely working out step 1-3, yields more, namely that there is A0 > 0 such that for X 2 A0 the operator -qm,6,a,b(x, D ) - Aid extends also to a generator of an L2-sub-Markovian Clearly on Cco(R) n L2(R) we semigroup which we denote by (T,(2)’x)t20. have e-~t~,(c= o )Tu, ( ~ ) J U
a,e.
(39)
44
Moreover, one can prove the estimate 2 l141H$ 5 cBx(u1 u)1
(40)
which implies, by Sobolev's embedding theorem (borderline case, compare D. Adams and L. Hedberg I, Theorem 1.2.4.(b), p. 14), that
lu112* 5 a ( u ,). for all finite p
2 2. Using Theorem llTt(2)
for any
K
(41)
8.7 in W. Hoh
l1
we find now
I c't-2
IILm-Lz
(42)
= s , p > 2.
We may ask the natural question whether for t > 0 the operator Tt = TiDc)) has a representation as pseudo-differential operator and if so, how we can calculate or approximate the symbol a(Tt)(z,<)of Tt. In case where all parameters are constant the answer is easy: 1
s,
Ttu(z)= ( 2 ~ ) - 7
e
ixc - t q m J A b
e
(43)
( E ) Q ( < ) d<
which holds (at least) for all u E S(R).Thus we should long for O t ) ( 3 : , < )=
e
-qm,6,a,b
+
(zL)t r ( t , 2 , <),
(44)
where r ( t ,z, E ) satisfies certain smallness conditions. To proceed further we need some preparations. The class Sk (R) as defined by (21) is a Frkchet space if topologized with the serninorms
WE HAVE COMPARE h. kUMANO-GO
~ q ( +(2)1= z,
(2n)-+
IJ,
e i x ~ z<)&(<) , d< 5 cm(u)plc,o(q)
A CONSTINUOUS LINEAR FUNCTIONAL
(46)
IS GIVEN ON s (r) BY
and from (46) we even get a uniform bound with respect
Now we arew in a position to solve our problem by a straIGHTFORWARD APPLICATION OF tHEOREM 4.1, cHAPTER 7.IN h. kUMANO-GO
45
Theorem 4.3. Let q = qm,6,a,b E MS(E%) be as in Theorem (4.1). Then the operator Tt has on S(R) the representation Ttu(x)= (27r)-3
eizEa(Tt)(z, J)G(J) dJ
(47)
weakly in S'(IW),
(48)
where a(Tt)(x,J) satisfies t-0 l i m a ( T t ) ( z , J )=
1
+
a ( T t ) ( zJ) , = e-q(z>E)t ro(t,z,[)
(49)
where ro(t,., .) E S-l(R) and t-io lim ro(t,z, <) =
o
weakly in s-'(R.),
(50)
and {?ro(t,x,J); 0 < t 5 T } is for each T > 0 a bounded set in So@). I t follows from Theorem (4.3) that
and
lii
eizEro(t,z, [ ) G ( [ ) dJ
= 0.
Finally, let us consider the result in a heuristic way. Using the semigroup property of (Tt),>0 - we arrive for small t > 0 at
~,+,u(xM ) (27r1-4
S,
eizEe-q(z>E)t(TsuT(5)
and assuming that for Ix - zol and t
> 0 small
dJ1
46
In particular, if
s is small and therefore p,(y,A) can be substituted by ~ ~ p ~ ~ y ~ ' " y ~d z~ we u ~find y ~now ~ bfor~ Iz y~ zoI ( ~as) well as s and t small that
should be a n approximation for p t f s ( z , A ) . For some further considerations in this direction we refere t o 16.
References 1. Adams, D. and L. I. Hedberg, Function spaces and potential theory. Vol. 314 of Grundlehren der math. Wissenschaften. Springer Verlag, Berlin 1996. 2. Barndorff-Nielsen, 0. and S. Z. Levendorski:, Feller processes of normal inverse Gaussian type. Quantitative Finance 1 (2001), pp. 318-331. 3. Barndorff-Nielsen, O., T. Mikosch, and S. I. Resnick (eds.), Le'vy Processes Theory and Applications. Birkhauser Verlag, Boston 2001. 4. Barndorff-Nielsen, 0. and N. Shephard, Modelling b y Le'vy processes f o r financial econometrics. In 3 , pp. 283-318. 5. Berg, C. and G. Forst, Non-symmetric translation invariant Dirichlet forms. Inventiones Math. 21 (1973), pp. 199-212. 6. Bottcher, B., PhD-thesis, University of Wales Swansea. (In preparation). 7. Eberlein, E., Application of generalized hyperbolic Le'vy motions to finance. In ', pp. 319-336. 8. Fraenkel, L. E., Formulae for,higher derivatives of composite functions. Math. Proc. Cambridge Phil. SOC.83 (1978), pp. 159-165. 9. Fukushima, M., Y . Oshima, and M. Takeda, Dirichlet forms and symmetric Markov processes, Vol. 19 of d e Gruyter Studies in Mathematics. Walter de Gruyter Verlag, Berlin 1994. 10. Grigelionis, B., Processes of Meixner type. Lithuanian Math. J . 39 (1999), pp. 33-41. 11. Hoh, W., Pseudo differential operators generating Markov processes. Habilitationsschrift. Universitat Bielefeld, Bielefeld, 1998. 12. Hoh, W., A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka 3. Math. 35 (1998), pp. 789-820. 13. Jacob, N., Pseudo-differential operators and Markov processes. Vol. 94 of Mathematical Research. Akademie Verlag, Berlin 1996. 14. Jacob, N., Pseudo-Differentia1 Operators and Markov Processes, Vol. I: Fourier Analysis and Semigroups. Imperial College Press, London 2001. 15. Jacob, N., Pseudo-Differential Operators and Markov Processes, Vol. 11: Generators and Their Potential Theory. Imperial College Press, London 2002. 16. Jacob, N. and R. L. Schilling, Estimates for Feller semigroups generated b y pseudo differential operators. In: Rakosnik, J. (ed.), Function Spaces, Differential Operators and Nonlinear Analysis. Prometheus Publishing House, Praha 1996, pp. 27-49. 17. Jacob, N. and R. L. Schilling, Le'vy-type processes and pseudo differential operators. In 3, pp. 139-168.
47
18. Kumano-go, H., Pseudo-differential operators. MIT Press, Cambridge MA 1974. 19. Ma, Z.-M. and M. Rockner, A n introduction t o the theory of (non-symmetirc) Dirichlet forms. Universitext. Springer Verlag, Berlin 1992. 20. Schilling, R. L., Conservativeness and extensions of Feller semigroups. Positivity 2 (1998), pp. 239-256. 21. Schoutens, W., T h e Meixner process: Theory and applications in finance. Preprint 2002. 22. Schoutens, W. and J. L. Teugels, Le'wy processes, polynomials and martingales. Commun. Statist.-Stochastic Models 14 (1,2) (1998), pp. 335-349.
SOME REMARKS ON IT0 AND STRATONOVICH INTEGRATION IN 2-SMOOTH BANACH SPACES
ZDZISlAW BRZEZNIAK Department of Mathematics University of Hull
Hull HU6 7RX, U.K. E-mail: z. brzezniakQmaths.hu11.ac.uk In this paper we study It8 integral in 2-smooth Banach spaces. Burkholder inequality is proved using It6 formula in certain subclass of such spaces. Relationship with an integral introduced recently by Mikulevicius and Rozovskii is discussed. Finally, Wong-Zakai type approximation for such integrals is proved.
1. Introduction
This paper has its origin in the author’s attempt to understand an important work by Mikulevicius and Rozovskii 28. In order to study stochastic Navier-Stokes equations in Rd for d = 2,3 in Sobolev space HS9P, the authors introduce a new type of It6 integral for some Banach space valued processes. One of the aims of the current presentation is to show that the Mikulevicius-Rozovskii integral is a special case of an integral in 2-smooth Banach spaces first introduced by Neidhardt in 31 and then extensively studied and used by the present author and his collaborators. The main object however is to present a concise and detailed exposition of the subject. The paper is organised as follows. In the section 2 we recall the basic definitions, i.e. of 2-smooth Banach space and of It6 integral with valued in 2-smooth Banach space. Section 3 is devoted to statement and proof of the Burkholder inequality for It6 integrals taking values in certain class of Banach spaces. Let us note here that Burkholder inequality is valid in 2smooth Banach spaces, see l6 and 32. The class of Banach spaces considered in this section is big enough as it contains important examples of L P , p 2 2 and Besov and Sobolev-Slobodetski spaces. In section 4 we show how the theory on It6 integration in 2-smooth Banach spaces can be used to solved certain nonlocal stochastic differential equations. In the section 5 we investigate the relationship of the integral introduced by Mikulevicius-Rozovskii with the one in 2-smooth Banach spaces. We show that the former is a 48
49
special case of the latter. For this we use a result of the author and Peszat on identification of y-radonifying operators with values in LP-spaces with certain class of integral operators. We conlude the paper with a discussion of dependence of the It6 integral on the Wiener process. We prove that the Stratovich integral is equal to limits of the Riemann sums with the mid-point approximations is repalced by interval averages. Our result should be seen as in conjunction with the authour's paper with A Carroll on Wong-Zakai approximation for stochastic differential equations in 2-smooth Banach spaces.
2. It6 integral in 2-smooth Banach spaces
I
In what follows X will be a real Banach space with norm . I. A modulus of smoothness of (X, I . I) is defined by
p x ( t ) := sup
Ix1=lyl=l
1
+ +
- (12 tyl 2
1 2
- tyl) - 1.
A Banach space (X, I . I) is called 2-smooth iff there exists a norm I . I on X , equivalent to I 1 and k > 0 such that the modulus of smoothness px of (X, I . I ) satisfies +
PX(t)
5 kt2,
t E (0,1].
The notion of a 2-smooth Banach space was introduced by Pisier in 34. Pisier proved there that X is a 2-smooth Banach space iff one of the following two conditions is satisfied
(i) There exists a constant A > 0 such that 12
+ yI2 +
15 - y12
I 2(212+ AIyI2,
5, y
E
X.
(1)
(ii) there exists a constant C = C z ( X )> 0 such that for any X-valued finite martingale { Mk} the following holds
In fact, the implication X is 2-smooth -----r. (i) had been earlier proven by Figiel & Pisier in 19, see also l4 p. 144. The proof of converse implication, only alluded to in 34, is rather straightforward. A Banach space X satisfying property (2) is usually called an Mtype 2 Banach space. Although an It6 type integral for 2-smooth Banach spaces was first introduced by Hoffmann-Jorgensen and Pisier l8 only for 1-dimensional square integrable martingales, a complete construction was
50
carried out by Neidhardt in 31, see also Belopolskaya and Daletskii 2 , Dettweiler 16, Brzeiniak and references therein. In order to introduce this integral we need one more new notion, i.e. of a y-radonifying operator. If H and X are separable real Hilbert and resp. Banach spaces, a bounded linear operator L : H -+ X is called y-radonifying iff L ( ~ H is ) a-additive, where Y H is the canonical Gaussian distribution on H . If this is the case, L ( ~ Hhas ) a unique extension to a a-additive Bore1 probability measure V L on X. One can then also show that VL is a centered Gaussian measure on X with Reproducing Kernel Hilbert Space (RKHS) (i.e. the Cameron Martin space) equal to H . In particular, in the spirit of L Gross 17, the triple ( H ,X , VL) is a Abstract Wiener Space (AWS). The set of all y-radonifying operators from H to X we will denote by R ( H , X ) . Note that in 31 and earlier papers this set is often denoted by R ( H ,X ) . For L E R ( H , X ) one puts
Neidhardt in 31 proved that 11 . 11 is a norm on R ( H , X ) , that R ( H , X ) with that norm is a separable Banach space and that the set Cfi,(H,X) of bounded linear operators L : H + X with finite dimensional range, is a dense subspace of R ( H ,X). It follows from Baxendale that R ( H ,X ) is an operator ideal, i.e. if L E R ( H ,X ) , A E C(G,H ) and B E C ( X ,Y ) (where G and Y is another separable Hilbert, resp. Banach space) then also B L A E R ( G , Y ) and JJBLAIIR(G,Y) I CIBIqx,r)IILIIR(H,x)IAILc(G,H) for some constant C independent of A, B and L. Let us fix an orthonormal basis (ONB) {ek}k of H and let us denote by IIn the projection onto the space spanned by e l , . . . ,en. Let us choose and fix an i.i.d. sequence of standard centered real valued Gaussian random variables ,&, k E N. It follows from the Itb-Nisio Theorem, see e.g. 23 then L E R ( H , X ) iff (IE)C,,&Lek1$)'/2 < 00. Moreover,
llLll = (IE I C , , P ~ L Q ~ $ ) One ~ / ~ can . also show that the exponent 2 above can be replaced by any p E (1,m). Denote, for n E N,by CCfin,(H, X ) be space of L E C ( H , X ) such that L = LHn. Note that U,LCfin,(H,X) is dense in R ( H , X ) . We fix a filtered and complete probability space 'u = (n,F ,(Ft)tc[O,~l,P). We have, see 26, and 13, the following Definition 2.1. An (3t)-adapted canonical cylindrical Wiener process o n H is a family W ( t ) ,t 2 0 of bounded linear operators from H into L2(R,F,P)such that: (i) for all t L 0, and $, cp E H , E W(t)+W(t)cp= t ( $ ,c p ) ~ ,
51
(ii) for each $J E H , W(t)+,t 2 0 is a real valued (Ft)-adapted Wiener process. One can show that if W ( t ) ,t 2 0, canonical cylindrical Wiener process on H iff there exists an orthonormal basis { e k } k of H and a sequence ,&(t), t 2 0, k E N of standard real valued (3t)-adapted Wiener processes such that W(t)+= C k P k ( t ) ( q b , e k ) , for all E H and all t 2 0. If W ( t ) ,t 2 0 canonical cylindrical Wiener process on H then by @(t)we will denote the series Pk(t)ek. If S is a normed vector space endowed with some a-algebra p, then for 05a
+
Ck
M P ( a ,b; S ) :=
{e E N(a,b; S ) :IE
I”
(e(t)l$d t < C O } ,
(4)
Then, we define MP(a,b; S ) to be the space of all equivalence classes of elements of &tP(a,b; S ) with respect to a natural equivalence relation, E q iff IE Jab Ic(t)-q(t)lP dt = 0. Note that MP(a,b; S ) is complete if S is. Denote finally by MfteP(a, b; &,(H, X ) ) the class of all E MP(a,6 ; R ( H ,X ) ) such that there exists rn E N and a partition 7r = { a = t o < tl < . . . < t, = b} of the interval (a,b) such that c(t) = [ ( t k ) & , t E [tk,tk+l), k = 1,.. . ,n - 1. One can show, similarly to 3 1 , that the latter space is dense in MP(a,b; R ( H ,X ) ) . Now we will define a linear map I : MZtep(a,b; Lfi,(H, X ) ) + L2(R;X ) by the standard way. Thus, if E Mstep(a, b; .&,(H, X)) with partition 7r = { a = t o < tl < . . . < t, = b } then we put, with F@(t)= C j, B j ( t ) e j , N
<
c
Since for L E Lfi,(H,X)), Ll/ir(t) E L’(R,X), I is a well defined
x:Ii
)
linear map. Denoting, M k = ( ( t j ) (@(tj+l) - @ ( t j ) then the sequence ( M k ) k is an X-valued martingale (with respect to filtration ( 3 t k ) k . Indeed, if an FSmeasurable L : R --+ R ( H , X ) is such that L = LII, and E : R + P,(H) is 3 measurable (and both are square integrable) then IE (L
L
E
IEl((tk) (@(tk+l)
-@(tk))
On the other hand, note that if
& , ( H , X ) ) then L = LII, for some n E N and so E(L@(t)12= = tl(L((2.Therefore, we have proved
E (LII,@(t)( = IE 1 Cj”=,/ ? j ( t ) L e j ( ’
52
that
One should mention here that in order to prove (6) both Neidhardt 31 and Dettweiler l5 used the property (i), while the author in and above has used the M-type 2 property (ii). The last inequality (6) shows that I is a bounded (obviously) linear map from I : Mstep(a,b;&,n(H,X))to L2(R;X ) . Since the former is dense in M 2 ( a ,b; R(H,X)), I has a unique extension t o a bounded linear map from the whole of M 2 ( a ,b; R ( H ,X ) ) with values in L2(R,IF,X). Moreover, this extension, also denoted by I satisfies b
W(t)I2I C 2 ( X ) E /
Ilt(t)ll&H,X) dt.
(7)
Let us recall, see e.g. 22 that a stopping time T is called accessible iff there exists an increasing sequence of stopping times r, such that a.s. T, < T and limn--tmrn = T . For a stopping time T we set Rt(7) = { w E L?: t < ~ ( w ) } [, o , ~x) R = { ( t , w ) E [0,00) x R : 0 5 t < ~ ( w ) } For . an admissible processa Q : [O,T) x R --t X we define I'S(s) dW(s)=w[o,T)r)? where I = la$. One can prove that for 0
Jit ( s ) d W ( s ) . We also have, see
4,
I r 5 t !E s," c(s) dW(s)lFT)=
the following
Proposition 2.1. If X is 2-smooth Banach space and 5 E ML,(O, a ;R ( H ,X)),then (1) The process x ( t ) :=
so[ ( s ) d W ( s ) , t 2 0 is an X-valued martingale, t
with almost all paths continuous; moreover x E MZ,(O, 00; X), (2) for any T 2 0 , T
ESUP t
In particular, x E L2(R, C(0,T ;X ) ) . ai.e. (i)qlnt : Rt + X is Ft measurable, for any t 2 0; (ii) for almost all w E R, the function [O,.(LO)) 3 t H q ( t ,w ) E X is continuous.
53
Proof. According to the statement preceding the Proposition, the process z ( t ) ,t 2 0 is an X -valued martingale. To prove the remaining two claims, :0 = we first assume that is an adapted step function with to < ... < t, = T , c(t) = c(ti) E L2(R,.Ftti,P;X)for t E [ti,ti+l). Then z ( t )= ~ , , < , [ ( t i (@(ti+l ) A t ) - @(ti)),and so z E C ( 0 , T ; X )a.e. Since also 1x1 is non-negative submartingale, applying the real version of Doob inequality, see 21 or Theorem IV.8.2 in
26,
we infer that
Therefore the operator 1 : M&,(O, T ;&,(H, X)) + L2(R,C(0,T ;X)) defined above for simple functions can be uniquely extended to the whole space M 2 ( 0 ,T ;R ( H ,X ) ) . We use the fact that the space of progressively processes in L2(R; C(0,T ,X)) is closed therein. We conclude this section with a statement of an It6 formula, see 31 and But first we define an important concept of a trace of a bilinear map. If X, Y,Z are Banach spaces and A : X x Y -+ Z is a bounded bilinear map and A E R ( H ,X ) , B E R ( H , Y ) ,then, see 8 , we put 8.
j
The series is absolutely convergent and its sum is independent of the choice of the ONB { e j } . If = Y and A = B we write trAA instead of trA,AA.
x
Theorem 2.1. (It6 Formula) Assume that X and Y are %smooth Banach spaces. Let 0 5 c < d 5 00. Assume that a function f : [c,d ) x X 4 Y is of C172class, i.e. f is Fre'chet differentiable, the Fre'chet derivative f ' : [c,d ) x X + C(R x X , Y ) is continuous and differentiable in the X-direction with the resulting derivative being continuousb. Let, f o r a E JV&(C, d; X) and E n/12c(c, d; R ( H , X)),
<
+
z ( t )= ~ ( c )
I " +L a(.) ds
((s)
d W ( s ) , t E [c,4.
(8)
Then for all t E [c,d ) , a.s,
~
~
g,
bSimply, appropriate space.
and
3
~~
exist and are continuous on [ c , d ) x X with values in the
54
Theorem 2.1. was proved in 31 in the case a Let us state an important
is a bounded bilinear map, then
3. Burkholder inequality In this section we assume that our
( H ) X is a real separable Banach space such that there exists p E [ 2 , c o ) for which the function ' p p : X 3 x H l x l p E IR is of C2 class and there are k 1 , kz > 0 such that for every zE X ,lp'(x)I 5 klIz(p-' and ('p''(x)(5 2k2 1 2 1 p - 2 . Note that the Sobolev Hsip-spaces with p E [2,co) and s E IR satisfy the condition (H). Moreover, a Banach space X satisfying (H) is 2-smooth7 see l4 If q 2 p , the following is a special case of Theorem 1.1 from 12.
Theorem 3.1. Assume that X is a Banach space satisfying the condition (HI. A s s u m e that 5 E M ~ , ( O , c o , R ( H , X ) .Let x ( t ) = J,"C(s)dW(s), t 2 0. Suppose q E ( 1 , a). T h e n there exists a constant Kq > 0 depending only o n q, H , X , and the constants kl, kz appearing in (H), such that for every T > 0 ,
Remark 3.1. With a slight modification of the proof below one can show that in fact the Burkholder inequality above is also valid for any accessible (and hence any bounded) stopping time.
55
Proof. The first step is to prove this result for q = p . We follow the above mentioned paper l2 where a more general result is studied. Suppose first that A is a bounded dissipative linear operator. Since ‘p(x)= 1xIP is of C2 class we can use It6’s formula of Neidhardt, see Theorem 2.1 above, and obtain
Consider a process y(t) defined by
1 t
Y(4
:=
9’( 4 s ) )
( E ( 4 ) dW(s),
t 2 0.
Obviously, y is an R-valued martingale with quadratic variation
1 (4 t
4(t)=
From the inequality (25) in
(4s)) as)) 31,
O
(4s))
CW* ds-
i.e. for L E L ( X ,W), B E R ( H ,X), 2
l(LB)0 (W*II lFl:(x,R)l%(H,E)
I: I~l~(X,,)lI~ll~(H,E,.
Therefore, by (H), we infer that
Applying next the Davis inequality, see also 33, we arrive at
Now, we shall deal with the second term on the RHS of (13). Since for L from R ( H , X ) and a bilinear mapping A : X x X 4 R, Itr A o ( L ,L ) ( I ]A1 . JJL))2, we have, again by ( H p ) ,
Combining this with the previous estimates we obtain
56
We shall study each term on the RHS of (13) separately. Let we have
E
> 0. First
where we have used Holder and Young inequalities. Similarly for the second term we have
Choosing now
E
> 0 such that p-1
(ski- P we obtain, for some generic K
p-2 kg P
+-I
&=-
1
2
> 0,
which proves (12) for q = p. The proof in the case q > p follows the same lines. It is enough to observe that if the Banach space satisfies the condition (H) with p 2 2 then it also satisfies this condition with q 2 2. In order to complete the proof we need to consider the case q < p. The proof in this case is motivated by the proof of the Burkholder inequality given by Revuz and Yor in 36. It is based on the following (see Proposition IV.4.7 therein)
57
Proposition 3.1. Suppose that a positive, adapted right-continuous process Z is dominated by an increasing process A, with An, i e . for every bounded stopping time 7, IEZ, 5 EAT. Then for any k E ( 0 ,l),
IE
sup
Olt
2-lc zk<- 1IEAL. -k
Let now q E (1,p). We apply Proposition (3.1) to processes zt = )xtlPl{tlT)
(s,"
P/2
lttST} with k = q / p . We have just proven and At = IIE(s)ll")) that Z is dominated by A . Since Z is continuous (by Proposition 2.1),
This proves completely Theorem 3.1.
0
4. An Example
This Example is motivated by a question raised by Terry Lyons. Let S1 be the unit circle (with normalized Haar measure) and let H = H19"(S1) and B = L2(S1). Let E = H"ip(S1) with < a < Let us recall that H"ip(S1) = [LP(S1);HIJ'(S1)la, the complex interpolation space. Then it is well known that E is a Banach algebra. Note also that E is 2-smooth Banach space. Consider three maps
3.
A : E 3 u H { H 3 H u . y E E } E R ( H ,E ) , A :E 3 uH { B 3 y H U * Y E E } E R(R,E), B : E 3 u H { E 3 y H u.yE E } E L ( E , E ) .
(15)
(16) (17)
Since E is a Banach algebra, B is a (well defined) bounded linear operator. Since A(u) = B ( u ) o i, where i : H E is the natural embedding, the map A : is well defined and bounded as well. Here we use the fact that
i E R(H,E).
Since for f E L'(S1) the map A, : u H f * u is bounded from LP into L P (by the Young inequality) and from HIJ'(S1) into H1+'(S1) (by the former fact and equality D ( f * u)= f * (Du)) we infer, by means of the interpolation theory, that As is a bounded linear map from E = H"J'(SI) into itself. Therefore, is a bounded linear map from E into L ( L 1 , E ) , hence into C ( f i , E ) . To prove that is a bonded linear map from E into R(I?,E) we argue as follows. Let u E E . Then, Dau E LP(S'), where D"
58
is the fractional power of the derivative operator D . We will show first that if u E El then
L2(S1)3 y H D"(u * y) = ( D " u ) * y E L p ( S 1 ) is y-radonifying. Obviously, it's enough to show that for linear operator
K : L2 3 y
-
E
LP(S1)the
v * y E Lp
is y-radonifying. Note that K is an integral operator with a kernel k ( z , y ) := v ( x - y ) (here we treat S1 as a group). Since
is finite, as v E LP C L2, the result follows, see l1 and Theorem 5.1 in the next section. In fact we have proven that the map A : E -+ M ( H ,E ) is well defined, linear and bounded. Hence the following result follows directly from 31 and 8 .
Theorem 4.1. Suppose in addition that W ( t ) ,t 2 0 and l%'(t), t 2 0 are two independent cylindrical Wiener processes with respect to Halbert spaces H and H respectively. Then for every uo E E there exists a unique continuous E-valued process that i s a solution t o
d u ( t ) = u(t)d W ( t ) u ( 0 ) =uo. Since H
L--)
+ u ( t )* d W ( t )
H we also have the following
Theorem 4.2. Suppose in addition that W ( t ) ,t 2 0 is an H - cylindrical Wiener processes. Then for every uo E E there exists a unique continuous E-valued process that is a solution to
d u ( t ) = u(t)d W ( t ) u ( 0 ) = uo.
+ u(t)* d W ( t )
(19)
Remark 4.1. The reason we used the H"1p spaces and not the SobolevSlobodetski W"tP was that for u E Wa,p,D"u may not be an element of L2. Thus Theorem 4.1 may be not true in this case. However, in Theorem 4.2 we can use E = WaJ'(S1).
59
5. Relationship with the approach of Mikulevicius and Rozovskii Suppose Y is a Hilbert space and W ( t )be a canonical Y-cylindrical Wiener process. Consider an LP(0, Y)-valued process g ( r ) , r 2 0. Consider a map * from LP(0, Y ) to R(Y, LP(0)) defined by the formula, with g E LP(0, Y)
where g j ( z ) := ( g ( z ) , e j ) ,x E 0. Let us recall the following result, first stated in l 1 (see also lo, where a complete proof is given).
Theorem 5.1. Suppose Y is a separable real Hilbert space and let p E (1,a)be fixed. Let (0,F ,u ) be a a-finite measure space. For a bounded linear operator K : Y 4 LP(0) the following assertions are equivalent: (1) K is y-radonifying; (2) There exists a u-measurable function
K.
: 0 --+ Y with
such that f o r all u-almost all x E 0 we have ( K ( Y ) ) ( X )= ( K ( X ) , Y ) ,
Y E Y.
Moreover, there exists a constant C > 0 such that for all
IC
E LP(0, Y),
In fact, the above Theorem means that the map LP(0,Y) 3 K. H K E M(Y,Lp(O)) is an isomorphism of Banach spaces. Since by the Parseval formula, for y E Y and x E 0
& A x ) ( Y , e j ) = C ( 9 ( 4 , e j ) ( Y , e j )= ( S ( X ) , Y ) j
j
we infer that the map A : g H 4 is nothing else but the isomorphism K. H K from Theorem 5.1. Therefore, for a process g E Mi,(O, a; LP(0, Y)) we can define an LP(O)-valued integral s,” g ( r )d W ( r ) simply by putting
l g ( r ) d W ( r )=
I’
i j ( r ) d W ( r ) , t 2 0.
This integral, being just a special case of the integral introduced earlier in section 2 satisfies all its properties. In particular, it satisfies the Burkholder
60
inequality (12), a special case of which in the present situation takes the following form. If p 2 2, then
Let us now show its another property whose a byproduct is that it coincides with the It6 type integral of Mikulevicius-Rozovskii,see 28 (subsection 5.1 in the Appendix) and 27.
Proposition 5.1. Under the above assumptions, af cp E LP(0) with 9j(., 5 , w ) = ( g ( r ,2,w ) , e j ) , then
Proof. This result is in fact a special case of the It6 formula, see Theorem 2.1. Indeed, cp can be identified with a bounded linear map on LP(0). Since then for E LP(0), = cp and cp”(J) = 0, we get that a.s. E (s,”g(r)d W ( r ) ,9) = J;(g(~)cp)dW(+ Denoting by E(r) = L(Y,R) = Y* E Y and observing that L(Y,R) S R(Y,R) the integral S,”c(r)dW(r)is again a special case of the It6 integral from section 2. Hence, S,”S(r)dW(r) = C,”=, tj(r)dWj(r)= C&(gj(r),cp)dWj(r) what 0 concludes the proof of the Theorem.
e
M r M
Remark 5.1. The above can be generalised to any Banach space X which is isomorphic with the space LP(O),in particular for the Bessel spaces He>p(Rd). Indeed, the isomorphism between the latter space and LP(Rd)is given by f H (1 - A)e/2f. 6. Approximation of the Stratonovich integral We conclude this paper with a brief discussion of the relationship between the It6 and Stratonovich integrals in the framework of 2 smooth Banach spaces. One should mention here a recent paper by Ledoux, Lyons and Qian 24, where a very novel approach this question for solutions of stochastic difhential equations in Banach spaces via rough path theory of T Lyons is discussed.
61
6.1. The result w e fix a filtered and complete probability space % = (a,F , (Ft)t@J-], IF'), a separable Hilbert space H and an (Ft)-adapted canonical cylindrical Wiener process o n H , see Definition 2.1. Let us fix an ONB { e k } of H . @(t):= CF1(W(t), e j ) e j . We suppose that the Wiener process @(t),t 2 0 , lives on a some Banach space E 2 H . With certain abuse of notation we will denote the latter simply by W ( t ) ,t >_ 0. Recall that for A E L ( E ,E ; X ) , trA = C A ( e j , e j ) j
and the sum is independent of the ONB { e j } .
t - t; Wn(t)= W ( t l ) ty+l - t;
+
(w(t:+A- w w )
7
(23)
where 0 = tt < t? < . . . < t"Nn, 5 T < t",n)+l < 00 is a partition of the interval [O,T]. Recall that L ( E ,X ) is the space of bounded linear maps from E t o X and that the imbedding C ( E , X ) 3 A H A o E Z R ( H , X ) is bounded. Here i : H 4 E is the canonical imbedding. Our main result is the following theorem.
Theorem 6.1. Suppose that the progressively measurable stochastic processes a(.) and b(s), respectively X and L ( E ,X)-valued, have almost all trajectories continuous, and for some fixed T < 00,
Assume that F : [O,T]x X -+ C ( E , X ) is of class C1 in t and C2 in x, with the second derivative bounded o n bounded sets. Finally, let c(t) be an X-valued process such that
Then
62
Remark 6.1. We state and prove this theorem for the second moment. A generalization to any p 2 2 is possible and will be discussed in ‘. For simplicity and clarity of exposition we only give a proof in a special case of one-dimensional Wiener process, i.e. H = E = R and (identifying C(W,X) with X ) F ( t , x ) = x , t E [O,T],x E X . We will also assume that a(t) = 0 for t E [O,T].The proof in the general case will also be discussed in ‘.
Corollary 6.1. Suppose E = X and [ ( t ) = W ( t ) ,t 2 0. Consider the following approximation sums
From Theorem 6.1 we infer their convergence t o the Stratonovich integral of F ( W ( s ) ) (see for the Definition of the Stratonovich integral). Therefore, the Stratonovich integral appears not only by the choice of midpoint values of the integrand but also by taking its integral averages. See also Mackevicius 2 5 .
6.2. The proof We show that if [ ( t )= s,” b ( s )d W ( s ) and (24) then
where W, is defined by (23). Proof of (27) Let us denote
m,(t) = Sup{k : tk 5 t }
In what follows we shall try to drop the sub-(super-)script n whenever we are not facing ambiguity. Moreover for simplicity we assume for the time being that t; = . Thus we have
m ( t )= m,(t)
= sup{k : tk
5 t } = SUp{k
:
k ; 5 t}
+
m(t) 1 ) - W C "n" ) ) = I2(t)
+ I;@) + I;@)
Lemma 6.1. Under the above assumptions and notations we have
Proof. From the uniform continuity (on interval [0,TI) of paths of both processes W ( t )and <(t)we have sup II;(t)l2 -+ o a.e.
o
Moreover,
From (24) and the Doob inequality, see Theorem IV.8.2 in tion 2.1 in this paper, we infer that
26
and Proposi-
We conclude the proof of Lemma 6.1 by applying Lebesgue dominated convergence theorem. 0
Lemma 6.2.
64
Proof. E o m Proposition 2.1 we have rt
in(.) = [(k) for 5 s < % Im(T) and & ( s ) = 0 for M ( T ) 5 5 T and C is a generic constant (which value can change from line t o
where s
line). We conclude the proof by observing that from Lebesgue dominated converT 0 gence theorem EJo l[(s) - t n ( s ) l 2 d s+ 0 as n -+ m. The main point in the proof lies in the following
Lemma 6.3.
Proof. From the integration by parts formula we have
Therefore
65
The last equality can be written in the following way
1 t
I z ( t )-
m(t)-1 k=O
+
b(s)d s = I i 1 ( t ) Iz2(t),
k+l
(36)
s)b(s)d s .
n
First we shall show that
*(w
Sk
Since n n
n
- s)d s = n
cn l * ( D c + c m(t)-1
k=O m(t)-1
k+l
m(t)-1 k=O
s
- s)b(s)ds=
n
nJ**(*
k=O
so&ds = & we obtain
n
- s) (6(s) - b ( k ) ) ds
n
k
n L * ( - - - rk) b+(1- ) d s n K
n
m(t)-1
lctl k=O
n
S)
(
'
b(s)-b(-)
ds+?
1
m(t1-1 k-0
1 k ,!I(--)
and thus in order t o prove (37) it is enough to prove
However (38) and (39) easily follow from continuity of paths of the process b(s) and assumption (24) by applying the Lebesgue Dominated Convergence Theorem. 0
66
The crux of the matter is to prove the following
Lemma 6.4.
Proof. Since 122(t)is constant on each time interval
(k, y]we have
where
m(T)-1 Now we are going t o show that for fixed n, the sequence (Y?,)k=Ois a m(T)-1 , where c k = F w .For martingale with respect to a filtration ( C k ) + O this it is enough to show
(43)
IE(Xp-1) =0
This follows from the following
In the last equality we used Corollary 2.1. Therefore, from (41) by the M type 2 property of X , see (2), we obtain m(T)-l
I IEIY,"(,)12 =El
c X1l2 s i=O
m(T)-1
C2(X)
c i=O
IEIXTI2(44)
67
Let us observe that each term on the right hand side of (44) can be estimated in the same way. Thus we may take i = 0 and get
By the Proposition 2.1 we have
where as usual, C > 0 is a generic constant. Similarly we have
From the last twoinequalities, (44), (45) and the fact that m(T)= m,(T) 5 cn we get (40). This concludes the proof. 0
Acknowledgments The authour would like to thank Marek Capihski, David Elworthy, Terry Lyons, Jan van Neerve, Martin Ondrejat, Szymon Peszat and Boris Rozovskii for their helpful discussion on various topics related to this paper.
References 1. Baxendale, P., Gaussian measures o n Function Spaces, Amer. J. Math. 98, 891-952 (1976). 2. Belopolskaya, Ya.L. and Daletskii, Yu.L., Diffusion processes in smooth Banach manifolds. I, Trans. Moscow Math. SOC.1, 113-150 (1980). Russian original: Trudy Moskov. Mat. Obsh. 37, 107-141 (1978) 3. Belopolskaya, Ya.L. and Daletskii, Yu.L., STOCHASTIC EQUATIONS AND DIFFERENTIALGEOMETRY,Mathematics and Its Applications vol. 30, Kluwer Academic Publishers, Dortrecht Boston London 1990. 4. Brzeiniak, Z., Stochastic PDE in M-type 2 Banach Spaces, BiBoS preprint (1991). 5. Brzeiniak, Z., Stochastic Convolution in Banach spaces, Stochastics and Stochastics Reports 61, p.245-295, 1997.
68
6. Brzeiniak, Z. and Capinski, M., Wong Zakai Theorem for stochastic integrals in Banach spaces, in preparation. 7. Brzeiniak, Z. and Carroll, A,, Wong Zakai Theorem on Loop Manifolds, 40 pages, to appear in SQminairede probabilitis XXXVII, edt. M. Ledoux. 8. Brzeiniak, Z. and Elworthy, K.D., Stochastic differential equations o n Banach manifolds; applications to diffusions on loop spaces, MFAT (a special volume dedicated to the memory of Professor Yuri Daletski), 6, no.1, 43-84 (2000). 9. Brzeiniak, Z. and van Neerven, J., Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem, Studia Math. 143, no. 1, 43-74 (2000) 10. Brzeiniak, Z. and van Neerven, J., Space-time Regularity for linear stochastic evolution equations driven b y spatially homogeneous noise, to appear in J. Math. Kyoto Univ. 11. Brzeiniak, Z. and Peszat, S., Space-time continuous solutions t o SPDEs driven b y a homogeneous Wiener process, Studia Mathematica 137, 261299 (1999), 12. Brzeiniak, Z. and Peszat, S., Maximal inequalities and exponential tail estimates for Stochastic Convolutions in Banach Spaces, pp. 55-64 in STOCHASTIC PROCESSES, PHYSICS A N D GEOMETRY: NEW INTERPLAYS. I A Volume in Honour of Sergio Albeverio, CMS Conference Proceedings, v. 28,Providence, Rhode Island (2000). 13. Brzeiniak, Z. and Peszat, S., Stochastic two dimensional Euler Equations, Annals of Probability 29, 1796-1832 (2001) 14. R. Deville, G. Godefroy and V. Zizler, SMOOTHNESS AND RENORMING IN BANACH SPACES,Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific and Technical, 1993. 15. Dettweiler, E. Stochastic Integration of Banach Space Valued Functions, in STOCHASTIC SPACE-TIME MODELSA N D LIMITTHEOREM, pp. 33-79, Arnold, L. and Kotelenez, P. edts., D. Reidel Publ. Comp. 1985 16. Dettweiler, E., Stochastic Integration Relative to Brownian Motion o n a General Banach Space, Boga - Tr. J. of Mathematics, 15,6-44 (1991). 17. Gross, L., Measurable functions o n Hilbert space Trans. Am. Math. SOC. 105 372-390 (1962). 18. Hoffman-Jorgensen J. and Pisier, G., The Law of Large Numbers and the Central Limit Theorem in Banach Spaces, Annals of Probabilityl, 587-599 (1976) 19. Figiel, T. and Pisier, G., Series aleatoires duns les espaces uniformement convexes ou uniformement lisses, C. R. Acad. Sci., Paris, Sir. A 279, 611-614 (1974). 20. N. Ikeda and s. Watanabe, STOCHASTIC DIFFERENTIAL EQUATIONS A N D DIFFUSION PROCESSES, North-Holland, Amsterdam - Oxford - New York 1981. MOTIONAND STOCHASTIC CALCU21. Karatzas, I., Shreve, S.E. BROWNIAN LUS, Springer Verlag, New York Berlin Heidelberg, 1988. 22. Kunita, H., STOCHASTIC FLOWS AND STOCHASTIC DIFFERENTIAL EQUATIONS, Cambridge University Press, Cambridge, 1990.
69
23. Kwapieri,
s. and Woyczyriski, W.A., RANDOM SERIES AND
INTEGRALS:
SINGLE AND MULTIPLE,
STOCHASTIC
Probability and Its Applications,
Birkhduser, Boston, 1992. 24. Ledoux, M., Lyons, T. and Qian, Z., Le'vy area of Wiener processes in Banach spaces, Ann. Probab. 30,no. 2, 546-578 (2002) 25. Mackevicius, V., O n polygonal approximation of Brownian motion in stochastic integral, Stochastics 13,167-175 (1984) 26. Metivier, M. and Pellaumail, J., STOCHASTIC INTEGRATION, Academic Press (A Subsidiary of Harcourt Brace Jovanovich, Publishers), New York, 1980 27. Mikulevicius, R.; Rozovskii, B., A note on Krylov's &-theory for systems of SPDEs, Electron. J. Probab. 6, paper No.12, 35 p., electronic only (2001). 28. Mikulevicius, R. and Rozovskii, R., Stochastic Navier-Stokes Equations for Tubdent FLOWS,74 pages, Warwick preprint 21/2001 29. J.M. Moulinier, The'orbme Limite Pour Equations Diffe'rentielles Stochastiques, Bull. Sc. math., 2e skrie, 112, 185-209, 1988. 30. Nakao, S. and Yamato, Y., Approximation theorem on stochastic differential equations, Proc. Intern. Symp. SDE Kyoto 1976 (ed. by K. Ito), pp. 283-296, Kinokuniya, Tokyo (1978). 31. Neidhardt, A.L., Stochastic Integrals in 2-uniformly smooth Banach Spaces, University of Wisconsin, 1978 32. Ondrejat, M., Uniqueness f o r SPDE's in Banach spaces, a manuscript, 33. Pardoux, E., INTEGRALES STOCHASTIQUES HILBERTIENNES, Cahiers Mathkmatiques de la Decision No. 7617, Universitk Paris Dauphine, 1976. 34. Pisier, G., Martingales with values in uniformly convex spaces, Israel Journal of Mathematics, 20:3-4, 326-350 (1976). 35. Pisier, G., Probabilistic Methods in the Geometry of Banach Spaces, in Probability and Analysis (Varenna 1985) pp. 167-241, LNM 1206, Springer Verlag, Berlin Heidelberg New York, 1986. 36. Revuz, D. and Yor, M., CONTINUOUS MARTINGALES AND BROWNIAN MOTION,Grundlehren der mathematischen Wissenschaften 293,Springer Verlag, Berlin 1991 37. Stroock, D.W., LECTURES O N STOCHASTIC ANALYSIS:DIFFUSIONTHEORY, Cambridge University Press, Cambridge 1987. 38. Sussman, H.J., On the gap between deterministic and stochastic ordinary differential equations, Ann. Prob., 6, 19-41 (1978). 39. Wentzell, A.D., A COURSE IN THE THEORY OF STOCHASTIC PROCESSES, McGraw-Hill, New York 1981.
THE LONG-TIME BEHAVIOUR OF STOCHASTIC 2D-NAVIER-STOKES EQUATIONS
T. CARABALLO Dpto. Ecuaciones Diferenciales y A n d i s i s Nume'rico, Universidad de Sevilla, Apdo. de Correos 1160, 41 080-SE VILLA, Spain, E-mail:
[email protected] Some results on the pathwise asymptotic behaviour of the weak solutions to a stochastic 2D-Navier-Stokes equation are established. In fact we prove some results concerning the asymptotic behaviour with general decay rate (exponential,sub and super-exponential).
1. Introduction
The long-time behaviour of flows is a very interesting and important problem in the theory of fluid dynamics, as the vast literature shows (see Temam 2 6 , Hale 18, Ladyzhenskaya 19, among others, and the references therein), and has been receiving very much attention over the last three decades. One of the most studied models is the Navier-Stokes one (and its variants) since it provides a suitable model which covers several important fluids (see Temam 24,26 and the references inside these). On the other hand, another interesting question is to analyze the effects produced on a deterministic system by some stochastic or random disturbances appeared in the problem. These facts motivated the analysis done in Caraballo et al. l 1 and the one in the present work. Therefore, our main objective is t o show some aspects of the effects produced in the long-time behaviour of the solution to a two dimensional Navier-Stokes equation under the presence of stochastic perturbations, since it is very interesting to investigate if a fluid subjected to random influences is asymptotically more or less stable than the deterministic unperturbed one. There exists a controversy concerning the different interpretations which can be given to the stochastic terms used to model our problem. Two formulations are the most commonly used for the noise in the literature: It6's formulation and Stratonovich's one. Each interpretation gives a different solution of the stochastic equation, so they provide different answers t o the 70
71
same problem. There exist several reasons which make reasonable both possibilities and there exists a rule which permits us to pass from one kind of equation to the other (see Arnold Oksendal 2 1 , Kunita 2 0 , among others). However, when one is analyzing the long-time behaviour of the solutions, special care should be paid to the choice of the model since the solutions of both stochastic equations can have totally different behaviour. We will comment again about this in the final section. In this work, we will first recall some results from Caraballo et al. l 1 concerning the exponential behaviour of the solutions to our stochastic 2DNavier-Stokes model. Then we will improve those results by giving some information concerning the general decay rate of solutions. To this end, we will consider the following stochastic 2D-Navier-Stokes equation:
+
i
d X = [vAX - ( X ,0) X f(X) divX = 0 in [O, co) x D , X = 0 on [O, co) x r, X ( 0 , z ) = X o ( z ) , J: E D ,
+ Vpldt +g(t,X ) d W ( t )
where D is a regular open bounded domain of R2 with boundary I?, u is the velocity field of the fluid, p the pressure, v > 0 the kinematic viscosity, Xo the initial velocity field, f the external force field and g(t,z)dW(t) the random field where W ( t )is an infinite dimensional Wiener process, i.e., if (Q, P, 3) is a probability space on which an increasing and right continuous family {3t}ZE~0,00)of complete sub-o-algebra of 3 is defined, and &(t) ( n = 1 , 2 , 3 , ...) is a sequence of real valued one-dimensional standard Brownian motions mutually independent on (0,P, S),then M
n=l where A; 2 0 ( n = 1 , 2 , 3 . . . ) are nonnegative real numbers such that C,"==, A; < +co, and {en} ( n = 1 , 2 , 3 , ...) is a complete orthonormal basis in the real and separable Hilbert space K . Let Q E L ( K , K ) be the operator defined by Qe, = xien. The above K-valued stochastic process W ( t )is called a Q-Wiener process. Our problem can be set in the usual abstract framework by considering the following Hilbert spaces:
H = the closure of the set {u E C p ( D ,R2) : divu = 0} in L 2 ( D ,R2) with the norm (u(= ( u ,u ) ; , where for u,u E L 2 ( D ,R2),
72
V = the closure of the set {u E C r ( D , R 2 ) : divu = 0} in Hi(D,R2) with the norm llull = ( ( u , v ) ) iwhere , for u,v E HA(D,R2),
Then, it follows that H and V are separable Hilbert spaces with associated inner products (., .) and ((., .)) and the following is safisfied:
V c H = H' c V', where injections are dense, continuous and compact. Now, we can set A = -PA where P denotes here the orthogonal projector from L 2 ( D ,R2) onto H, and define the trilinear form b by
As we shall need some properties on this trilinear form b, we list here the ones we will use later on (see Temam 26): Ib(u,v,w)l I c1 I u l i l l 4 + 11v11 I w l b ( u , 21, ?= I) 0, vu,v E ~ ( u , u , wU-) - ~ ( w , w , w- u ) = -b(v
v,
i
IlWll+
,vu,v,w E
v, (1)
-U,U,W
-
u),'~u,v E V,
where c1 > 0 is an appropriate constant which depends on the regular open domain D (see Constantin and Foias 13). Furthermore, we can define the operator B : V x V + V' by
( B ( u , v )W, ) = b(u,w,w),Vu,v, w
E
V,
where (., .) denotes the duality (V',V ). We also set
B ( u ) = B ( u , u ) ,vu E
v.
Thus the stochastic 2D-Navier-Stokes equation can be rewritten as follows in the abstract mathematical setting:
d X ( t ) = [ - v A X ( t ) - B ( X ( t ) )+ f ( X ( t ) ) ld t + S ( t , X ( t ) ) d W ( t ) ,
(2)
where f : V + V', g : [ O , c o ) x V + L ( K ,H) are continuous functions satisfying some additional assumptions (see conditions below). Also we consider the deterministic version of this equation, namely,
+
d X ( t ) = [ - v A X ( t ) - B ( X ( t ) ) f ( X ( t ) ) ]dt.
(3)
First, we give the definition of the weak solutions to stochastic 2D-NavierStokes equation (2).
73
Definition 1.1. A stochastic process X ( t ) , t solution of (2) if
2 0,
is said to be a weak
( l a ) X ( t ) is Qt-adapted, ( l b ) X ( t ) E L”(0, T ;H ) n L2(0,T ;V ) almost surely for all T > 0, (lc) the following identity in V’ holds almost surely, for t E [0,a)
X(t)=X(0)
+ s,” [ - - Y A X ( S ) B ( X ( s ) )+ f(X(s))]ds -
+ s,”ds, X(s))dW(s). As we are mainly interested in the analysis of the asymptotic behaviour of the weak solutions t o the problem (2), we will assume the existence of such weak solutions (see, for instance, Bensoussan or Capinski and Gatarek for some results on the existence and uniqueness of solutions). We also recall some definitions from Caraballo et al. l 1
Definition 1.2. A weak solution X ( t ) to (2) is said to converge to z , EH exponentially in mean square if there exist a > 0 and Mo = M o ( X ( 0 ) )> 0 (which may depend on X ( 0 ) ) such that E ~ x (-t ) z,12 I MOePat,t 2 0, In particular, if ,z is a solution to (a), then it is said that z , is exponentially stable in mean square provided that every weak solution to (2) converges to z , exponentially in mean square with the same exponential order a > 0.
Definition 1.3. A weak solution X ( t ) to (2) is said to converge to z, almost surely exponentially if there exists y > 0 such that 1 lim - log I X ( t ) -1,z
t-m
t
5
EH
-7, almost surely.
In particular, if ,z is a solution to (a), then it is said that z , is almost surely exponentially stable provided that every weak solution to (2) converges to IC, almost surely exponentially with the same constant y. 2. The exponential stability of solutions
In this section we will deal with the moment exponential stability and almost sure exponential stability of weak solutions to stochastic NSE (2). Let A1 > 0 be the first eigenvalue of A . We remark that 11u112 2 A 1 luI2,Vu E V. We also denote by
Ildt, u)l12;
= t r ( d t ,u ) Q d t ,u>*>.
74
Throughout this section we will use the following condition:
Assumption A.
There exists
p > 0 such that
II f (u)- f (v) Il"& P It 'u. - II, u, 2)
E
v.
We first recall a result ensuring existence of stationary solutions, i.e., solutions to the next equation
v ~ +u~ ( u=)f(u) (equality in v').
(4)
Indeed, we have the following lemma (see Caraballo et al.
'I)
Lemma 2.1. Suppose that Assumption A is satisfied and the function f satisfies that f(v,) converges to f(v) weakly in V' whenever {v,} c V converges to v E V weakly in V and strongly in H . Then,
( a ) if v > p, there exists a stationary solution u, E V to (4); ( b ) furthermore, if v > clI'f(o)I' f i ( y - p " , ' + p, then the stationary solution to (4) is unique. Now, in order t o study the long-time behaviour of weak solutions X ( t ) to the stochastic Navier-Stokes equation (2) under some conditions including that the kinematic viscosity v is sufficiently large, we will assume that there exists a unique stationary solution u, E V to (4). Also, we will need the following hypothesis. 2 Assumption B. Ilg(t,U)ll;q I (5 W ) )l'1L -Iu, ,
+ +
b
where 5 > 0 is a constant and y ( t ) ,6 ( t ) are nonnegative integrable functions such that there exist real numbers 0 > 0, M y , M6 2 1 with
y ( t ) 5 Mye-et, b(t) 1. Mge-et, t
2 0.
Theorem 2.1. Let u, E V be the unique stationary solution to (4) and 2p 11 u, 11 . Suppose that assumptions A assume that 2v > A(', and B are satisfied. Then, any weak solution X ( t ) to (2) converges to the stationary solution u, to (4) exponentially in mean square. That is, there ex& real numbers a E (0, e),MO= M o ( X ( 0 ) )> 0 such that
+ +3
E I X ( t ) - u,I2 1. MOe-at, t 2 0.
Proof (sketch). Since 2v > A;'5+2p+3 llu,ll, we can take a positive + a ) 2p + llu,ll . Then, real number a E (0, 0) such that 2v > A;('<
+
2 2
by applying the It6 formula to the function eat IX(t) - u , I , taking into account assumptions A and B and Gronwall's lemma, we can prove the statement (see Caraballo et al. ll).
75
Now using the energy equality, Burkholder-Davis-Gundy's lemma, Borel-Cantelli's lemma and the previous result, it can also be proved in a standard way the following result.
Theorem 2.2. Suppose that all the conditions in Theorem 2.1 are satisfied. Then, any weak solution X ( t ) to (2) converges to the stationary solution u, of (4) almost surely exponentially. In the particular case in which the stationary solution to (4) is also solution t o the stochastic equations, it holds the following result.
Theorem 2.3. Let u, E V be the unique stationary solution to sume that condition A and the following ones hold:
( a ) g ( t , urn) (b)
II S ( t ,).
(4). As-
-- 0, t 2 0, -
+
d t ,). ; ,1 5 cg II 21 - 21 II,
cg
> 0, u,v
E
v.
&
If 2v > 2p c i + 11 u , 11, then any weak solution X ( t ) to (2) converges to u, exponentially an mean square and so u, is exponentially stable in mean square. That is, there exists a real number y > 0 such that
Furthermore, pathwise exponential stability with probability one of u, also holds.
3. Exponential stabilizability and stabilization In the previous sections, the exponential pathwise stability has been proved as a by product of the mean square stability. However, i t may happen that a solution of a stochastic equation can be pathwise exponentially stable and not exponentially stable in mean square. Indeed, let us consider the following scalar ordinary differential equation to illustrate this fact,
d z ( t )= az(t)dt+ bz(t)dW(t), where a , b are real numbers and W is a one dimensional Wiener process. The solution is then given by
{
z ( t )= z(0)exp ( a
-
T) + t
bW(t)}
76
Thus, the zero solution is pathwise exponentially stable with probability one if and only if a - $ < 0. Also, we have that
E Jx(t)I2= E Jz(0)I2exp{ (2a
+ b2) t } ,
and therefore, the zero solution is exponentially stable in mean square if and only if a < 0. So, we observe that there exist many possibilities of being the zero solution pathwise exponentially stable and, at the same time, exponentially unstable in mean square. In Caraballo et al. l1 it is proved a result ensuring pathwise exponential stability without using the previous mean square analysis but under more restrictive assumptions on the terms appearing in the model. To this end let us firstly state the following assumption
+
Assumption C .
If
f :H
( u )- f I).(
g ( t , .) : H
5
-i
c
H , and satisfies
121 - 211
I
c
+ L ( K ,H ) , and
> 0, u , 21 6 H , satisfies
Ildt, u ) - g ( t , 2 " ) l l L ( K , H )I c, Iu - 4,B E
P
I
m),Vu,v E H .
Observe that if vX1 > c and f ( 0 ) = 0, then the zero solution to (3) is exponentially stable (see Temam 25). But when vX1 5 c and f (0) = 0 we do not know, in general, if the zero solution is exponentially stable or not. The following theorem is going to state that, under some particular conditions, any weak solution of the stochastic Navier-Stokes equation converges t o zero almost surely exponentially stable. So, in a sense, we can interpret that a kind of stabilization could have taken place in the system, i.e., the stochastic perturbation implies that the model exhibits better stability properties than it had.
Theorem 3.1. In addition to Assumption C, assume that f(0) g ( t , 0 ) = 0 for all t 2 0 , and that there exists po > 0 such that a$(s,
x) := tr [($,(x)
XI*)]
@ $z(x))(ds,z)Qds,
2d
lb14
=
0 and
,
where $(.) = 1xI2 (recall that ($z(.) @ $Z(X))(h)= &(.) (&(x), h ) I for E H ) . Then, there exists 00 c 0 ,P(R0) = 0 , such that for w @ 00 there exists T ( w ) > 0 such that any weak solution X ( t ) to (2) satisfies
x,h
cz + + 9). 2
where y := ~ ( X I U- c In particular, exponential stability of sample paths with probability one holds i f y > 0. We omit the proof since this result is a particular case of Theorem 4.1.
77
4. Pathwise stability and stabilizability with general decay
rate It may happen in some occasions that some systems are asymptotically stable but not exponentially, so it is very interesting to determine what is the actual decay rate of solutions. Now we will prove some results in this way concerning our Navier-Stokes model by adapting the techniques used in the papers Caraballo et al. to this case. First we will prove a general theorem which extends Theorem 3.1 and then we will comment about its consequences. ‘1’
Theorem 4.1. Assume that f : H + H , g ( t ,.) : H that f(0) = 0 and g(t,0 ) = 0 for all t 2 0 , and that
If(.)
-+
L ( K ,H ) are such
f(u)l 5 c I u - 211 , h u E HI < S ( t ) 1 2 ~- ‘ ~ 21 ,V U ,TJ E HIt 2 0 , lldtl.) - g(tluU)IIL(K,H)tr [(u8 u ) ( g ( t u)Qg(t, , u)*)l 2 ~ ( tI2l4 ) ,vu E H , t 2 0, where c > 0 and S ( . ) , p ( . ) are integrable nonnegative functions such that there exist 60 2 0, po > 0 satisfying -
2
where A(.) is a nonnegative continuous function such that A ( t ) +co as t goes to +co. Then, if Alv - c > 0 , it follows for any weak solution X ( t ) to (2) defined for every t 2 0 and such that IX(t)I > 0 for all t 2 0 and P-as., that
Let us apply Ito's formula for our soution X(t) satisfying the assumptions mentioned in the theorem. Then, it follows
78
and, applying once again ItG’s formula to the function log lX(t)I2, and taking into account the hypotheses, it follows
+q t
(X(S)Ig(s1 X ( s ) ) d W ( s ) )-
Ix;s)lz
2l
t
p(s)ds.
so
Now, observe that M ( t ) = !x!s,l ( ~ ( s g(s, ) , ~ ( s ) ) d ~ ( is s )a )real continuous local martingale and it is not difficult t o prove, by means of the law of iterated logarithm, lim
t++m
* log X ( t )
= 0, P
-
almost surely.
Indeed, if we denote by ( M ( t ) )the quadratic variation process associated t o M ( t ) , we deduce from the assumptions that
and, as po > 0, it follows that limt++m ( M ( t ) )= +m, what implies, by 0 means of the strong law of the large numbers, that limt+foo ( M ( t ) ) and, consequently
Dividing now in both sides of (5) by log X ( t ) we obtain
79
and the proof is finished by taking limits when t goes to +m.
Remark. Observe that when Xlv - c > 0 the null solution to the deterministic Navier-Stokes equation is exponentially stable, i.e. , every weak solution approaches zero with exponential decay rate. Then, by means of Theorem 2.1, we have that when the perturbation term tends t o zero exponentially fast, the weak solutions to the stochastic Navier-Stokes model also approach the null solution with the same decay rate. But, what in principle can be much more surprising is that when the perturbation is large enough (in a suitable way), we also have asymptotic behaviour with a decay rate which is similar to the growing of this perturbation term. To illustrate this idea assume for simplicity that this term is linear and is given by g ( t , z ) = a(t)z and W ( t )is a standard Wiener process. Now we can easily check that 6 ( t ) = p ( t ) = a2(t).If there exists X(t) such that lim
t+m
a2(s)ds = a0 > 0, logX(t)
then, Theorem 4.1 implies
(for some positive ao) we Consequently, if for example we take a ( t )= can take X ( t ) = t and it holds exponential stability of the null solution. If a(t) has exponential decay to zero, then Theorem 2.1 ensures exponential stability for the zero solution with probability one, and if a ( t ) grows to infinity with certain rate, say a ( t ) = t1/2, then choosing X ( t ) = expt2 it follows that a0 = 1 and therefore the weak solutions to (2) converge to zero with superexponential decay rate. So, we deduce from the previous analysis that certain stochastic perturbation may improve the stability properties of stable solutions to the deterministic equation.
Remark. However, even much more can be proved in the case that we do not know what happens with the null solution t o the deterministic problem, i.e, when Xlu - c < 0, we do not know whether the stationary solution of the deterministic problem is exponentially stable or not, but if the growing rate of the perturbation is super-exponential, then we can obtain superexponential decay asymptotic behaviour for the solutions t o the perturbed problem. See the next corollary.
Corollary 4.1. Assume that f : H 4HIg ( t , .) : H that f(0) = 0 and g ( t , 0 ) = 0 for all t 2 0 , and that
If(.)
- f(v)l
---f
L ( K ,H ) are such
I c Ju- 211 ,vu, v E HI
80
where A(.) is a nonnegative continuous function such that X ( t ) T +m as t goes to +m. Then, if Xlu - c 5 0 , and
t
lim
-
~
logX(t) - O1
t-02
it holds that
Proof. Proceeding as in Theorem 4.1 we get 1% lX(t)I2
1% IX(0)l2
1% X(t>
logX(t)
+
2s:
[-uX~+c+
"i"3 ds
1% X(t>
+--logM X( (t t))
2
J," log X ( t )
+
log IX(0)l2 + 2 (-vX1 c) t logX(t) 1% X(t) 2 P ( S ) ) ds +-logM X( (tt)> + sot (S(s)1% Vt> -
I
and, taking into account the super-exponential growth of X(t), the result follows immediately.
5. Conclusions, comments and open problems What we have first tried to point out in this work is that the theory of stability for linear and semilinear stochastic differential equations is so general that can be applied to the stochastic Navier-Stokes ones. Also, we have proved that the stochastic versions of Navier-Stokes equations satisfy similar stability properties to the deterministic unperturbed models. On the other hand, we also have pointed out that, when the noise is appropriately chosen, the perturbed stochastic model may exhibit better
81
stability properties than its deterministic counterpart. However, one obvious question is the following. The interpretation we have given to the noisy term has been in the sense of Itd, so is it possible that happens the same if we consider it in the sense of Stratonovich? This is of course an interesting and challenging problem for which we can only give some partial results and comments. In the finite dimensional case, there exits a wide literature on this topic (see Arnold and the references therein) which proves that some kind of multiplicative noise may produce a stabilization effect on deterministic unstable systems. However, for the infinite dimensional case, a similar result has not been proved yet, mainly due to the fact that the technique developed in the finite dimensional framework cannot be extended to this case or, a t least, it is not known how to do that. The main result proved in Arnold l ensures that an unstable linear differential system in Rn,namely k ( t ) = A z ( t ) with trace A < 0, can be stabilized by adding a multiplicative noise in the Stratonovich sense containing a suitable skew-symmetric matrix. One interesting remark is that when the stochastic multiplicative perturbation is considered in the It6 sense, this uses to imply a general stabilization effect on the system. In a limit sense, the It6 equations with multiplicative noise correspond to deterministic equations with a mean-zero fluctuating control plus a stabilizing systematic control. This would mean that only the stabilization produced by Stratonovich terms could be considered as proper stabilization produced by random noise. However, in the infinite-dimensional case we have been able to prove in the linear framework that, if some kind of commutativeness holds, the deterministic systems and their stochastic perturbed versions have the same behaviour when the noise is considered in the sense of Stratonovich, while if the noise is considered in It6’s sense, persistence of stability, stabilization and even destabilization may happen (see also Caraballo and Langa l o for an analysis on these topics). Finally, we would like to mention that from a global point of view, the analysis of the effects produced by random perturbations in deterministic systems is being investigated right now by many authors within the framework of the theory of random attractors recently introduced, among others, by Crauel and Flandoli 14. On the one hand, existence of random attractors is only known for specific random terms (see, for instance, Crauel and Flandoli 14, Capinski and Cutland 7 , Flandoli and Lisei 1 7 ) . On the other hand, almost nothing is known on the structure of these random sets, so that many challenging open problems, as those related to stability and instability, are still open.
82
Acknowledgments
I would like t o thank Aubrey Truman and Ian Davies for the kind invitation t o take part in this Conference on Probabilistic Methods in Fluids. I finished this work during my stay in the Mathematics Institute (University of Warwick, June-August 2002). I would like to thank the Royal Society of London for their generosity, and especially, to James Robinson and Tania Styles for the hospitality and friendship they offered me, what made me feel as if I would have been at home. This paper has been partially supported by Secretaria de Estado de Universidades e Investigacibn (Spain).
References 1. L. Arnold, Stabilization by noise revisited, Z. angew. Math. Mech. 70 (1990), 235-246. 2. L. Arnold, Stochastic Differential Equations: Theory and Applications, J. Wiley and Sons, New York, (1974). 3. A. Bensoussan, Stochastic Navier-Stokes equations, Acta Applicandae Math., 38 (1995), 267-304. 4. Z. Brzezniak, M. Capinski and F. Flandoli, Pathwise global attractors for stationary random dynamical systems, Prob. Th. Rel. Fields, 95 (1993), 87102. 5. M. Capinski and D. Gatarek, Stochastic equations in Hilbert spaces with application to Navier-Stokes equations in any dimension, J. of Funct. Anal. 126(1994), 26-35. 6. M. Capinski and N. Cutland, Measure attractors for stochastic NavierStockes Equations, Electronic J. of Prob., 3(1998), 1-15. 7. M. Capinski and N.J. Cutland, Existence of global stochastic flow and attractors for Navier-Stokes equations, Prob. Th. and Rel. Fields 115(1999), 121-151. 8. T. Caraballo, M. J. Garrido-Atienza and J. Real, Asymptotic stability of nonlinear stochastic evolution equations, Stoch. Anal. Appl., to appear. 9. T. Caraballo, M.J. Garrido-Atienza and J. Real, Stochastic stabilization of differential systems with general decay rate, Systems and Control Letters, to appear. 10. T. Caraballo and J.A. Langa, Comparison of the long-time behaviour of linear It6 and Stratonovich partial differential equations, Stoch. Anal. Appl. 19(2001), 183-195. 11. T. Caraballo, J.A. Langa and T. Taniguchi, The exponential behaviour and stabilizability of stochastic 2D-Navier-Stokes equations, J . Diff. Eqns. 179(2002), 714-737. 12. T. Caraballo and K. Liu, On exponential stability criteriaof stochastic partial differential equations, Stochastic Processes and their Applications 83 (1999), 289-301. 13. P. Constantin and C. Foias, "Navier-Stokes Equations", The University of
83
Chicago Press, Chicago and London, 1988. 14. H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Th. Rel. Fields, 100(1994), 365-393. 15. G. Da Prato and J. Zabczyk," Stochastic Equations in Infinite Dimension;, Cambridge, 1992. 16. F. Flandoli and D. Gatarek, Martingale and stationary solution for stochastic Navier-Stokes equations, Prob. Th. Rel. Fields, 102(1995), 367-391. 17. F. Flandoli and H. Lisei, Stationary conjugation of flows for parabolilc SPDEs with multiplicative noise and some applications, preprint (2002). 18. J. Hale, Asymptotic behaviour of dissipative systems, Math. Surveys and Monographs 25, (1988). 19. 0. Ladyzhenskaya, Attractors for semigroups and evolution equations, Cambridge University Press, (1991). 20. H. Kunita, Stochastic Partial Differential Equations Connected with Nonlinear Filtering, in Lecture Notes in Mathematics, S.K. Mitter adn A. Moro, Eds. Springer Berlin 1982, Vol 972, 100-169. 21. B. Oksendal, Stochastic Differential Equations, Springer-Verlag, Berlin (1992). 22. E. Pardoux, Equations aux dhrivhes partielles stochastiques non linhaires monotones. Etude de solutions fortes de type It8, These,1975 23. T . Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics and Stochastics Reports, 53(1995), 41-52. 24. R. Temam, Navier-Stokes Equations, North-Holland, 1979. 25. R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, second edition, SIAM, 1995. 26. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988.
SEMILINEAR STOCHASTIC WAVE EQUATIONS
P.L. CHOW Wayne State University Detroit, Michigan 48202, USA E-mail:
[email protected] The existence and uniqueness of solutions to a class of semilinear stochastic hyperbolic equations in Ed are considered. First an energy inequality for a linear stochastic hyperbolic equation is established. Then it is proven that there exists a unique continuous local solution for the associated nonlinear equation in the Sobolev space H I ( R d when ) the nonlinear terms are locally bounded and Lipschitzcontinuous. Under an additionalcondition on the energy bound, the solution exists for all time. The results are shown to be applicable to stochastic wave equations with polynomial nonlinearities of degree m with m 5 3 for d = 3 , and for any m 2 1 for d = 1 or 2.
1. Introduction
Consider the stochastic wave equation:
8,".
=v2u
+ o(u)atW(x,t )
(1)
where at denotes the partial derivative in t , V2 the Laplacian; W ( . ,t ) is a Wiener random field. For d = 1 or 2, Mueller proved that the equation Eq. (1) has a unique long-time global solution pointwise in (5, t ) E Rd x [0, m), provided that o(u)grows no faster then lul(logIu1)' with T E ( 0 , a ) . In the case d > 1, the Weiner field W ( x , t )must be smooth in z, because nonlinear equations such as Eq. (1) and Eq. (2) below are not well defined if a t W ( x ,t ) is a space-time white noise (see Walsh '). In view of Mueller's result, the following question arises naturally. That is, if o(u)grows like u ' for a sufficiently large T > 1, whether a solution to Eq. (1) may blow up in a finite time. This question is still open. Here we consider a related problem as follows:
+
+
f ( u ) .(u)atw(x, t ) , z E Rd, t > 0 0) = g(z), dtu(z,0) = h ( x ) ,
a,zu = v 2 u
{ u(x,
(2)
where nonlinear terms f ( u ) and a(u) are assumed to grow like polynomials in u,and the initial data g and h are given functions. In general 84
85
such an equation admits only a local solution, if it exists. For example, when f(u)is cubically nonlinear, we showed (see Chow that, under some conditions on the data, the solution can explode in finite time. We also proved the existence and uniqueness theorems for local and global solutions in the case when f(u)is a polynomial of degree m under suitable conditions, where m depends on the space dimension d 5 3. In this paper we will consider the case where the Laplacian is replaced a second-order strongly elliptic operator and the nonlinear terms are locally Lipschitz continuous in a Sobolev space. In particuIar we will show that such results are applicable to equations with polynomial type of nonlinearities mentioned above. To be specific, we shall first derive the basic energy inequality for a linear stochastic hyperbolic equation in section 2. Then, for a class of nonlinear hyperbolic It6 equations in section 3, the existence and uniqueness of a continuous local solution will be presented in Theorem 3.1 under the assumptions that the nonlinear terms are locally bounded and Lipschitz continuous in the Sobolev space H1(Rd).As stated in Theorem 3.2, if an additional energy inequality can be established, the solution will become global. In section 4, the theorems are applied to some polynomially nonlinear stochastic wave equations to yield the existence and uniqueness results obtained in the paper by Chow 2. Linear Stochastic Hyperbolic Equation Let H := L 2 ( R d )with the inner product and norm denoted by (., .) and 11. 1 1 respectively. Let H1 = H1(Rd)be the L2-Sobolev space of order one with norm ((.111. Let (0,F,P ) be a complete probability space for which a filtration .Ft is given. Let M ( z , t ) ,t 2 0 , x E R d ,be a continuous martingale with a spatial parameter s E Rd and M ( z , 0) = 0 in the sense of Kunita ‘. Let its covariation function q(s,y, t ) be defined as in
< M ( z , .), M(y, .) >t=
I’
q(s,y, s)ds,
x,y
E Rd,t E
[0,TI
a.s.
(3)
Regarding Mt = M ( . , t ) as a continuous H-valued martingale with covariation operator Qt defined by
Let W ( x , t )be a continuous Wiener random field with mean zero and covariance function ~ ( z9,) defined by
E W ( x ,t>W(Yl,s> = (t A
S ) T ( Z , !/)I
x,y
E Rd,
86
= min(t, s) for 0 5 t , s 5 T . Let a(x,t)= o ( x , t , w ) for t 2 0,s E Rd and w E R , be a continuous predictable random field such
where (t A s) that
a2(x, t ) d t < oa,for each x E Rd a.s..
As a special case, let M be the stochastic integral M ( x ,t ) =
lt
o(x,s ) W ( x ,ds), t > 0 , x E
Ed,
which is a continuous Wiener martingale with spatial parameter x and covariation function given by
d x ,Y,t ) = T ( 2 , y)a(x, t k ( Y , t ) for x , y E Rd,tE [O,T]. Now we consider the Cauchy problem for the linear hyperbolic equation with a random perturbation:
i
+
[a; - A(x,D ) ] u ( x t, ) = f (z, t ) & M ( x ,t ) , 0 < t < T , u ( z ,0) = uo(x), &u(x,0 ) = vo(2), 2 E Rd,
(4)
where A ( x ,D ) is a strongly elliptic operator of second order of the form:
c d
A(& D)cp(x)=
~ z , [ ~ i j ( ~ ) ~ z , cp (b(z)cp(x), 41
(5)
i,j=l
where the coefficients aij ao(1
+ m2)5
= u j z and
c
b are smooth functions such that
d
U%)&<j
+ ls1)2), t , x E Rd,
I
i,j=l
for some constants a1 2 a0 2 0. This condition implies that ( - A ) is a self-adjoint, strictly positive linear operator on H = L 2 ( R d )with domain D(A)= H z ( R d )and its square root B = is also a self-adjoint, strictly positive operator with domain D ( B ) , which is a Hilbert space under the inner product (9, h ) := ~ (Bg,Bh). Since the norms 11 . I I B and 11 . 111 are equivalent, we have D ( B ) 2 HI.
a
Let ut system:
=
u ( . , t ) ,ut = &u(.,t) and rewrite the equation Eq. (2) as a
rt
87
or equivalently, rt
(7) where we set
and
with I being the identity operator on H . Introduce the Hilbert space ‘FI = ( H I x H ) . As a linear operator in ‘FI, A generates a strongly continuous semigroup etA on ‘H. Now regarding Eq. (7) as a stochastic evolution equation in ‘FI in a distributional sense, we have the following lemma:
Lemma 2.1. For $0 = ( U O ,UO) E ‘H, let f t be a continuous predictable process in H , and let Mt be a continuous H-valued martingale with covariance operator Qt such that
Then the equation Eq. (7), or Eq. (6) has a unique (mild) solution $t = (ut, ut) which is a continuous predictable ‘FI-valued process. Moreover at satisfies the energy equation: rt
rt
for t E [0,TI. Moreover, if in addition to Eq. (8),
where the constants C1, C2 depend on p , T and the initial conditions. Notice that, due t o the lack of required smoothness of solutions, the general It6 formula does not hold here. The energy equation Eq. ( 9 ) can be proved by a smoothing technique, such as the Yosida approximation (Yosida 5 , as done in (Chap. 5, Da Prato and Zabczyk 6 ) , and then taking a proper limit.
88
The energy inequality Eq. (11) can be shown to hold by applying the It6 formula to the energy equation and by invoking Burkholder's submartingale inequality. The proof is similar to the special case given in Chow and will be omitted. 3. Semilinear Stochastic Hyperbolic Equations
Let us consider the Cauchy problem for the following hyperbolic equation:
+
i
(8,"- A)ut = f t ( J u t ) & M t ( J u ) , t > O uo = 9, &uo = h.
(12)
In the above equation, we assume g E H I and h E H , and set J u = (21,&,u,..., aZdu, at.), f(x,J u ( ~ ) , t = ) f t ( J u ) ( x )and M ( z , J u ( x ) ,t ) = Mt ( J u )(x)defined by rt
where, for x E Rd, E E Rd+',f(x,[, t ) and o(x,6 , t ) are continuous predictable random fields, and Wt = W ( . , t )is a continuous Wiener random field with covariance operator R of kernel ~ ( xy), , for x,y E Rd.Let C t ( J u ) be defined by
[ C t ( J ~ ) h ] (=z U ) (Z, Ju(z), t)h(z) h E H. For brevity, let F t ( J u ) be a stochastic integral defined by
Again we rewrite the equation Eq. (12) as a stochastic system in the Hilbert space IH:
ut Vt
+ J, usds = YO + s,' Ausds + F t ( J u ) , = uo
t
which, similar to Eq. (7), yields the simple form: $t = $0
where
dt
+
1 t
A$&
+ 3t(4),
and A are defined as before and
(15)
89
Let (p = ( u , v ) E ‘FI and set J u = (u0,u1,...,ud+l) E ( H ) d f 2 ,with the convention: uo = u,uj = dZju,j = 1, ...,d , and ud+’ = &u . Introduce the energy function e ( 4 ) defined by
j=O
Theorem 3.1. Suppose the following conditions hold: (1) Let f t ( J . ) : H1 4 H such that
llft(Ju)l12 I Cl(1 + 11~113 and IlfdJu)
-
ft(Ju’)JI _<
C2llu - u’JJ1 a s . ,
f o r all u, u’ E H I ,t E [0,TI, and f o r some constants C1, C2 > 0 . (2) For any u E H I , the map C . ( J u ) : [O,T] L ( H ) is continuous a s . , where L ( H ) denotes the space of bounded linear operators on H . There exist positive constants C3 and C4 such that ---f
T r [ C t ( J u ) R C ; ( J 7 4I C3(1
+ llull:),
and T r { [ C t ( J u )- C t ( J u ’ ) ] R [ C t ( J u) Ct(Ju’)]*
I C4llu - U’III: a s . , f o r any u , u‘ E H I ,t E [0,TI, where * denotes the adjoint. (3) W, is a H-valued process with covariance operator R such that
Then, f o r g E H I ,h E H , the system Eq. (15) or Eq. (16) has a unique (mild) solution ut on [0,T ] with u.E C ([0,TI,H I ) and u.E C ([0,TI, H ) . Moreover the following energy equation holds e ( u t , vt) = e(uo,uo) +2
I”
+2
(w,, C,(Ju,)dW,)
I”+
(‘us, f
I”
(Jus))ds (19)
Tr[C,(Ju,)RC; (Ju,)]ds.
90
Under the above conditions (1)-(3), it is easy to check that the coefficients of the evolution equation (3.5) in IFI satisfies the usual global Lipschitz continuity and linear growth conditions. Theorem 3.1 follows from a standard existence theorem (Theorem 7.4, Da Prato and Zabczyk 6 , for stochastic evolution equations in a Hilbert space. To be able to apply the theorem to equations with coefficients of a polynomial growth, we relax the global conditions (1) and (2) to the local ones. To this end, replace the constants by functions of the form bl(s), b z ( s , t ) ,b3(s), b4(s, t ) , for s, t E R such that they are positive, locally bounded and monotonically increasing in each variable. Then the following theorem holds. Let conditions (Nl)-(N4) be given as follows:
(Nl) Let f t ( J . ) : H I + H such that
and
lIft(J.1
-
ft(Ju')ll
i bz(lluIl1,
ullll
11~1111)11~ -
a.s*>
for all u,u' E H I , t E [O, TI. (N2) For any u E H I , the map C . ( J u ) : [O,T]+ C ( H ) is continuous a s . such that
and
T r { [ C , ( J u )- C t ( J u I ) ] R [ C t ( J u-) C t ( J u I ) ] * )
for any u,u' E H I , t E [O,T]. (N3) Wt is a H-valued process with covariance operator R such that
and sup X E R d
T(2,X)
< 00.
91
(N4) Suppose that, for any u.E C([0,TI,HI)n C'( [0,TI,H) with &u v, there exist constants c l , c2 > 0 and
IE
<
t E [O, TI,
=
1 - such that, for any 2
t
< c1+ c2
J, e(us,v,)ds + IEe(ut,vt)
as..
Theorem 3.2. If the conditions ( N l ) - (N3) hold, then, foruo E H1,vo E H, the Cauchy problem Eq. (12) has a unique continuous local solution u ( . , t )E H I with &u(.,t)E H . If, in addition, condition (N4) is satisfied, the solution u(., t ) exists on (0, T] for any T > 0. The main idea of the proof, similar to that in Chow 3 , is to show that, by a smooth HI-truncation, the conditions (Nl)-(N3) reduce to the conditions (1)-(3) in Theorem 3.1. Therefore the truncated problem has a continuous solution uN(., t ) E HI for t < (TN A T ) , where ( s A t ) = rnin.{s,t } , and TN is a stopping time defined by
> 0 : 11ur111 > N } , with N being a cut-off number. Hence, for t < (TN A T), u(., t) = uN(., t) TN
= inf{t
is the solution of Eq. (12) with &u = v N . Noting that TN increases with N , let T = lim T N . Define u ( . , t ) for t < ( T N A T ) by u ( . , t ) = u N ( . , t ) t+m
if t < TN 5 T . Then u ( . , t ) thus defined is the unique local solution. To obtain a global solution, it is necessary to have an energy bound. This can be established by imposing condition (N4). Then it can be shown that Prob ( r < m) = 0. Therefore the solution exists on any finite time interval [0,TI as claimed.
Remark: In the above theorem, for simplicity, we assumed that the Wiener random field W ( x ,t ) is scalar or, in the integral Eq. (13), Wt is a H-valued process. Theorem 3.2 still holds true when W ( x ,t ) and ~ ( x5,,t ) are both random vector-fields such that the product in the integrand of Eq. (13) is regarded as a scalar product. 4. Applications
R3: + at(J..t)atwt, t > 0 ,
Let us consider the following initial-value problem in
(a; - c 2 v 2+ y2)ut = ft(.t) 210
= g,
dtuo = h,
92
where c and y are positive constants, while ft and ut are nonlinear (deterministic) functions of polynomial type. In comparison with Eq. (12), we have A = (c2V2- y2),f t ( J u )= ft(u)and M t ( J u ) is defined by Eq. (13). In particular, we assume that the following conditions hold:
ft(s)(x) = f(x,s, t ) ,x E Rd,s E R d ,t form:
> 0, is a polynomial of the
j =O
where aj(x,t ) is bounded and continuous on Rd x [0,T ] for each ...,m. The function .t(E)(x) = (~(x, 6 ,. . . , & + I , t ) , for x E Rd,[ E Rd+2 and t > 0 , is continuous. There exist positive constants C,,C2 such that, with k 5 2m,
j=O,l,
j=1
and
for x E Rd;,$,q E Rd+' and t E [0, TI. Let W ( x , t )be a continuous Wiener random field as given before with covariance function r(x,y) such that
Tr R = and To
=
s
r(x,x)dx < 00
sup ?-(x,x) < m. z€Rd
To apply the previous theorem to Eq. ( l a ) , it is necessary to show that the nonlinear terms are dominated by a HI-norm. For polynomial nonlinearities, one appeals to the Sobolev imbedding : H1(R3) c LP(Rd) (p.112, Adams '). In particular we recall the following useful lemma (see, e.g. p.21, Reed s). Denote the LP(Rd)-normby I . I p and let C r stand for the set of Cm-functions on Rd with a compact support.
93
Lemma 4.1. For u,v E C,W and 1 5 k ~ 1~2,
5 m, there exist positive constants
such that
and
where m = 3 f o r d = 3, and m 2 1 ford = 1 or 2. With the aid of this lemma and conditions (P1)-(P3),we can apply Theorem 3.2 to give a local existence theorem for Eq. (20) with polynomial nonlinearities. Theorem 4.1. Suppose that conditions (Pl)-(P3) given above hold true. Then, for g E H I and h E H , the Cauchy problem Eq. (20) in Rd,for d _< 3, has a unique continuous local solution ut E H I with &u, E H , provided that m 5 3 for d = 3, and m 2 1 for d = 1 or 2. The proof of this theorem under the stated assumptions is to verify the conditions (Nl)-(N3) in Theorem 3.2 are satisfied for d 5 3 . We will sketch the proof in steps: Step 1) : In view of condition (Pl) and Lemma 4.1, we have,
where a0 = max
m
m
1
j=1
sup
\ a j ( x , t ) l .Hence, for u E H1 and t
E
l l j l m tE[O,T],ZERd
[O, TI, we have Ilft(u)1125 b l ( l l ~ l l ) l l ~ l l f l
(21)
where m
bl(r) =
(~OC~)~(C ~ j - ' ) ~ .
j=1
Step 2) : Similar to Step 1, it can be shown that, for u,v E H I and t E [0,TI,
Ilft(.)
I b 2 ( l l ~ l l 1 l, l ~ l l 1 ) l -l ~ Vll?,
- ft(v)l12
(22)
94
where b 2 ( ~s) , is a polynomial of degree 2(m - 1) in T , s E R with positive coefficients. For instance, consider the case d = 3 and m = 3. By condition ( P l ) , we have
cllJ 3
IIft(.)
- ft(41I2I
c1
j=1
-
412,
(23)
for some constant C1 > 0. By invoking Lemma 4.1,
+
)lu2- v2112I c211u vll?ll. - ull: I 2C2(ll.llf + Il4lf)ll. - vll4,
(24)
and
q = II(.~+ uw+ G)(.- .)II~ I 8(IIu2(~ v)(I2+ IIu2(u - u)l12) 1 1 ~3
(25)
-
L C3(ll4I! + 1 1 ~ 1 1 ~ ) 1-1 ~414.
In view of Eq. (23)-Eq. (25), condition ( N l ) holds for d = 3. For d < 3, it can be verified in a similar fashion. Step 3) : By making use of conditions (P2) and (P3) together with Lemma 4.1, we get, for u E H I and t E [O,T],
TT[o~(Ju)R~~(Ju)] = J T ( X , z)o2(z,J u , t ) d z
c;::
I c4 J.(z, z)(l + IU12k + lujl2)dz I Cq(T7-R+ r o l ~ l $ k+ ~ 0 1 1 ~ 1 1 ~ ) IK ~ ( + I IIullf(k-l))(l + ~ ~ u for~ k~I ~m. ) ,
(26)
The inequality Eq. (26) implies that t.[
where
b 3 ( ~= )
( J U ) Rat
K1(1+ T
(J.)1
I b3 ( /I11 II 1) (1 + IIuI I 3,
~ ( ~ - for ~ )some )
(27)
constant K1 > 0.
Step 4) : Similar to Step 3, by means of conditions (P2), (P3) and Lemma 4.1, we deduce that
T r { [ g t ( J u) at(Jv)]R[n(Ju -).t(J.>]*} = J?-(z,.>[cJt(Ju)- at(Jv)]2dz I c2 J.(z,z)([l+ J ? q ( k - - 1 ) 1w12(k--1)]Iu - 2112
+
+ C;f:
IUj12)dz
(28)
< KZ[l + (Iu112(k-1) + ~ ~ w ~ / 2 ( ~ - 1.u11?, )]~~u for some constant K2 > 0. I t follows from Eq. (28) that -
TT{[gt(JU)-gt(JV)IR[gt(JU)-gt(J21)1*} F b4(11UII1,
+
+
11~111)11~-~11~,(29)
) ( ~ - ' )In ] . view of Eq. (27) and with b 4 ( ~s), = Kz[l T ~ ( ~ - ~s ~ Eq. (29), the condition (N2) is valid.
95
Clearly condition (P3) is similar to (N3). Therefore, by Theorem 3.2, the Cauchy problem Eq. (20) has a unique continuous local solution as stated.
Remark: As pointed out in the remark following Theorem 3.2, in Eq.(4.1), the noise term may assume a more general form, such as
where W j ,j = 1, ..., n, are n independent Wiener processes in H with covariance functions r j . Then Theorem 4.1 will hold if, for each j , the conditions (P2) and (P3) are satisfied, (see the example given below). To obtain a global solution, it is necessary to impose further conditions on the functions f and CT so that the condition (N4) will be met. For convenience, introduce the function G defined by
f ( x ,r, t)dr = -
“
2
-aj(x, j=1 3 + 1
t)uj+l
Then the next theorem holds true, and its proof can be found in Chow
3.
Theorem 4.2. Suppose that all conditions in Theorem 4.1 are fulfilled. In addition to ( P l ) and (P2), assume that
+
( l a ) Given m = (2n 1) for a positive integer n, there exist constants a 2 0 and ,L? 2 0 such that G ( x ,r , t ) 2 ( a + PrZn)r2 for each x E Rd,r E R and t E [0,TI. (2a) Condition (P2) holds with k = ( n 1).
+
Then the solution obtained in Theorem 4.1 exists in any finite time interval (0, TI. As an example, consider the cubically nonlinear wave equation in R3 under a random perturbation:
+
(8,”- c 2 v 2 y2)u u(., 0) = g,
a,.(.,
= xu3
+ atMt(Ju)
0 ) = h,
x E
723,
t > 0,
(31)
96
where c, y and X are real parameters, the functions g, h are given as before, and M t ( J u ) is assumed to be of the bilinear form:
c/ 3
M t ( J u ) ( z ):= M ( z , J u , t ) =
j=1
t
[aZju(z,s ) ] W j ( zds). ,
0
Here W j ,j = 1, 2,3, are independent Wiener random fields with covariance functions r j ( z ,y) such that
~ j ( z , z ) ] d z sup + r j ( x , z )< m. j=1
.€R3
Clearly Eq. (31) is special case of Eq. (20) with f t ( J u ) = Xu3,d = m = 3 and a more general noise term. In view of Eq. (32) and the remark following Theorem 4.1 , it is easy t o check that, for any XI conditions (Pl)-(P3) are met so t h a t there is a unique continuous local solution as stated. By definition given in Eq. (30), we have
-A 2 so that, if X 5 0, condition (la) in Theorem 4.2 holds with a = 0 and p = ($). Condition (2a) is obviously true. Therefore, for X 5 0, the Cauchy problem Eq. (31) has unique continuous solution u E C ([O, TI, H l ) n C1( [0, TI , H ) on any finite interval [0,TI.
G(z,T , t ) = (-)r4,
Acknowledgments This work was supported in part by the NSF Grant DMS-9971608.
References 1. C. Mueller, A n n . Probab. 25, 133 (1997). 2. J.B. Walsh, Lect. Notes in Math., Springer-Verlag, Berlin, Heidelberg, New York, 1180, 265 (1984). 3. P.L. Chow, A n n . Appl. Probab. 12,361 (2002). 4. H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Univ. Press, Cambridge, England, 1990. 5 . K. Yosida, Functional Analysis, Springer-Verlag, New York, 1968. 6. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, England, 1992. 7. R.A. Adams, Sobelev Spaces, Academic Press, New York, 1975. 8. M. Reed, Abstract Non-linear Wave Equations, Springer-Verlag,Berlin, 1976.
STOCHASTIC NAVIER-STOKES EQUATIONS: LOEB SPACE TECHNIQUES & ATTRACTORS
NIGEL J. CUTLAND Department of Mathematics, University of Hull Hull, HU6 7RX, UK E-mail: n.j.
[email protected] We survey the use of Loeb space methods in stochastic fluid mechanics, with particular emphasis on recent results concerning the existence of attractors for the stochastic Navier-Stokes equations.
1. Introduction
A general version of the stochastic Navier-Stokes (sNS) equations in a bounded domain D c EXd takes form:
{
+
+
du = [VAU- (u, V ) U f ( t , u)- V p ] d t g ( t , u ) d ~ t divu = 0
(1)
Here u ( t , x , u )is the (random) velocity of the fluid a t the location x E D a t time t , so that we have
u : [O, m) x
D
xR
---f
Rd
where R is the domain of an underlying probability space. The initial condition u(0) = u g is prescribed (and may be random); the boundary condition is either u(t,x) = 0 for x E 8D or, occasionally, when d = 2 we assume periodic boundary conditions. These equations have been the subject of considerable study since they were first solved in [6], for d 5 4, using Loeb space methods. Some time after the publication of [6] a number of alternative proofs of existence appeared (see below) so that now there is considerable interest in more delicate issues such as the existence of a stochastic flow and attractors for the sNS equations. Loeb space methods have continued to prove powerful in this field, in combination with the well-developed techniques of LLc1assica177 infinite dimensional stochastic analysis. The purpose of this paper is to survey what 97
98
has been achieved, with particular emphasis on recent work on attractors for the sNS equations in d = 2,3. 2. Existence for the stochastic Navier-Stokes equations The equations (1) with additive noise - that is, with g independent of were first discussed by Bensoussan & Temam in [3]- where they were solved in 3 dimensions with 20 a 1-dimensional Wiener process and g = Identity. Later contributions t o the additive noise case were made by Viot [29] and Vishik & F’ursikov [30]. For multiplicative noise the general equations (1) in dimensions d 5 4 (with only natural growth conditions on f , g ) were first solved in 1991 by the author and Marek Capinski [6] using the Loeb space techniques to be elaborated upon below. Just prior to this (though published later) Brzezniak, Capinski & Flandoli [2] obtained solutions for d = 2 with a special form of multiplicative noise, and only for small initial conditions; around the same time Bensoussan [4]established general existence ford = 2. Some three years later, alternative proofs of existence for the general equations in higher dimensions began to appear, beginning with the papers of Capinski & Gatqrek [15] and followed by Bensoussan [5]. The latter paper, curiously, has exactly the same title and appears in the same journal as [6],which is not acknowledged even though the author of [5]was an editor of the journal at the time. Around the same time Flandoli & Gatqrek [24] proved existence of solutions to a number of different formulations of the general sNS equations (1) for d 5 4, as well as stationary solutions in each case. 2.1. Mathematical formulation
To proceed it is first necessary t o note the precise mathematical formulation of the equations (1). We adopt the usual Hilbert space setting as follows. Denote by H the closure of the set {u E C r ( D ,Rd):div u = 0} in the L2 norm 1ul = ( ~ , u ) lwhere /~,
The space V is the closure of {u E C r ( D ,Wd): div u = 0) in the stronger u I llull where /lull = ((u,u ) ) ~and /~ norm I
+
99
H and V are Hilbert spaces with scalar products (., .) and ((., .)) respectively, and 1 . I 5 cII . 11 for some constant c. By A we denote the self adjoint extension of the projection of -A in H; A has an orthonormal basis { e k } of eigenfunctions with corresponding eigenvalues X k r X k > 0 , X k m. For u E H we write U k = ( u , e k ) , and write Pr, for the projection of H on the subspace H, spanned by { e l , . . . , em}. Since each ek E V then H, C V. The trilinear form b defined by d
avi
(whenever the integrals make sense) has the well-known and crucial property b(u,w , w) = --b(u,w, w) so that b(u,w,w) = 0. In this framework, the stochastic Navier-Stokes equations (1) may be formulated as a stochastic differential equation in H as follows:
+
+
du = l-vA.1~- B ( u ) f ( t ,u ) ] d t g ( t , u)dwt
(2)
where B ( u ) = b(u,u,.). This is initially regarded as an equation in V’ (the dual of V) although it turns out that the solution lives in H (and in fact in V for almost all times). Compared to ( l ) ,note that the pressure has disappeared, because V p = 0 in V’ (using divu = 0 in V and an integration by parts). The equation (2) is really an integral equation, with the first integral being the Bochner integral and the second an extension of the It6 integral t o Hilbert spaces, due t o Ichikawa [25]. The noise is given by a Wiener process w : [0, m) x R t H with trace class covariance, and so the noise coefficient g belongs t o L(H, H). It is assumed that g : [ O , o o ) x V + L(H,H)
while
f : [O,m) x
v -+ V’.
(The restriction to V in the domains is sufficient because we will have the solution in V for almost all times.)
2.1.1. Definition of solutions to the stochastic Navier-Stokes equations The following makes precise what is meant by a solution t o the stochastic Navier-Stokes equations as formulated above. In fact there is a range of
100
solution concepts of varying strength, each of which is appropriate in certain circumstances.
Definition 2.1. Suppose that u g E H and f,g as above are given, together with a probability space R carrying an H-valued Wiener process w. A weak solution of the stochastic Navier-Stokes equations is a stochastic process u : [0,co) x R 4 H such that for 8.8. w (i) u E L 2 ( 0 ,T ;V) n L"(0, T ;H) n C(0,T ;Hweak)for all T < co , (ii) for all t 2 0 rt
rt
A strong solution" has in addition that for a.a. w
for all T The notion of solution for the deterministic case is given by taking g = 0 and removing the random parameter w throughout, so a weak solution is a single function u E L2(0,T ;V) n Loo(O,T ;H) n C(0,T ;Hweak) for all T . The classical approach to the solution of the Navier-Stokes equations (deterministic or stochastic) is to begin with an approximate version in the finite dimensional space H, for each n , the so called Galerkin approximation, which can be solved easily using standard techniques from ODES (or SDEs in the stochastic case) to give Galerkin approximate solutions u"(t). The hard part is then to find some way to pass to the limit t o obtain a solution to the Navier-Stokes equations. First, some specialized compactness theorems are required t o show that there is a subsequence of (u"(t)),cn that converges in an appropriate sense to a limit ~ ( tsay. ) Second it is necessary to show that this limit u ( t ) actually is a solution. The difficulties are compounded in the case of the stochastic equations especially in dimension 2 3 because it seems necessary to work with a probability space that is bigger than the Wiener space. 2.2. Loeb space methods
The methods used in [6] and subsequent papers are expounded in full detail in the book Capiriski & Cutland [ll]and also in the monograph [16], so "Some authors require a strong solution to have the stronger property
+ JT 1Au(t)l2dt) < ca for all T ; we prefer to call this strictly strong.
( s u p t 5 ~llu(t)112
101
here we only convey the key ideas. A Loeb space is essentially an ultraproductb of probability spaces; the particular Loeb space used in [6,11]is an ultraproduct of finite dimensional Wiener spaces. This is simply a rather rich conventional probability space with filtration] that happens to be constructed as an ultraproduct. The power of Loeb spaces comes from the combination of their richness and the fact that they are tractable: their richness can be easily exploited using the ideas of Robinson’s nonstandard analysis. The heart of this is a transfer principle that means that properties of the original spaces in the ultraproduct are inherited in a precisely defined way by the Loeb space. In the appendix we put a little more flesh on this idea, and point the reader to sources where a full exposition may be found. The methods of [6,11], which apply to dimensions d 5 4, can be informally described now as follows. Take R, to be the canonical Wiener space of dimension n and let R be the Loeb space that is the ultraproduct of the spaces ( R , ) n E ~ .Let u, be a solution on R, to the n-dimensional Galerkin approximation of the sNS equations (2). Then the ultraproduct U of the solutions ( u , ) ~ € N is a “nonstandard” approximate solution to (2) that lives on R. Formally] U has values in the ultraproduct of the spaces ( H n ) n E ~ , which is denoted HN,where N is an infinite natural number. That isc
U :R x R
+ HN
Most importantly] U inherits] via the transfer principle, the properties of the Galerkin approximations (u,),€~, especially the usual energy estimates. This enables the definition of a process u : R x R -+ H by u(tl w ) = “ U ( t w , ) using a mapping O : HN -+ H called the standard part mapping, that is defined for certain nearstandard members of HN. The energy estimates inherited by U are crucial in showing that U(tl w ) is nearstandard for a.a. w E 0. Once u is defined it is fairly routine to check that it is a solution to (2). In 2-dimensions, if the noise g(tlu)has a special form - essentially that it is orthogonal to the solution process u - it was shown in [8] how the above method provides a construction of a global stochastic flow for (2). The techniques developed in [6,11] for the stochastic equations originated in the paper [7] where the idea of Galerkin approximations of dimension N , with N an infinite natural number, were used to give a very bThat is, a quotient of a product by an equivalence relation that is given by an ultrafilter. =In fact U : *R x R -+ HN where *R is the hyperreals, the extension of R given by the ultraproduct of countably many copies of R,but in particular U is defined on all of R.
102
simple proof of existence for the deterministic Navier-Stokes equations in dimensions d 5 4. An almost trivial extension of the method gave existence of statistical solutionsd in dimension d 5 4 with an arbitrary initial measure. The idea applies also to the sNS equations to provide Foias and Hopf statistical solutions of the stochastic Navier-Stokes equation - see [9]. The framework sketched above for solving the sNS equations allows a more radical approach to solving the Foias equations. At the penultimate stage of the construction, the “nonstandard” solution U ( 7 ,w)lives in HN, which is isomorphic to RN and carries a nonstandard version of Lebesgue measure. Thus the Foias equation for evolving measures may be recast as an equation for an evolving density against Lebesgue measure on HN. In the stochastic case this is a second order (nonstandard) PDE whose solution readily gives a solution to the Foias equation using a simple Loeb measure construction. Details may be found in (lo] or [ll]. A further extension of the basic existence theory and techniques developed in [6] gave one of the first solutions to the stochastic Euler equations (that is, equation (2) with v = 0 ) in dimension d = 2 with periodic boundary conditions [13]. It is also shown that the laws of solutions to (2) for 0 < v 5 1 are relatively compact and that for any convergent sequence of laws for solutions with v, 4 0 there is a solution of the stochastic Euler equations with the limiting law. An alternative (but more or less equivalent) approach to solving the sNS equations using Loeb space methods is to apply Keisler’s theory of neocompact sets and rich adapted probability spaces [22,23]. A rich adapted probability space is one that has those features of a Loeb space that are at the heart of existence proofs such as in [6]. The theory captures these key features as intrinsic properties of the space itself, rather than properties that are derived from its construction. A typical existence result in this theory is proved using a property called neocompactness - weaker than classical compactness - to show for example that an intersection of sets of approximate solutions to a stochastic equation (for example Galerkin approximations) is non-empty, and contains a solution. The paper [17] shows how to recast the basic existence proof of [6] in the setting of a rich adapted spaces, and moreover proves the existence of a wide range of optimal solutions to (2) for d 5 4. For example, there is a
dA statistical solution is a time-evolving family of probability measures that solves the so-called Foias equation. This is derived heuristically from the Navier-Stokes equations, and describes the evolution of the probability distribution of a solution to the equations on the assumption of a random initial condition and uniqueness of solutions.
103
solution that minimizes the expected energy integral
E(u) = $lE:J lu(t)12dt and there is a solution that minimizes the expected enstrophy integral
This may well have a bearing on the uniqueness question.
3. A t t r a c t o r s for stochastic Navier-Stokes equations There are several ways to formulate the idea of an attractor for a system of stochastic differential equations - for example by considering measure attractors (see [12,27]),or by working with the notion of stochastic attractor developed by Crauel & Flandoli [19]. A third approach is to extend the approach of Sell [28] that was used for deterministic Navier-Stokes equations to overcome the problem of nonuniqueness. In each case, to avoid unnecessary additional complications, the drift and noise coefficients f , g in (2) are taken to be time-independent, so the equations considered are
3.1. M e a s u r e attractors This approach is currently applicable only to d = 2 since it is necessary that the equation (4) has a unique solution. Thus it is assumed that f , g satisfy an appropriate Lipschitz condition, to ensure that for each initial condition u E H there is a unique solution u(t) = v(t,u)with u(0)= u (so w(0,u) = u).A semigroup St is now defined on Ml(H),the set of Bore1 probability measures on H, by putting Stp = pt where
s,
d(u)dCLt(u) =
s,
IE 29(v(t1U))dP(UZL)
for all bounded weakly continuous functions 0 : H -+ R. An attractor for the dynamical system ( M l ( H ) , St) is called a measure attractor. The existence of measure attractors for the sNS equations was first investigated by Schmallfufi in [27] for example. The paper [12] with Capiriski establishes existence of a measure attractor for (4) under quite general conditions:
104
Theorem 3.1. Suppose that f l g are Lipshitz and satisfy an appropriate growth conditione. Then there is a measure attractor A c Ml(H) for the stochastic Navier-Stokes equations (4). That is (a) A is weakly compact; (b) StA = A for all t ; (c) for each open set 0 2 A , and for each r > 0 StBr
0
for all suficiently large t , where Br = { p E X :
IuI2dp(u) 5 r }
The methods in [12] do not make essential use of Loeb spaces although a t some points they can be employed to assist the construction. 3.2. Stochastic attractors
For a stochastic system such as (4) the idea of a stochastic attractor developed by Crauel & Flandoli [19] takes into account the fact that a t all times new noise is introduced into the evolution of each path of any solution t o (4). A stochastic attractor is defined to be a random set A ( w ) that, a t time 0, attracts trajectories “starting a t -m” (compared to the usual idea of an attractor being a set “at time m” that attracts trajectories starting at time 0). This idea is spelled out below, and involves the introduction of a one parameter group Bt : R + R of measure preserving maps, which should be thought of as a shift of the noise to the left by t. In proving the existence of a stochastic attractor for the system (4) the nonstandard framework makes it particularly easy to consider -co. Making this precise, suppose that cp is a stochastic flow of solutions to (4). That is, cp is a measurable function cp: [O,co) x H x
R +H
such that cp(.,.,w) is continuous for 8.8. w, and for each fixed initial condition uo the process u ( t , w ) = c p ( t , u 0 , W ) is a solution to (4) with u(0,w)= uo. The notion of a semigroup in the usual definition of a deterministic attractor, along with the notion of an attractor itself, is now replaced by the following.
5 c + 61IIuII and Ig(u)IH,H 5 < 2u, where Q is the covariance of the
eFor example, a sufficient condition is that l j ( u ) & I
+
c 621bll for Some &,& > 0 with 261 H-valued Wiener process w.
+ 6;.trQ
105
Definition 3.1. (i) The flow cp is a crude cocycle if for each s E such that for all w E R, 4 s
+ t ,z, w ) = cp(t,cp(% 2 , w ) ,
R+ there is a full set R, Q3w)
holds for each z E H and t E R+. (ii) A cocycle is perfect if R, does not depend on s. (iii) Given a perfect cocycle cp, a global stochastic attractor is a random compact subset A ( w ) of H such that for almost all w
cp(4 A ( w ) ,w ) = A(Qtw), t L 0, lim dist(cp(t, B , Q P t w ) ,A ( w ) ) = 0
t+m
for each bounded set B
c H.
Note that the existence of a perfect cocycle is necessary for the possibility of having a stochastic attractor. Constructing a perfect cocycle is difficult for infinite dimensional systems, particularly for those that are truly stochastic (as compared to random dynamical systems in which paths may be treated individually). 3.2.1. Existence of a stochastic attractor for the Navier-Stokes equations
A stochastic attractor was constructed for the stochastic Navier-Stokes equation with d = 2 by Crauel & Flandoli [19], but their version of (4) reduced to a random equation that could be solved pathwise, giving essentially a pathwise construction of the random attractor A ( w ) . The first example of a stochastic attractor for a truly stochastic version of the NavierStokes equations was constructed in [14] using Loeb space methods, seemingly in an essential way. In the following, for simplicity the Wiener process was taken to be one dimensional. Theorem 3.2. (Capiliski & Cutland[l4]) (a) Suppose that ( g ( u )-g(v), uv) = 0 and (g(u), u ) = O.f With appropriate Lipschitz and growth conditions o n f,g , there i s a n adapted Loeb space carrying a stochastic flow of solutions to the system (4) that is a perfect cocycle, and there i s a stochastic attractor A ( w ) (compact in the strong topology of H) for this system. (b) If g has the additional property that ((g(v),v)) = 0 for ZI E V the stochastic attractor is bounded and weakly compact in V. fFor example g(u) = (h,0 ) u for some h E H.
106
The proof of this result is quite long and complicated, and uses heavily the fact that solutions to (4) may be obtained as standard parts of Galerkin approximations of dimension N , infinite. A delicate extension of the Kolmogorov continuity theorem as adapted to a nonstandard setting by Lindstrom [l]is at the heart of the construction of the perfect cocycle. An outline of the main steps and ideas of the proof is given in Chapter 2 of [16]. 3.3. Process attractors
Sell’s radical approach [28]to the problem of attractors for the deterministic Navier-Stokes equations for d = 3, bearing in mind the possible nonuniqueness of solutions, was to replace the phase space H by a space W of entire solutions to the Navier-Stokes equations. That is, each point in W is the complete trajectory in H of a solution. The semigroup action St on W is simply time translation. That is, if u = u(.) E W then Stu = w E W is given by
w(s) = u(t
+ s).
Clearly this is well defined, and has the crucial semi-flow property
st, st, = Stlft2 0
along with Sou = u. Using this idea, Sell was able to establish the existence of a global attractor for the 3-dimensional (deterministic) Navier-Stokes gquations. For the 3-dimensional stochastic case, Sell’s idea was used by Flandoli & Schmalfufi in the paper [20] for the Navier-Stokes equations with a special form of multiplicative noise, using a mild solution concept. The equation considered allowed essentially a pathwise solution, and then a random attractor was obtained by combining Sell’s approach with the idea of pulling back in time to -00, as developed by Crauel & Flandoli [19]. In a later paper [21] Flandoli & Schmalfufi consider in the same framework the Navier-Stokes equations with an irregular forcing term, but no feedback. In the paper [18] with HJ Keisler we consider 3-d stochastic NavierStokes equations with a general multiplicative noise g ( u ) as in equation (4) above. The idea is to use Sell’s approach at the level of processes rather than paths. In this way the idea of an attractor is formulated in the conventional sense, examining the long term behaviour of solutions as t 4 m. To do this, it is necessary to have a single underlying probability space, rich enough t o carry a supply of solutions to the 3-d stochastic Navier-Stokes equations
107
that is sufficient for the concepts to make sense. For this an adapted Loeb space is needed. A precise formulation of the notion of a process attractor and the main result of [18]is as follows. On an arbitrary space R carrying a 1-dimensional Wiener process (wt)t20 suppose that a class X of solutions to the sNS equations (4)is defined. Suppose further that R is equipped with a family of measure preserving maps Ot : R -+ R for t 2 0 with the following properties:
(el) B0 =identity and 8, 0 BS = 8t+s; (82) 8 t 3 s = Ft+s for all s , t 2 0, where ( F t ) is the filtration on R; (83) w ( t s, &w) - w ( t , Otw) = W ( S , w) for all s 2 0.
+
Note that the property (83) tells us that for a fixed t the increments of the process w ( t + s , 8,w) are the same as those of the process w ( s , w ) . Thus Ot can be thought of as a shift of the noise to the right by t. The family (8,) allows the following definition of a semiflow S, of stochastic processes. Definition 3.2. (Semiflow of Processes) Suppose that u = u ( t , w ) is a stochastic process defined for t > 0. Then for any r 2 0 the process u = S,u is defined by
v(t, w)= u ( r
+ t ,8,w)
It is clear that S, is a semigroup, and if u is adapted so is Stu. Suppose now that X is closed under St. Then a process attractor for the class X can now be defined. In the following, if u is a stochastic process then Law(u) is defined to be the probability law (on path space) of the coupled process ( u ,w ) . Definition 3.3. (a) A set of laws A
c Law(X) is a Law-attractor if
(i) (Invariance) & A = A for all t 2 0, where St is the mapping of laws induced by the semigroup St. (ii) ( A t t r a c t i o n ) For any open set 0 2 A and bounded 2 c Law(X), $2
c0
eventually (i.e. this holds for all t (iii) ( C o m p a c t n e s s ) A is compactg
2 to(O,2)).
108
(b) A (process) attractor for the semiflow St on X is a set of processes A 5 X such that (i) Law(A) is a Law-attractor (in particular Law(A) is compact and so A is bounded); (ii) (Invariance) StA = A for all t 2 0; (iii) (Attraction) For any bounded set Z c X and compact set K
limt+,d(StZ,
K ) 2 d(A, K )
(iv) A is closedh. Remarks on Definition 3.3. (1) Since existence results for the stochastic Navier-Stokes equations require a rather large probability space, it is to be expected that any space carrying a whole class of solutions X as above will be too big to allow an attractor A c X that is compact in the usual sense. However, the attractor A of the following theorem is neo-compact, the key notion developed in [23] It is a consequence of neo-compactness that Law(A) is compact. (2) The attraction property 3.3(b)(iii) is equivalent to the following:
stz 5 0
(5)
eventually for any bounded Z and any open 0 3 A of the form 0 = L2(R,M)\KsE, with K compact. Property 3.3(b)(i) means that in addition (5) holds eventually for any open set 0 of the form 0 = LawP1(0’) where 0’ is an open set of laws with Law(A) 2 0’.The usual attraction property for attractors, namely that StZ C 0 eventually for any bounded 2 and any open 0 2 A is probably too much to expect. However, the attractor in the following theorem has property (5) for a smaller class of open sets namely those that are neo-open, a further key notion of [23]. Sets 0 of the form L2(R,M ) \ K s Eor Law-l(O’) as above are neo-open. We can now state the main theorem of [18]. Theorem 3.3. There is a Loeb space R (which carries solutions to the stochastic Navier-Stokes equations for all L2 30-measurable initial conditions) with a process attractor A for the class of solutions X described below. where do is the Prohorov metric and pi (i = 1,2) is the projection of X i onto the first coordinate- that is, path space for the solutions of (4). hHere and in (iii) the topology is the L2 norm topology on processes in H given by lu12 = IE lu(t)12exp(-t)dt.
som
109
The class of solutions in the following definition depends on the constants k ~k, ~k3, , a , ,B, a , b. In the proof of Theorem 3.3 in [18] an explicit choice of these is identified that ensures that X # 8.The condition (X5) is the only one that needs explanation - see the remarks below.
Definition 3.4. (i) Denote by X the class of adapted stochastic processes u : [0, 00) x R -+ H with the following properties.
(Xl) For a.a. w the path u(., w ) belongs to the following spaces:
L:~(O,00; H) n L?~,[O, 00; H) n &(o, (X2) For all t l
00;
V)n ~ ( 000;,
2 to > 0
u(to)+lo tl
4tl) =
Hweak)
[--vAu(t)-B(u(t))+f(u(t))ldt+
(X3) For a.a. to > 0 and all tl
9(u(t))dwt
2 to,
IE(Iu(t1)12)I ~ ( I u ( t o ) Iexp(-kl(tl ~)
(X4) For a.a. t o > 0 and all tl
1:
- t o ) ) + k2
2t o ,
(X5) For a.a. to > 0 and all t~ 2 to, for all n 2 1
(X6) IE
J Iu(t)12dt i < co
(ii) Denote by
xk
the set of u E
x with
(X6k) E J i Iu(t)I2dt i k Remarks 1. The above conditions tell us nothing about u ( t , w ) at t = 0 and there may be a singularity there. In this sense the class X is a class of generalized weak solutions to the stochastic Navier-Stokes equations (cf. [28] p.12). 2. It follows from (X6) that IE(Iu(t))I2)< 00 for 8.8. t E ( 0 , l ) . Thus, from (X3) we see that IE(Iu(t))I2) is bounded on m) for all n. 3. In condition (X5), the function cpn(u)is an explicit smooth approximation to the function \~1~1{1..12,).The inequalities (X5) follow heuristically from the equation (4) as a particular instance of the Foias equation corresponding to (4). The choice of the functions qn makes (X5) a kind
[A,
110
of uniform integrability condition for the random variables lu(t,u)lz for
t
E
Ito,m).
The proof of Theorem 3.3 proceeds as follows. First show that X # 0 by the construction outlined in Section 2.2. The heuristic argument for the inequalities (X5) can be made precise for the approximate solution U living in HN and it is this that gives (X5) for the solution u = " U . The other properties in the definition of X follow naturally. Next it is necessary to define an internal ("nonstandard") set of approximate solutions X t o (4) that is wider than the Galerkin approximations on HN: X includes processes U where the equality (X2) is replaced by an infinitesimal approximation. Then it is shown that X is precisely the set of processes u such that u = "U for some process U E X that is nearstandard as a process: in symbols
X
= " ( X nNS)
Finally, after defining a semigroup operation T, on X corresponding to S,, the set
c = 0Tnxk nEN
is defined for a certain k (for which Xk is absorbing). It is easily proved that T,C = C for finite times T and that C attracts bounded sets in X - so C is a nonstandard attractor. The key now is that C is non-empty (this follows from a kind of compactness property of Loeb spaces) and also that C c N S . In consequence the set A = "C is nonempty and in fact neocompact. The properties required for A to be an attractor follow from the corresponding properties of the nonstandard attractor C. In the final part of [18] the class X of two-sided solutions to (4) is discussed. It is shown that # 0, and the attractor A is simply the restriction of solutions in X to the nonnegative time interval [0, co[.
x
Appendix Here we give a concise but mathematically complete construction of the Loeb space used in the paper [18] and discussed in the previous section. This is to take some of the mystery away from the notion of a Loeb space, and to show that its construction is entirely algebraic. What we are not able to do here is to expand on the properties of Loeb spaces that make them so useful. For this see any of the introductions such as [1,11,16].
111
A . l . The hyperreals To define the extension of the reals known as the hyperreals *R first fix a nonprincapal ultrafilter U on N. That is, U is a collection of subsets of N that is closed under intersections and supersets, does not contain any finite sets, and is maximal with this property. This means that for every set E C N either E E U or N \ E E U (but not both). The hyperreals *Rare defined by
*R= RN/U meaning the quotient of RN by the equivalence relation ( ~ i )-u
{ i : ai = bi} E
(bi)
U
We say that ai = bi a.c.' Write [(ai)]for the equivalence class (ai)/U and identify r E R with the constant sequence [ ( r ) ]so , that *R2 R. Operations of addition and multiplication are defined on *R pointwise, and it is easy to check that this makes *Ra field. A hyperreal a = [(ai)] is said to be finite if there is n E N with (a1 5 n, which means that Jail _< n a.c. The standard part " a E R of a finite hyperreal is now defined byJ
" a = inf{r E R : ai 5 r a.c.} It is easy to check that " ( a
+ b ) = ' a + " b and the same for products.
A . 2 . Construction of a Loeb space Let W be two-sided Wiener measure on Co(R) = {x : The set R is defined by
R
R
4
R;x ( 0 ) = 0).
= CO(R)"/IA
just like *R (so we could write R = *Co(R)). An algebra 6 of subsets of R is given by sets of the form
A where Ai
= r I i e ~ A i / U=
[(Ai)]
C: Co(R). That is, for x = [(xi)]E R we define xEA
@
xi E Ai a.c
'A property Pi is said to hold a.c. if {i : Pi holds } 6 U . jIn the terminology of the subject, the standard part O a is the unique real number that is infinitely close to a , written ' a N a
112
It is easily checked that 4 is indeed an algebra, and in fact the operations n, U, \ are given pointwise.k A finitely additive probability measure POis now defined on Q by
P o ( 4 = "[(W(Ai))l for A = [(Ai)] E 4. Checking that POis finitely additive is straightforward.' The Loeb measure P on SZ is now the unique a-additive extension of PO given by Loeb's fundamental result [26] which in this context is the following. Theorem A . l . (a) If (A,) is a sequence of sets from Q with A, = 0 then there i s a m E N with A, = 0 . (b) Hence, by Carathe'odory's Extension Theorem, there is a unique uadditive extension P of Po to the a-algebra a(G). Proof (a) Without loss of generality we may assume that the sets A, are decreasing. Suppose for a contradiction that A, # 0 for each n. Then we have for each n
nnEN
On<,,
0 # Anfl
c A,
which means that if A, = [(A,,i)]we have 0 # A,+l,i C A,,i a.c. For n = 1 , 2 , 3 , .. . in turn, systematically modify A,,i on a smallm set of indices i , so that for each n
0 # A,+I,, 5 A,,%
for all i E N
This does not alter the sets A, themselves. Now pick xi E Ai,i for each i and note that z i E A,,i for n 5 i. Consider the element x = [ ( x i ) ]Then . xE A, because for each n
nnEN
{i : xi E A,,i} 2 {i : i 2 n } E ZA. Thus A # 0 , the required contradiction. (b) Carathkodory's extension theorem shows that the finitely additive probability POon the algebra 4 extends uniquely to a a-additive probability on a(G) provided that whenever n n E N A n= 0 for a decreasing sequence of sets from G then Po(A,) 4 0 with n. In our case this follows trivially from (a). The above construction gives a probability space
( 0 , 4 4 ) P, ) kFor example, [ ( A i )U ] [ ( B i )= ] [(Ai U B i ) ] . ' I n fact for disjoint A , B we have Po(A U B ) = Po([(Ai)] U [ ( B i ) ]= ) Po([(AiU Bi)]) = O[(W(AiU B i ) ) ]= " [ ( W ( A i ) W ( B i ) )= ] O [ ( W ( A i ) ) ]O[W((Bi))] = Po(A) Po(B). mThat is, a set not in the ultrafilter U .
+
+
+
113
The Loeb space resulting from the above construction is now the completion of this space with respect t o the measure P (that is, adding in the P-null sets) which is still denoted P , giving the space (0,F,P ) say. The a-algebra F is a Loeb algebra, and t o indicate its origin i t is often denoted F = L(G). Similarly we often write P = Q L , the Loeb measure constructed from Q, where Q is the *R-valued function defined on 4 by
Q(A)= [(W(Ai))], so that Po = " Q . The key to the use of Loeb spaces hinges on two main facts. T h e first is due t o Loeb [26] and shows t h a t L(G) = G modulo null sets: for any B E L ( 4 ) there is A E 6 with P ( B A A )= 0. T h e second is t h a t the sets in 4 and their measures inherit (in a way made precise by the Transfer Principle) the properties of the measurable subsets of Co(R) and their Wiener measure. This makes the algebra 6 tractable, as expounded in any of the references cited above.
References [l].S.Albeverio, J.-E.Fenstad, R.H@egh-Krohn,and T.Lindstr@m,Nonstandard
Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York 1986. [2]. Z.Brzeiniak, M.Capinski, and F.Flandoli, Stochastic Navier-Stokes equations with multiplicative noise, Stochastic Analysis and Applications 105 (1992), 523-532. [3]. A.Bensoussan and R.Temam, Equations stochastiques du type NavierStokes, J. Functional Analysis 13 (1973), 195-222. [4]. A. Bensoussan, A model of stochastic differential equation in Hilbert space applicable to Navier-Stokes equation in dimension 2, in: Stochastic Analysis, Liber Amicorum for Moshe Zakai, eds. E.Mayer-Wolf, E.Merzbach, and A.Schwartz, Academic Press 1991, pp.51-73. [5]. A. Bensoussan, Stochastic Navier-Stokes equations, Acta Applicanda Mathematicae 38 (1995), 267-304. [6]. M.Capiriski and N.J.Cutland, Stochastic Navier-Stokes equations, Acta Applicanda Mathematicae 25 (1991), 59-85. [7]. M.Capifiski and N.J.Cutland, A simple proof of existence of weak and statistical solutions of Navier-Stokes equations, Proceedings of the Royal Society, London, Ser.A, 436 (1992), 1-11. 181. M.Capiriski and N.J.Cutland, Navier-Stokes equations with multiplicative noise, Nonlinearity 6 (1993), 71-77. [9]. M.Capifiski and N.J.Cutland, Foias and Hopf statistical solutions of NavierStokes equations, Stochastics and Stochastic Reports 5 2 ( 1995), 193-205. [lo]. M.Capihski and N.J.Cutland, Statistical solutions of stochastic NavierStokes equations, Indiana University Mathematics Journal 43(1994), 927940. [ll]. M.Capiriski & N.J.Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore, London, 1995.
114
[la]. M. Capiriski & N.J. Cutland, Measure attractors for stochastic NavierStokes equations, Electronic J. Prob. 3(1998), Paper 8, 1-15. [13]. M. Capinski & N.J. Cutland, Stochastic Euler equations on the torus, The Annals of Applied Probability, 9(1999), 688-705. [14]. M. Capiriski & N.J. Cutland, Existence of global stochastic flow and atractors for Navier-Stokes equations, Probability Theory & Related Fields 115(1999), 121-151. [15]. M.Capinski and D.Gqtarek, Stochastic equations in Hilbert space with application to Navier-Stokes equation in any dimension, Journal of Functional Analysis 126(1994), 26-35. [16]. N.J. Cutland, Loeb Measures in Practice - Recent Advances, Springer Lecture Notes in Mathematics 1751(2000), Springer, Berlin. [17]. N.J. Cutland & H.J. Keisler, Neocompact sets and stochastic Navier-Stokes equations, in Stochastic Partial Differential Equations, (Ed. A. Etheridge), LMS Lecture Notes Series 216, CUP, 1995, 31-54. [18]. N.J. Cutland & H.J. Keisler, Global attractors for 3-dimensional stochastic Navier-Stokes equations, in preparation. [19]. H.Craue1 and F.Flandoli, Attractors for random dynamical systems, Probability Theory and Related Fields 100 (1994), 365-393. [20]. F. Flandoli and B. Schmalfui.3, Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative white noise, Stochastic & Stochastics Reports, 59(1996), 21-45. [21]. F. Flandoli and B. Schmalfui.3, Weak solutions and attractors for threedimensioanl Navier-Stokes equations with nonregular force, J . Dynamics and Differential Equations, 11(1999), 355-398. [22]. S. Fajardo and H.J. Keisler, Existence theorems in probability theory, Advances in Mathematics, 120(1996), 191-257. [23]. S. Fajardo and H. J.Keisler, Neometric spaces, Advances in Mathematics, 118(1996), 134-175 [24]. F. Flandoli and D. Gatqrek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probability Theory 63 Related Fields 102(1995), 367-391. [25]. AJchikawa, Stability of semilinear stochastic evolution equations, Journal of Mathematical Analysis and Applications 90 (1982), 12-44. [26]. P.A.Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. SOC.211(1975), 113122. [27]. B. Schmallfui.3, Measure attractors of the stochastic Navier-Stokes equation, Bremen Report No.258, 1991. 128). G.R. Sell, Global attractors for the three-dimensional Navier-Stokes equations, J. Dynamics and Diff. Equations 8 ( 1996), 1-33. [29]. M.Viot, Solutions Faibles d 'Equations aux Derivees Partielles non Lineaires, Thesis, Universite Paris VI (1976). [30]. M.I.Vishik and A.V.Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht - London 1988.
THE 2D-NAVIER-STOKES EQUATIONS PERTURBED BY A DELTA CORRELATED NOISE.
A. DEBUSSCHE
ENS d e Cachan, a n t e n n e d e Bretayne Campus de K e r L a n n 351 70 Bruz cedex France E-mail: arnaud. debussche@bretayne. ens-cachan.fr We study the two-dimensional Navier-Stokes equations with periodic boundary conditions perturbed by a space-time white noise. We prove that, for almost every initial data with respect t o a measure supported by spaces with negative regularity, there exists a unique global solution in the strong probabilisticsense. The nonlinear term is defined thanks t o techniques borrowed from Wick renormalisation and paraproducts in Besov spaces. Note however that no renormalisation is made here and the nonlinear term is not modified. This result was given in g , here we give simplified proofs. Then we prove ergodicity of the Gaussian invariant measure.
1. Introduction
We consider the two dimensional incompressible Navies-Stokes equations in a periodic domain driven by a space time white noise:
{
du = (vAu - (u. V ) u- V p ) d t div u = 0, in [0,T ]x 0 , 4 0 , t) = uo(t), 5' E 0, u is periodic with period 27r,
+ dW, in [0,TI x 0 ,
(1)
where 0 = [0,27rI2. The unknown are random processes: the velocity field u(t,6) = ( u l ( t ,<), uz(t,6)) and the pressure field p ( t , 6); these are defined for (5'1, (2) E 0 and t 2 0. The kinematic viscosity v has no importance in this work and we will take it equal to 1. The equations are forced by a space time white noise. It is delta correlated in time and in space, i.e. we formally have
It is the time derivative of a cylindrical Wiener process associated to a stochastic basis (Q3, P,(.Tt)tzo) (see 12). 115
@ on
(L2(0))2
116
Stochastic Navier-Stokes equations have been investigated in many articles In most cases, the noisy forcing term is white in time and correlated in space. Recently much progress has been obtained in the study of the associated invariant measures for noises which are very smooth in space. Uniqueness and ergodicity properties have been proved Also in 18, the singularities of the solutions in the three dimensional case are studied. In the work 16, a space-time white noise is also considered and equation (1) has been studied through the associated Kolmogorov equation. They prove directly the existence of a solution to this latter equation but are unable t o connect it t o the original equation. The main difficulty is that, as is well known, with such a rough noise it is expected that a solution of (1) is not regular. In this work, we first observe that, using ideas borrowed from the theory of the Wick product the nonlinear term can be defined for random variables whose law is absolutely continuous with respect t o a certain Gaussian measure. Note however that we do not use renomalisation here. Then, we split the unknown into a part whose law satisfies this property and a smoother part. Using the bilinearity of the nonlinear term and using product rules in Besov spaces, we show that the nonlinear term can be defined for a sufficiently large class of random variables which contains a solution of the equation. Local existence is proved by a fixed point argument and an a priori estimate is proved to get global existence. This a priori estimate is based on the fact that we explictly know an invariant measure for equation (l),it is a Gaussian measure. Note that the idea to use an invariant measure is used by J. Bourgain in the context of the deterministic nonlinear Schrodinger equation. (See and the references therein). Finally, we prove that this invariant measure is ergodic. A space time white noise might not be relevant for the study of turbulence where it is usually accepted that a spacially correlated noise should be taken into account. However, in other circumstances, when a flow is subjected t o an external forcing with very small time and space correlation length, a space-time white noise can be considered. 1,2,516114115122.
4117,20,21.
11,22124,
2. Notations
We introduce standard notations used for the Navier-Stokes equations (see for instance 2 5 ) . The subspace of ( L 2 ( 0 ) ) consisting 2 of periodic divergence
117
free functions with zero average is denoted by H :
z2(51,0) = X2(E1,2.rr)
Zl(O1 E 2 ) = ZlP.rr, E2)r
1 1
and P is the orthogonal projection onto H . The inner product of H is the same as in ( L 2 ( 0 ) ) 2and is denoted by (., .). It is convenient to use the complexification Hc of the space H . For k = ( k l , k 2 ) E Zi := Z2\{0, 0}, we write
kL = ( k 2 , k . E = klEl
lkl
+ k2E2,
=
(k? + k 22 )112
ek(E) =
1
' k ik.E - -e E 27T
IN
7
=
( E l , E2)
E 0.
Then (ek)kcq (resp. (Re(ek))kEZ;)is a complete orthonormal system of Hc (resp. H ) . We also use the space (R)";:= 7-t. We shall consider H as a subset of
x. The unbounded operator A is defined by AX
=PAX,
z E
D ( A ) = ( H ; ( O ) ) n~ H .
We have Aek = -Ikl2ek, k E Z,.2 Here and in the following H&(O) is the subspace of the Sobolev space H T ( 0 )consisting of all periodic functions. For T E R, we use the fractional power ( - A ) T on the domain D ( ( - A ) T ) .It is classical that D ( ( - A ) T ) is the closure in ( H 2 ' ( 0 ) ) 2of the space spanned by (ek)&Z,2. Moreover .I is a norm on D ( ( - A ) T ) equivalent to the usual norm on (H2'(0))2. For any T E R, P can be defined on ( H z ( 0 ) ) 2and its image is D ( ( - A ) T ) . We set
W
=PW.
(2)
It is not difficult to see that W is a cylindrical Wiener process on H thus, for any complete orthonormal system (ek)kcz; in H , we can write
W=
Pkek k€Z;
118
where (,&)kEZ; is a sequence of independent Brownian motions on the stochastic basis (Q, F ,PI (Ft)t?o). As is well known, thanks to the incompressibility condition, we can rewrite the nonlinear term as
(u. 0 ) u = div (u@ u), where
We will use this form which is better suited to the case of non smooth velocities. Whenever it makes sense, we set
b(z,y)
=P
div (z 18 y), b(z)= b(z,z).
(3)
When projecting equations (1) on HI we get
{
+
+
du = (Au b(u))dt dW, (4)
u(0)= uo.
We wish to solve (4) and t o find a solution which is a D((-A)') valued process. Implicitly this means that we restrict our attention t o zero average initial data. This is no loss of generality since we can change the unknown in (1) and consider only such initial data.
3. Preliminaries It is not expected that (4)has a solution in D ( ( - A ) T )for not even true for the linear equation
dz
= Azdt
T
2
0. This is
+ dW, (5)
4 0 ) = 201 whose solution is given by
~ ( t=)etAzo +
hrn
1 t
e(t-s)AdW(s).
The second in the right hand side is a continuous process with values in D ( ( - A ) T )for any T < 0 but does not take its values in D((-A)') for any T 2 0. This follows from
119
for any r
< a < 0 and rt
for r 2 0, see 12. We have denoted by L H S(K1, K2) the space of all HilbertSchmidt operators from a Hilbert space K1 on a Hilbert space K2. It follows that we have to work with non smooth processes and this creates difficulties when working with the nonlinear term. Here we proceed as is usual when dealing with parabolic equations in negative Sobolev spaces. We use Littlewood-Paley decomposition and paraproduct to define the nonlinear terms, see However, working in the context of negative Sobolev spaces introduces some technical difficulties and i t is convenient t o consider Besov spaces. We define, for N E N,PN as the orthogonal projector in Hc onto Span (ek)lklSN, PN is also orthogonal in D ( ( - A ) T ) ,r E R,and it can be easily extended to 3-1. We also set, for q E N,6, = P,,- P2,--1. Then 6,u i s defined for all u E ‘H and contains the Fourier components of u between 2q-l and 24 : 718123.
For cr E R , p 2 1, p
2 1 we define
it is a Banach space with the norm 1lP
Iulr3;,p = (~2~q~16qul;.;.))
The following result is crucial in our argument and is the main motivation for working in Besov spaces, see 7,8.
+
Proposition 3.1. Let p , p 2 1, cy p > 0 , a < 2 / p , p < 2 / p . Then zf 2 u E B& and v E i3t,,p we have uv E Bz,P where y = a P - p, and 14B;,p
i C14Bpqpl~lBg,p.
+
(6)
Let us also recall that the nonlinear term verifies the following identities,
see
1i26
(~(x),= x )0, ( b ( z ) ,A X )= 0.
(7)
120
These are true for any x such that the quantities on the left hand side make sense. Let us denote by p the product measure on 'H, P=
I-I"0,
1/(2142)).
kEZ:
We write p = N(0,Q). Notice that p ( D ( ( - A ) ' ) ) = 1 if and only T < 0, so the support of p is included in D((-A)'). This follows from the fact that (-A)-l+" = Q(-A)z' is trace class if and only if T < 0. Also, it is not difficult t o prove that p(f?&) = 1 for any (T < 0, p , p 2 1. This can be done using similar ideas as in lo. Moreover, it is well known that in the case of periodic boundary conditions considered here, thanks t o (7), the measure 1-1 is formally invariant for equation (4). We use techniques borrowed from the theory of Wick renormalized product to extend the definition of the nonlinear term. We shall denote by H,, n = 0 , 1 , ... the Hermite polynomials defined by the formula
It is convenient here to work on the space Hc as well as in the complexification of D((-A)') which for simplicity is still denoted by D((-A)'). For x E 'H we write X N = PNX =
C (x,el)et,
1 2 XN = (zN,zN),
IlllN
and
b N ( x )= b ( P N z ) . We also define for x E If, and N E
where
and
N:
121
As easily checked we have : (XZ,), : (I$) = ( X p ( I $ ) - p$, 2 = 1 , 2 ,
< E 0 , N E N.
We will see that : X N @ X N : converges in some sense, the limit can be defined as a renormalized tensor product. A key observation is that
bN (x)= b ( X N ) = P div
(XN@XN)
= P div (: X N €9
XN
:).
Thus b ( x N ) converges without any renormalization and the limit is a natural definition of b(z). More precisely, we have
Lemma 3.1. For any o < 0 , p 2 1, p 2 1, k 2 1, the sequences (: ) N E W , (: (x$), : ) N E W , ( X ~ X $ ) N E M are Cauchy in Lk(('FI, p; B&).
(xL),:
This result is proved in the context of Sobolev spaces in ', and for the classical Wick product in Besov spaces in lo. Using the techniques developed there, it is not difficult to prove this lemma. Using the continuity properties of P , we deduce Proposition 3.2. For any 0 < 0 , p 2 1, p ( b N ) N E n is convergent in L'"('FI, p; B:,;~).
2
1, k
2 1, the sequence
Corollary 3.1. Let X be a random variable with a law vx which is absolutely continuous with respect to p and such that E L'(('FI;p),1 > 1, then the sequence ( b N ( X ) ) N E Nis convergent in Lk((R;BE,;') f o r any 0 < 0, p 2 1, p 2 1, and k 2 1. W e denote by b ( X ) its limit.
%
Proof : It suffices to write
+
with = 1. So that the result follows from Proposition 3.2. Let us now set t
z(t)=
.I,
e ( t - S ) A d W ( s ) , t E R,
which is the stationary solution of
dz
= Azdt
+dW(t)
with invariant law C ( z ( t ) ) = p.
Lemma 3.2. For any
0
< 0 , p 2 p 2 2, we have z E C(R;
f?g,p), Pa.s.
(8)
122
Proof : It is not difficult to prove that for any S < 0 the trajectories of (-A)&. are continuous with respect to t , II: on R x 0. This uses for instance the Kolmogorov criterion of continuity, see 12. Since for a n y p 2 1, (C(a)>'c 13:,m, it follows that z has trajectories in C(R; a;,,) for any 0 < 0. Moreover, it is easy to see that z has trajectories D((-A)"/')) = C([O,T];Bz,,), < 0, so that by interpolation in C([O,T];
z E C([O,T]; BE,,),0 < 0,p 2 p 2 2, P - a s . .
0
By Corollary 3.1, b(a(t)) can be defined for each t E R so that we have a well defined process (b(z(t))tEw.
Lemma 3.3. For any T
2 0 , lc 2 1, p, p 2 1 and
0
< 0 , we have
b ( z ) E Lk(R x [0, TI; Proof : We have by Fubini theorem, since L ( z ( t ) )= p for any t E R,
and this is a finite quantity by Proposition 3.2. 0 We now want to extend the definition of b in a suitable way so that b(u) makes sense for a solution of (4). The idea is that if we define v = u - z then v is expected to be smoother than both u and z . The following result states that, if this is the case, b(u) can be defined in a nonambiguous way.
Proposition 3.3. Let X and 2 be random variables such that tke law of Z is p and Y = X - Z E Lb(R;a),: where
2 b>2, ->a>O, P
then the sequence of random variables ( b N ( X ) ) N E N converges in L ~ / ~ (B:R,,) ; for any a < a - 1 - ;.2
If moreover, the law of X , ux, is absolutely continuous with respect to % E L'(7-i;p ) with 1 > 1 then the limit coincides with b ( X ) defined
p and
in Corollary 3.1 and
+
+
b ( X ) = b(Y) 2 b ( X , Y ) b ( 2 ) .
(9)
5.
Proof : Let a < a - 1 - 2 and set (T = a - a - 1 Clearly, 0 < 0 P and 2 E Lb(R;a,",,).Thanks t o Proposition 3.1, b(Y,2 ) is well defined in Lb12((R;a:,,) and
bN(y, Z ) -+ b ( ~Y), , in L ~ / ~ ( (B;,,). R;
123
Similarly, since a
> u , B;,, c t3&
P(Y)
--t
and
b ( ~ ) in, ~ ~ ' ~B;,,). ((0;
We have :
b y x ) =byY)
+ 2 b N ( X , Y )+ b N ( 2 ) .
Using Corollary 3.1 the last term of the right hand side also converges. We deduce that the left hand side converges. The last statement is clear. 0
Remark 3.1. It follows that b ( X ) can be defined whenever the hypotheses of Corollary 3.1 or of Proposition 3.3 are satisfied. Moreover, if instead of the assumptions of Proposition 3.3 we only have X - 2 E Pas., by a standard localization argument, we easily deduce that ( b N ( X ) ) ~ E ~ converges almost surely in a,; .
a;,,
4. Existence and uniqueness The main result of this work is the following.
Theorem 4.1. Let
CT
< 0, p 2 p 2 2, ,O 2 1, and a > 0 such that
2 1 1 a l u < a < -, and - - - < - - - < -, P P 2 2 P 2 then for any T 2 0 there exists a unique mild solution u to -g
(4) such that
u - z E C([O,TI;B;,,) n La(O,T ;B;,,).
Moreover, for any 1 E N,
Remark 4.1. By Remark 3.1, the solution has the required properties to ensure that b(u) is well defined. Indeed, u ( t )- z ( t ) E B;,, P a s . w E 0 for for almost every t E [O, TI. Remark 4.2. Note the condition - $ < 5 implies that p > 2. This is the reason for working in Besov spaces. Recall that Sobolev spaces correspond to Besov spaces with p = 2. Proof : We split the proof in two parts. We first prove local existence on a random time interval depending on the initial data. Then an a priori estimate enables us to get global solutions. First step : Let uo E B;,,. We fix w and solve (4) pathwise, w is taken in a set of probability one such that the various properties on z and b ( z ) proved above are true.
124
Let us fix
and consider the mapping defined on
Thanks to Propossition 3.1 we know that
we deduce that
since, by assumption Similarly, we have
since, by assumption
S9imilarly, we have
Furthermore,
125
so that, choosing k sufficiently large and using Lemma 3.3, we obtain
1 1
eA(t-s)b(z(s))dslETo
Finally, it is clear that t
o".(I
H
5
e"(u0
- Z(0))lETo
C(Q1
-
PI
T)lb(z)Ilk(o,T;a~,,')'
z ( 0 ) )is in E T ~and
1.4%
(TIPI
P,T)bo - "(0)lJ3,.,,.
This shows that 7 maps ET,, into itself. It is standard to deduce from the above estimates that there exists T*(IUO - "(O)lB,.,,I Ib(z)la;,1, Izlc([o,T];a,.,,)) > 0 and " 0 z(O)la;,,, I ~ ( Z ) I ~ ; , ~ I , IzIc(p,q;~;,,))> 0 such that, for TO5 T * ,7 is a strict contraction on the ball of center 0 and radius R in E T ~ We . deduce that there exists a unique solution on [O, T*]. Second step : It is clear that it is sufficient to obtain an a priori estimate in f3& in order to have global existence. We will in fact prove that, if u(t,uo)is a solution to (4) r
This implies that for IJ. almost every uo we have
thus the local in time construction of the first step can be iterated leading to a global solution. Thus Theorem 4.1 is proved. Let us prove that (11) holds. We use a formal argument which can be easily justified by a Galerkin approximation. Indeed, it is not difficult to prove that the local solution constructed above is the limit of Galerkin solutions. We have u(tl uo)= e"(u0 - z ( 0 ) )
+
1"
e(t-S)Ab(u(s, u0))ds
therefore
lu(k ~o)If?;,,
5 C ( P , PI ~)(IuOlB,.,,
+ Izola;,,)
+z(t),
126
We have, by Holder inequality in time and then in the expectation,
Then, integrating with respect to p, using again Holder inequality and the invariance of p, we obtain
+C(P, p, a)W2(Jx/b(uo)l;;,;l)~P(~O))
1/3
.
By Lemma 3.2 and Proposition 3.2, we know that the right hand side is finite. This proves our claim (11). The last statement (10) is proved in the same way. 5 . Ergodicity
As already mentionned, using a galerkin approximation, it is not difficult to prove that the Gaussian measure p is invariant for (4). We now study some of its properties. Given a functional cp defined on 'Ti, we denote by (p its average with respect to p r
We have the following result of exponential convergence which clearly implies ergodicity.
127
Theorem 5.1. There exists a constant X > 0 such that
for any p E L2('FI,p) and t
> 0.
Proof : Again the proof is formal and could be justified by approximation. Replacing 'p by p - p, we can assume that (p = 0. Let U ( t ,u g ) = E ( p ( u ( t ,ug))),then U is formally a solution to the Kolmogorov equation (see 13) d t - 1TrD2U 2
{ U(0,
+ (Au + b(u),D U ) ,
uo) = 4 . u . o ) .
Recall that thanks to the invariance of p , we have
Jx ( ; T r D 2 U ( 4 + (Au + b(u),W 4 ) U) ( U ) d P ( U ) =
-;J,
IDU(u)l2dp(u).
Therefore
I t is well known that the Gaussian measure satisfies the spectral gap inequality
for any $ E W1i2(7d,p ) , with X > 0. I t follows
Hence, the result follows by integration.
0
References 1. Albeverio S., Cruzeiro A. B. (1990) Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two dimensional fluids, Commun. Math.
Phys. 129,431-444. 2. Bensoussan A., Temam R. (1973) kquations stochastiques du type NavierStokes, J. Funct. Anal., 13,195-222. 3. Bourgain J. (1999) Nonlinear Schrodinger equations, in Hyperbolic equations and frequency interactions, Providence, RI, edited by Caffarelli et a]., AMS. Park City Math. Ser. 5,3-157. 4. Bricmont J., Kupiainen A., Lefevere R. (2000) Exponential mixing f o r the 2D Nauier-Stokes dynamics, Preprint.
128
5. Brzezniak Z., Capinski M., Flandoli F. (1992) Stochastic Navier-Stokes equations with multiplicative noise, Stoch. Anal. Appl., 10, 523-532. 6. Capinski M., Gatarek D. (1994) Stochastic equations i n Halbert spaces with applications to Navier-Stokes equations in any dimension, J. Funct. Anal., 126, 26-35. 7. Chemin J.-Y. (1995) FLUIDES PARFAITS INCOMPRESSIBLES, Astbrisque, 230. 8. Chemin J.-Y. (1996) About Navier-Stokes system, Prepublication du Laboratoire d’Analyse Numbrique de l’Universit6 Paris 6, R96023. 9. Da Prato G., Debussche A. 2D-Navier-Stokes equations driven b y a space-time white noise, J. Funct. Anal., to appear. 10. Da Prato G., Debussche A. Strong solutions t o the stochastic quantization equations, Annals of Prob., t o appear. 11. Da Prato G., Tubaro L. (1996) Introduction to Stochastic Quantization, Pubblicazione del Dipartimento di Matematica dell’Universit8 di Trento, UTM 505. 12. Da Prato G., Zabczyck J. (1992) STOCHASTIC EQUATIONS IN INFINITE DIMENSIONS. Encyclopedia of Mathematics and its Applications, Cambridge University Press. 13. Da Prato G., Zabczyck J. (2002) S E COND ORDER PARTIAL DIFFERENTIAL EQUATIONS IN HILBERT SPACES, London Mathematical Society, Lecture Note Series 293, Cambridge University Press. 14. Flandoli F. (1994), Dissipativity and invariant measures for stochastic Navier-Stokes equations, Nonlin. Diff. Eq. and Appl., 1, 403-423. 15. Flandoli F., Gatarek D. (1995) Martingale and stationary solutions for stochastic Navier-Stokes equations, Prob. Theory Relat. Fields, 102, 367-391. 16. Flandoli F., Gozzi F. (1998) Kolmogorov equation associated to a stochastic Navier-Stokes equation, J. Funct. Anal., 160, 312-336. 17. Flandoli F., Maslowski B. (1995) Ergodicity of the 2 0 Navier-Stokes equations under random perturbations, Comm. Math. Phys., 172, no l , 119-141. 18. Flandoli F., Romito M. Partial regularity for the stochastic Navier-Stokes equations, Preprint. 19. Gallagher I., Planchon F. On infinite energy solutions to the Navier-Stokes equations : global 2 0 existence and 3D weak-strong uniqueness, Preprint. 20. Kuksin S., Shirikyan A. (2000) Ergodicity for the radomly forced 2D NavierStokes equations, Math. Phys. Anal. and Geom., t o appear. 21. E W., Mattingly J.C., Sinai Y. G. (2000) Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equations, Preprint. 22. Mikulevicius R., Rozovskii B. (1998) Martingale Problems for Stochastic PDE’s, in Stochastic Partial Differential Equations: Six Perspectives. R. A. Carmona and B. Rozoskii editors. Mathematical Surveys and Monograph n. 64, American Mathematical Society. 23. Ribaud F. (1998) Cauchy problem for semilinear parabolic equations with initial data in H;(Rn)spaces, Rev. Mat. Iberoamericana, 14,1-46. (QUANTUM) FIELD THEORY, Prince24. Simon B. (1974) THEP(4)2 EUCLIDEAN ton, NJ: Princeton University Press.
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25. Temam 26. Temam
R.(1977) THENAVIER-STOKES EQUATIONS, North-Holland. R.(1983) NAVIER-STOKES EQUATION A N D N O N L I N E A R F U N C T I O N A L
ANALYSIS,
SIAM, Philadelphia.
INVARIANT MEASURES OF LEVY-KHINCHINETYPE FOR 2D FLUIDS
S. ALBEVERIO Institut f u r Ang. Mathematik, Universitat Bonn, Wegelerstr. 6, 0 - 5 3 1 15 Bonn; S F B 61 1, Bonn; BiBoS, Bielefeld; CERFIM, Locarno; Accademia d i Architettura, USI, CH-6850 Mendrisio; Dipartimento d i Matematica, Universitci di Trento, I-38050 Povo; E-mail:
[email protected]
B. FERRARIO Institut fur Ang. Mathematik, Universitat Bonn, Wegelerstr. 6, 0 - 5 3 115 Bonn; Dipartimento di Matematica, Universitci d i Pavia, via Ferrata 1, I-271 00 Pavia; E-mail:
[email protected];
[email protected] A survey of results on invariant measures of the L6vy-Khinchine type for 2D Euler and stochastic Navier-Stokes equations is given. Uniqueness results of the corresponding Liouville respectively Kolmogorov flows are discussed. Stochastic dynamics associated with the invariant measures are also discussed (stochastic Stokes equation for the vorticity in the Gaussian case, Doob's independent Brownian motions process in the compound Poisson case).
1. Introduction
Let us begin considering the classical motion of an ideal incompressible fluid, that is the Euler equations
with suitable boundary conditions: u . n = 0 on OD, where n is the exterior normal t o the boundary dD of the smooth domain ED or the periodic boundary condition for w . n when ID is the torus. The unknowns are the velocity vector u = w(t,z) and the pressure (scalar) p = p ( t , z ) . Here z = (21,. . .,zd), V = (%, a . . . , &)and w . w is the scalar product in I@. An equivalent formulation can be expressed in terms of the vorticity w := V A u. For space dimension d = 2, w is a scalar field w = z V' .w
2 2
130
131
and applying the V' we obtain
{
=
(-&,&-) operator to the first equation in (l),
~ + u . V w = 0 w=vL.u
(t1x) E ( O I T I x
JD
(2)
with the tangential boundary conditions for the velocity. In (2) there is no pressure, but this has t o be recovered from the velocity field u: applying V to the first equation (1) we get -Ap = V . [(u. V)u]. Since w evolves according to a transport equation, the solution is w ( t , x) = w ( 0 , E t x ) where Et is the flow of material points in the fluid ( & x ( t ) = v ( t , x ( t ) ) ) .Since the vector field w is divergence free, any solution to system (2) corresponds to a volume preserving flow Et (i.e. the Lebesgue measure on ID is preserved in time). Moreover there are two other conserved quantities
,
energy enstrophy
E = L2 J I +)I2da: S= J , w(z)'dz
This has t o be understood as follows: if an Euler flow (2) with finite energy is defined, then the energy is indeed constant. The same holds for the enstrophy in a two dimensional spatial domain. The computations showing this invariance in time are easily checked in these cases, e.g. dS - --
dt
==
J, w ( t ,).
a t 4 t 1). dx
J, ~ ( X) t ,~ ( 2) t , . V w ( t ,X) dx J , V U ( X) ~ ,. U ( t , X) w ( t , X) dx + J , ~ ( x)V t , . u ( t ,X) w ( t , X) dx
2
Since V . w(t, x) = 0, then = 0. Notice that all the quantities have been assumed t o be well defined, i.e. the solution w is regular enough. By means of these conserved quantities, heuristic expressions of invariant measures can be given. In the next section, we will deal with probability measures m of Lkvy-Khinchine type. They are supported on distribution spaces. Therefore the Euler dynamics with initial data in the support of the measure rn (if it exists) is not a classical one. An overview on the study of a deterministic dynamics having m as invariant measure will be presented in section 3. According to the Koopman-von Neumann theory, as soon as a (candidate) invariant measure m is known, any flow S t ,t E R, in the space of distributions S' a gives rise to a flow in C2(rn),represented aBy S' we denote the vector space of continuous linear functionals on C,"(ID) the spatial domain is the torus, CpMer(T).
or, when
132
E R. And viceversa, under by a unitary group: (Utf)(w)= f(Stw),t some assumption on the group Ut,it is possible to construct a flow St, m a s . . For this reason the infinitesimal generator B of the unitary group (Ut = eitB) will be analyzed. On the other hand, in section 4 a stochastic dynamics having m as invariant measure will be introduced, as the Markov process associated to the classical Dirichlet form given by the measure m. In contrast to the deterministic nonlinear case, this is an easy (linear) problem t o study. The corresponding flow in C 2 ( m )is represented by a contraction semigroup Tt,t E R+; an analysis of its infinitesimal generator Q will be given (Tt = etQ). Finally, in section 5 we merge the deterministic and the stochastic frame. Partial results about the stochastic nonlinear problems, which arise in this case, will be given and open problems will be presented.
2. Invariant measures of LQvy-Khinchinetype One important feature of the evolution (2) is that the underlying flow Et in D preserves the Lebesgue measure. This allows to construct a family of probability measures on the space of distributions S‘ which are invariant for any given flow (2). In fact, let m be a probability measure on S’; this corresponds to a family of random variables realized canonically as random variables on the measure space (9, B(S’))
{x,},,~,
Assume now that the random variables are identically distributed and
X, independent
of
X,
’dp,II,E S such that pII, = 0
This expresses, in a distributional sense, independence in distinct points. Assuming some continuity (e.g. in the sense of X,,, -+ 0 in probability if pn -+ 0 in S), this is the definition of white noise (see, e.g., Gel’fand&Vilenkin’‘). Then the law of the random variable ( w ( t , .), p) = (w(0, .), p o is independent of time, where Et is the pointwise flow in D corresponding to (2). (Of course, this is rigorous if Et : D + D is “smooth enough”.) Because of independence, the knowledge of the law of each ( w ( t , .), determines any joint distribution for the family { ( w ( t , .), p ) } r p E ~ .Hence the white noise m is a time invariant measure. It is well known (see Gel’fand&Vilenkinl‘) that any Lkvy-Khinchine probability measure is a white noise in the sense specified above. The LkvyKhinchine representation for infinitely divisible laws gives the characteristic
133
functional
where the characteristic exponent is
with a E R, b 2 0, &,(lA v2)de(v)< 00 and RO= R \ ( 0 ) . When a = 0, b > 0,O = 0, we have a centered Gaussian measure p. When b = 0 and a = J,, vl{lvl
0); for negative index a < 0, the Hilbert space is defined by duality: 3-la(D) = ( ' W U ( D ) ) ' . Hence,
p('Hb(D))= 1
V b < -1
that is, the support of the Gaussian measure p is given by nb<-lRb(IO). Similarly when D is the torus. Compound Poisson measure The support of the compound Poisson measure IT is the space r of configurations. More precisely, for any n = 1 , 2 , . . . l let
;i(", = {((m Z l ) , . . ., (vn,4)E.
(Rlx D ) n : 51 # 21,for 1 # I C )
The space of n point configurations is defined as n
r(n) = { w = C qszl : vl E R O ,
z1 E
D~z1
+
Xk
for 1 # I C )
1=1
where 6, is the Dirac measure concentrated in z. For each index nl there is a bijection j(n)
;i(n)/s(n) r(n) +
where S(") denotes the permutation group over (1,.. . ,n). Consider on the Bore1 a-algebra of subsets of i ( n ) / S ( n the ) measure o @ = ' ~ (dO(v)dz)@'n, where for simplicity we assume the LBvy measure 6 to be finite. The image
134
measure on I'(n), under the bijection J("), is denoted by on Set = (8) and a 0 = Si0). The space of configurations
r = u=;W , is defined as disjoint union of topological spaces, with the corresponding Bore1 a-algebra B ( r ) . The compound Poisson measure II is defined by
Remark. Notice that the above measures are not the only invariant measures
known for the Euler equation (2). For instance, Albeverio et al.'l5 considered more general Gaussian white noises p ~(y ,> 0, ~ p > -y), expressed by means of the enstrophy and of the renormalized energy. For other types of invariant measures related to the Gaussian ones, see Capiriski&Cutland", Ciprianoll. Anyway we consider here only white noise distributions for the vorticity w , in order to have a unified approach ( p and II as particular cases of a Lkvy-Khinchine measure). Infinitesimal invariance of measures of Gaussian and Poisson type for the Euler equation has also been discussed in Boldrighini&F'rigiog. 0 3. Deterministic dynamics
From now on, we choose the spatial domain D to be the torus T = [0,27rI2; hence periodic boundary conditions are assumed. In this section this choice is done for mathematical convenience; in the next one it will anyway appear necessary for a right physical interpretation. Let w be a periodic distribution; it can be developed in Fourier series with respect to the complete orthonormal basis { & e i k ' z } k E Z z in the (complex) L2(T). Let denote for short by P k the k-th element in this basis. Then 1 W = 2;; x k E Z 2 w k p k with w k := s ' ( W , p - k ) S . The Coefficients w k E and w k = W - k , because w is real. Adding a constant to the velocity, solving (l), yields again a solution of (1). We select that one of zero mean value. Hence also the mean value of the vorticity is assumed to vanish: wo = O.b bThe starting problem indeed is formulated in t h e real framework. Now the complex structure arises in a somewhat artificial but practical way - via Fourier transform. Actually t h e relevant variables are {%~k,Swk}~.~~,~>~ or { w k } k E Z ~ , k > where O, k > 0 means either !q > 0 or kl = 0 , Icz > 0. Anyway, whenever t he whole sequence { ~ k } appears, t h e condition wl, = w-k is assumed.
~
~
~
135
Albeverio et a1.2 show that equation (2) can be rewritten as a system of infinite equations for the Fourier coefficients W k , for any k E Z2, k > 0
dwk(t) ChkWh(t)wk-h(t) dt h#k,h#O
--
(4)
where the r.h.s. Bk is a quadratic expression of the Fourier components with coefficients C h k = This is obtained formally from equation (2). We point out that the “Euler dynamics” with state space the support of the measure p or II has to be understood in the generalized sense, i.e. this is not a classical dynamics with function valued solution w ( t , .). When dealing with the compound Poisson measure II, equation (2) is actually the equation of vortices (see Marchioro&P~lvirenti~~). For any integer n 2 2, the vorticity w at time t is concentrated in n distinct points ~ l ( t .). ., , xCn(t) of T with given intensity vj of each vortex xj ( w ( t ,11:) = C:=, ~ j S , ~ ( ~ ) ( 1 1),: )the z j ( t )evolving according to
d dt
Y.-
Xj(t) =
vL ”j
l#j,l,j=l
-&
&
Ckfo eik’y, Here g is the Green’s function of -A on T: g(y) = 0. For n = 1 ( w ( t , x ) = v1SX,(,~(z) ), the single vortex moves Y f as if there were two vortices a t the points x1 and - X I , with intensity v1 and -v1 respectively, namely the vortex point moves according to d 1 %11:1(t) = - v 1 m CkZo eik.2xl(t) and the velocity field in any point 11: E T distinct from the vortex is w ( t , z) = Vig(11: - z l ( t ) ) . The main properties of the B k are given in the following
6
P r o p o s i t i o n 3.1. If m = II, assume that the finite Le‘vy measure satisfies
Then for both cases m = p or m = II, we have that Bk E
P(m)
dBk auk -
-(w)=O
for any 1 5 p
m-a.e.
< 00
w
B k ( u )= B - ~ ( w ) m - a.e. w
for any IC
E
Z2,k # 0 .
136
For the proofs, we refer to Albeverio et al.2,334;the LP-summability comes from Ciprianoll. What is the meaning of the functions Bk? If equation (4) would give a flow St (t E EX) in the support of the measure VI,then this would induce a flow in the Hilbert space L2(m)by (Utf)(W) = f(Stw),
f
E
C2(m)
(6)
The strongly continuous unitary group Ut in C 2 ( m ) (unitarity is given by the fact that the measure m is invariant) is characterized by its infinitesimal generator B , which is a self-adjoint operator with domain D ( B ) = { f E L2(m): 3 L2 - limt+o Its expression when acting on the dense subset 3 C r of smooth cylindrical functions ( 3 C r 3 f : f(w)= F ( w j l , . . . ,wj,) for some integer N and F E C F ( ( C N ) ) is the following
v}.
Utf - f 1 Bf = L 2 - l i m v = - ~ B k t-+O zt i
dF k
auk
(7)
This is a well defined expression in Cz(m),because the sum is finite and, according to Proposition 3.1, each B k is square summable. Let us call ( B ,3 C p ) the Liouville operator. The Liouville operator is symmetric, i.e.
and has self-adjoint extensions (according to von Neumann theorem, since B commutes with the conjugation J defined in C2(m)by J f ( w ) = T(-w) ). Actually, one of the self-adjoint extensions is B (when it exists, that is when a flow St is given). The question of uniqueness of the self-adjoint extensions of the Liouville operator was posed in Albeverio et a1.' (this is formulated as essential self-adjointness of the Liouville operator). This is interesting since any self-adjoint extension Be generates a strongly continuous unitary group. Among these groups, the positivity preserving ones are in one-to-one correspondence with a dynamics St, in the sense that there is a one-to-one correspondence between positivity-preserving unit-preserving (Utl = 1) unitary groups Ut in L 2 ( m )and weakly measurable measure preserving flows St in the support of the measure m (see Goodrich et al.17, Albeverio&Ferrario4). At this point, we have to distinguish which measure m is considered. For m = p, it has been proven by Albeverio&Cruzeiro' that there exists a flow (4) for p-a.e. initial data. Hence, for the Euler problem, the essential self-adjointness of the Liouville operator is equivalent with the uniqueness
137
of this generalized Euler flow, having p as invariant measure. But so far, the essential self-adjointness of the Liouville operator in L 2 ( p ) has not been proven. On the other hand, for m = II, a unique flow St, t E R, to equation (4) exists for II-a.e. initial data (see Albeverio&Ferrario*). II-a.e. is justified, indeed for some initial data the vortices can collapse. DUrr&Pulvirentil4 prove that for any number of vortices n and for any choice of the vortex intensities, there exists a unique (global in time) flow of (5) for each initial data in the complementary set of a (Lebesgue-)negligible subset of Ti". Therefore, keeping in mind the definition of the pre-image measure II on each for any TI L 1, there exists a set N" E with IT(N") = 0, such that there exists a unique (globaI in time) flow of (4) for each initial data w ( 0 ) in the complementary set of N" in I'("). This holds also for negative time t . Hence we can define, II-a.s., a flow St : w ( 0 ) H w ( t ) , r --t r for t E R. More precisely St : r(n)4 I'(");for each w(") the vortex intensities u j , j = 1,.. . , n, do not change in time, only the points xj on which the vorticity is concentrated evolve in time. This flow is volume preserving, since the point flow given by equations (5) preserves the Lebesgue measure. Therefore, St of (4) gives a IT-measure preserving flow on each component of r. This is expressed by
f?(r("))
ITost=n,
tER
(8)
Hence there exists a unique strongly continuous positivity preserving unitary group U,, defined by (6). Let us call this a Markov uniqueness result, adopting the same terminology of Markov uniqueness as used t o denote a second order (Kolmogorov) dissipative operator which has a unique extension generating a Markov strongly continuous semigroup in a Banach space (see, e.g., Albeverio et aL8, Eberle15, Stannat'O). We have therefore
Proposition 3.2. The Liouville operator ( B ,FCT)in L 2 ( r ,II) is Marlcov unique, that is there exists only one self-adjoint extension Be 2 B which generates a positivity preserving strongly continuous unitary group in P(r,n). Remark. The measure
II is invariant for the group Ut, i.e.
or, equivalently, the measure
(B, D(B))
I I is invariant
for the infinitesimal generator
138
(The equivalence of infinitesimal and full invariance is due to the fact that 1E D(B).) Finally, let us notice that even if any Gaussian measure can be approximated by a sequence of Poisson measures, the II-a.s. well posed dynamics (in S') is not helpful to define in the limit a p a s . dynamics. Indeed, p and II are singular measures: supp II c supp p and II(F) = 1, p ( r ) = 0 (see the proof by Colella and Lanford, in the modified version in Albeverio&Ferrario4). 4. Stochastic dynamics
One way to define a stochastic dynamics with a given invariant measure, is by means of the theory of Dirichlet forms. We consider the two cases separately. Gaussian measure Let & be the classical pre-Dirichlet form given by p:
where the measure p is the infinite product of centered Gaussian measures p k on @; since w k = x k + z y k ( X k , lJk E R),each measure p k is in fact defined as a measure on R x lR
Therefore the integration of a function f : { W k } k # O .+ C with respect to the measure p has to be understood as the integration of a complex function of the real variables Z k , Y k , through W k = X k i y k . In particular the preDirichlet form can be rewritten as
+
because -2auk = l 2( a pi&), for k > o . This form is closable, as easily seen by integration by parts, rewriting & as the positive symmetric sesquilinear form associated with a densely defined positive symmetric operator (see, e.g., Albeverio&Rockner7, Ma&Rockner" for this technique). The closure is a classical Dirichlet form, quasi-regular and local; moreover, the minimal and the maximal extension coincide (we refer, e.g., t o Ma&Rockner" for results of this type in the Gaussian case). Its associated classical Dirichlet operator is the
139
Ornstein-Uhlenbeck operator
in f ? ( p ) , which is the closure of
This Dirichlet operator generates a strongly continuous Markov semigroup dolt E R+, in ,C2(p);the Markov process properly associated solves the stochastic linear differential equation
where P,(t)
=
WL"'(t)
+ iW,"'(t),
with {WL"),W,("1
}k>O
a sequence of
independent standard real-valued Brownian motions. The measure p is invariant for this process. This is a stochastic Stokes equation
in which b / 2 represents the viscosity of the fluid ( b > 0). This corresponds to the following equation for the velocity vector fields
b dv(t,x) = ,A v(tl x)dt
+ V p ( t ,x)dt
v . v ( t , 2 )= 0 Therefore, the noise is defined by means of a Brownian motion, cylindrical in Lz(T) for the velocity. Interpretation of (10) as an equation of motion of a viscous fluid is possible only in the frame of periodic boundary conditions. Indeed, the boundary conditions in a bounded domain ID are v . n 1 a D = 0 for an ideal fluid (viscosity b = 0) and ~ 1 =8 0 ~for a viscous fluid ( b > 0). Hence the torus is the only case in which the boundary conditions for the two different fluids coincide, and therefore the functional spaces introduced for the Euler problem fit also for the Stokes problem (and in the next section for the Navier-Stokes problem). It is worth a t this point to say that all the results of section 3 require the spatial domain ID to be bounded. (For a formulation of the Euler problem as an infinite system of nonlinear equations (4) when ID is a bounded domain in IR2 with piecewise C1 boundary 8D1see Albeverio&H@egh-Krohn5.) Compound Poisson measure Similarly as before, we introduce the classical pre-Dirichlet form given by
140
the compound Posssion measure II
We refer to Albeverio,Kondratiev&Rockner' for the definition of the intrinsic gradient Vr on r, of the tangent bundle Tu(l?),as well as for the basic framework and results used in the following. We have
Any pre-Dirichlet form ( E n J , F C T ) is closable, as seen "by integration by parts". Thus the pre-Dirichlet form ( & , F C T )is closable, being the sum of closable forms. Let us denote by ( z , D ( z ) )the closure form. It is easy to see that this is a classical Dirichlet form, local quasi-regular. The corresponding classical Dirichlet operator is the closure of (Ar, F C F ) , the Laplacian on the space of configurations r; there is no drift term, since the reference measure on T is the (flat) Lebesgue measure. Therefore the Markov process properly associated to is a Brownian motion on the space of configurations, i.e.
r
X€W(O)
where { W:}zE~ are independent standard Brownian motions on the torus and W$ = x (x E w ( 0 ) means that the sum runs over all z on which w ( 0 ) is concentrated). The measure II is invariant for this process (which has been originally discussed, in other terms, by Doob, see Albeverio et al.')). 5 . Final remarks
Given the Liouville operator B and the diffusion operator Q, it is possible to merge them together in the following sense. Since both the operators B and Q are well defined on the dense subset FCF of C 2 ( m ) ,we can consider the sum operator (Kolmogorov operator)
K
=Q
+ iB,
D ( K )= 3 C r
(12)
which corresponds to a non-symmetric sesquilinear form. The operator (Q, FC?) is negative definite, the operator ( B ,F C T ) is skew-symmetric;
141
hence, the operator ( K ,F C T ) is dissipative in C 2 ( m ) , and therefore closable. The measure m is infinitesimally invariant for K
J K f d m
=0
Vf E F C ~
since it is so separately for Q and for B. Our analysis is based on the following: if the closure generates a strongly continuous Markov semigroup in C z ( m ) , then there would exist a unique Markov process (a diffusion) solving the associated stochastic nonlinear equation. This property is called C2(rn)uniqueness (or strong uniqueness) of the Kolmogorov operator ( K ,FCF). For the Gaussian case, this Markov process would be the (unique) weak solution to the stochastic Navier-Stokes equation
b d W k ( t ) = [ - -Ik12wk(t) 2
lkl + B k ( ~ ( t )d)t]+ Jz d P k ( t ) ,
k E Z2, k > 0 (13)
having p as invariant measure. Albeverio&Cruzeirol have proven that there exists a weak solution to equation (13); Da Prato&Debu~sche'~ (see also Debussche in these proceedings) have proven the existence of a strong solution. (Here, weak and strong are to be understood in the probabilistic sense). But there is still no proof of uniqueness. Results of Cp-uniqueness have been proven for some approximation operators. Preliminarily, we remark that the operator ( K ,F C T ) can be considered as an operator in any space P ( p ) ( p < a). First, consider the finite dimensional (Galerkin) operators K N, defined restricting the variables indices to vary in the subset I N = { j E 2 ' : 0 < Ijl 5 N} of Z2 (hence B[ (w)= x h , k - h , k g I N ch kwhwk-h)
For any N , the operator ( K NF, C F ) is CP-unique. For 1 5 p < 2, a proof can be given according to the following footnote (c). But for the finite dimensional case, it can be proven directly that there exists a unique solution of the stochastic (Galerkin-) Navier-Stokes equation, having p as invariant measure. (For a proof, see, e.g., Cruzeiro". In fact, the coefficients of the equation are locally Lipschitz and therefore there exists a unique solution, local in time. Conservation of the energy and It6 formula yield mean-square a priori estimates, hence the solution does not explode in finite time. This fails for the infinite dimensional problem, since the covariance of the noise is not trace class.)
142
Moreover, CP-uniqueness (1 5 p Kolmogorov operator
< 2) holds for the following approximated
which is still an infinite dimensional operator, but only a finite number of components B k appear. This is proven by Albeverio&Ferrario3, based on a result by Eberle15, since the Bk are smooth (quadratic expression of the w j ' s ) and Lq(p)-integrable for any q < 00 '. Let us point out that, if the C2(p)-normsof the components B k would decay fast enough so that
then this same technique would give C1-uniqueness for the Kolmogorov operator K defined in (12). Unfortunately, the best estimates are (see Albeverio&Ferrario4)
1
IBkI2dp
N
llc13+&
for (any) E
> 0,
as ~kl-+ co
We remark that for a different "regularization" of B in (12) (as well as in (7)), C1-uniqueness as been proven by Stannat21. For the Poisson case, the dynamics obtained merging the motion of vortices and Brownian motion is a stochastic inviscous equation of vortices; for any n 2 2, the n points in which the vorticity is concentrated evolve according to the following equation v j d z j ( t ) = V$
2
v j . l g ( z j ( t ) - z l ( t ) ) c t t + d ~ ~ " ' ( t )j, = 1 , .. . , n(14)
l#j,l,j=l We write the Kolmogorov operator with respect to the scalar product given by the symmetric part as
so that the k-th component of the first order perturbation operator is 2 2 3 Ikl
'
Then, in our setting the LP-uniqueness result of Eberle15 (Th. 5 . 2 ) holds true if the components satisfy the integrability condition
a Ikl
for 1 5 p
< 2.
143
L1-uniqueness of the corresponding diffusion operator would give a unique
weak solution of this problem, for II-a.e. initial data. Not even the existence in known so far; a n analysis of this problem is postponed t o future work. Acknowledgments We would like t o thank the organizers of the Conference on Probabilistic Methods in Fluids for the interesting meeting and for arranging a very pleasant stay in Swansea. The second author gratefully acknowledges financial support from the Alexander von Humboldt Stiftung.
References 1. S. Albeverio and A.B. Cruzeiro, Comm. Math. Phys. 129,431 (1990). 2. S. Albeverio, M. Ribeiro de Faria and R. H0egh-Krohn, J . Statist. Phys. 20 No. 6, 585 (1979). 3. S. Albeverio and B. Ferrario, J . Funct. Anal. 193 No. 1, 77 (2002). 4. S. Albeverio and B. Ferrario, Infin. Dimens. Anal. Quantum Probab. Relat. Top. (2002) in press. 5 . S. Albeverio and R. H0egh-Krohn, Stochastic Process. Appl. 31,1 (1989). 6. S. Albeverio, Yu.G. Kondratiev and M. Rockner, J . Funct. Anal. 154,444 (1998) and 157,242 (1998). 7. S. Albeverio and M. Rockner, J . Funct. Anal. 88,395 (1990). 8. S. Albeverio, M. Rockner and T.S. Zhang, Markov uniqueness for a class of infinite dimensional Dirichlet operators, in Stochastic processes and optimal control Stochastics Monogr. 7 (eds. H.J. Engelbert, I. Karatzas and M. Rockner) Gordon and Breach, Montreux, pp. 1-26 (1993). 9. C. Boldrighini and S. F'rigio, Comm. Math. Phys. 72 , 55 (1980); Errata: ibid. 78,303 (1980). 10. M. Capiriski and N.J. Cutland, Nonstandard methods for stochastic fluid mechanics, World Scientific Series on Advances in Mathematics for Applied Sciences, Vol. 27 (1995). 11. F. Cipriano, Comm. Math. Phys. 201,139 (1999). 12. A.B. Cruzeiro, Expo. Math. 7, 73 (1989). 13. G. Da Prato and A. Debussche, J. Funct. Anal. (2002). To appear. 14. D. Diirr and M. Pulvirenti, Comm. Math. Phys. 85,265 (1982). 15. A. Eberle, Uniqueness and non-uniqueness of semigroups generated b y singular diffusion operators LNM 1718,Springer, Berlin (1999). 16. I.M. Gel'fand and N.Ya. Vilenkin, Generalized Functions Vol. 4, Academic Press (1964). 17. R. Goodrich, K. Gustafson and B. Misra, Physica 102A,379 (1980). 18. Z.M. Ma and M. Rockner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer, Berlin (1992). 19. C. Marchioro and M. Pulvirenti, Vortex methods in two-dimensional fluid mechanics, L N P 203,Springer (1984). 20. W. Stannat, Ann. Scuola Norm. Sup. Pisa C1. Sci. (4) 28,99 (1999). 21. W. Stannat, Preprint Bielefeld (2002).
SOME REMARKS ON A STATISTICAL THEORY OF TURBULENT FLOWS
FRANC0 FLANDOLI Dipartimento d i Matematica Applicata, Universitci d i Pisa Via Bonanno 25b, 56126 Pisa E-mail: [email protected] Some recent notions and results, like invariant memures for the Navier-Stokes equations, random attractors, random invariant measures and vortex filaments are reviewed. Some conjectures about their relation are expressed.
1. Introduction The statistical theory of turbulent fluids contains a number of scaling laws derived on the basis of phenomenological arguments and experimental results, like the Kolmogorov K41 scaling law for the energy spectrum which asserts that E ( k ) behaves as k - 3 for wave numbers in the inertial range (between the integral scale and the dissipation scale); here E ( k ) is the mean value of J',,! 1C(k)l2dk,where S ( k ) is the sphere of wave numbers k of modulus k and u(k) is the Fourier transform of the velocity of the fluid. Moreover, in some cases the experiments and certain pieces of the theory have some discrepancies, like the scaling of the pmoments of velocity increments, $ ~ ~ (=r (lu(z ) r ) - u(z)Ip), that are not correctly described by Kolmogorov theory and seem to require proper intermittency corrections. The Kolmogorov theory would predict for the structure function 4 p ( r )a scaling of the form r c ( p ) with < ( p ) = f (for small r in a suitable range), but the experiments clearly show different exponents q ( p ) for p > 2. A number of models have been proposed to recover exponents close t o the experimental ones but a final model is not known. See the review of F'rischZ0for an extensive discussion of these topics. The most rational approach to the analysis of fluids is by means of the Navier-Stokes equations, but the previous facts and theories on the statistical properties of fluids have not been explained on such a basis. Of course a number of attempts to fill the gap between the Navier-Stokes equations and the statistical theory of turbulence have been performed, but the present understanding of this subject is very incomplete.
+
144
145
In the last ten years there has been some interest in the concept of statistics of vortex filaments. It is quite clear from numerical simulations and experiments that the vorticity field of a turbulent fluid presents some degree of geometrical organization and the concept of coherent structure has been introduced. Particularly interesting seem to be structures having the shape of filaments] therefore called vortex filaments. The importance of these structures for the statistics of turbulent fluids is not clarified yet, but the question whether a relation exists between them must be considered. In addition, the 3D geometric concreteness of these objects with respect to the vague concepts of eddies (K41 theory and many others) or fractal sets of singularities (multifractal models) and others, usually advocated in phenomenological studies of turbulence, may open the door to a more rigorous connection with the Navier-Stokes equations. In other words, there is some hope that vortex filaments (and maybe other structures not identified yet) may constitute the bridge between the Navier-Stokes equations and the phenomenological laws of turbulence. Whether the typical scalings of turbulence can be derived from statistical models of vortex filaments is still an open problem, with some preliminary indications in the works of Chorin4, and some work in progress. See also She et a1 26 and Boyer et a1 We devote this note t o the other question] namely the possible connection between the Navier-Stokes equations and the ensembles of vortex filaments. We describe just a few rigorous results that could build up such a bridge with the addition of.many other still unclear ingredients. In a sense, we meet in turbulence the same situation as in statistical mechanics] as described by R. Feynman. The theory of statistical mechanics is like a mountain: the ascent is the path from the Hamiltonian dynamics of particles (or other miscroscopic models) to the Gibbs measures, the descent goes from Gibbs measures to macroscopic predictions and laws. In fluid mechanics we see the ascent from the Navier-Stokes equations to statistical ensembles of vortex structures (filaments or others) and the descent from the latter to the laws of turbulence. This note is restricted to a few fragments of a possible path of the ascent. The main tools will be SPDEs, random dynamical systems and stochastic analysis. 2. Vortex filaments
We first describe the concept of vortex filaments following Flandoli et a1 12,15,161 which is a generalization to continuous processes of the ideas of Chorin4. In the next sections we re-start from the Navier-Stokes equations]
146
as promised in the introduction. Vortex filaments have been seen in numerical simulation of turbulent fluids, see the references in Chorin4, Frisch2'. The regions of space where the vorticity field is particularly intense seem to have the form of filament (instead of blobs or other geometric shape that appear in other sectors of Physics). Idealizing, we may think that the vorticity is concentrated on lines, around which the fluid rotates. This seems to be the 3D analog of the observed vortex points of 2D fluids (or fluids with reasonable 2D ~ ~ a lot of symmetry). As point vortex statistics, following O n ~ a g e rand subsequent work, proved to be interesting for 2D fluids, there is a similar hope for the statistics of vortex filaments. However, we want immediately to point out the transient aspect of this picture, in contrast to other statistical models. Vortex filaments are not stable objects. New vortex filaments continuously arise from various kind of instabilities (the Kelvin-Helmoltz is most famous one, but also others may be very important, see Pradeep et a1 2 4 ) . They persist for some time, but they undergo a number of modifications that eventually destroy them, producing for instance larger scale structures (see a mechanism described by Bonn e t a1 '). It is more like in Biology than in Physics. We presumably have a number of different structures, some of them more eddy-like, others more sheet-like, others like filaments, and maybe others; they may have different scales and different properties of scaling; and we observe a continuous evolution where new structures arise, evolve, and disappear into other structures. How this picture is correct we do not know exactly, but it may be a first intuitive approximation. From this viewpoint, a concept of statistical ensemble of vortex filaments cannot represent the long time statistics of certain eternal structures, like particles are in classical statistical mechanics. They do not have a long-time existence. Therefore we see two directions. One is to consider statistical ensembles like the grand-canonical, where the number of objects is not given a priori (here in each realization of the ensemble we shall see certain objects instead of others, depending on the realization). The other is that the ensemble of vortex filaments represents some sort of quasi-stationary measure, or another concept of measure having a meaning just for short or transient times. We shall explain this second appealing possibility in the next sections, with the help of dynamical systems. 2.1. Random 1-currents Following Flandoli et a1 16, we base the definition of vortex filament on the one of current.
147
We denote by D1 the space of all infinitely differentiable and compactly supported 1-forms on Rd. Such forms can be identified with vector fields cp : Rd + Rd. A 1-dimensional current is a linear continuous functional on V ' . We denote by V1 the space of 1-currents. A common example is the mapping T : V1 + R defined as T (cp) = ( c p ( X t ) , X t ) dt.
Jt
Definition 2.1. Given a complete probability space (R, A, P ) , a random 1-current is a continuous linear mapping from the space 23' to the space Lo (0) of real valued random variables on (R, A, P ) , endowed with the convergence in probability. Example 2.1. Given a continuous semimartingale (Xt)tGIo,llin Rd, the It6 and Stratonovich integrals I (cp) =
Jlo
1
(cp ( X t ),d X t )
I
s (9)=
Jlo
1
('p ( X t ) I
OdXt)
are typical examples of random 1-currents.
Definition 2.2. We say that the random 1-current cp wise realization if there exists a measurable mapping
w
I-+
H
S (cp) has a path-
S(w)
from (0,A, P ) to the space V1 of deterministic currents (endowed with the natural topology of distributions) , such that
[ S (p)](w)= [S(w)] (cp)
for P-a.e. w E R.
(3)
for every p E V'.
A general theorem of Minlos in nuclear spaces implies that the usual It6 and Stratonovich integrals have a pathwise realization. A direct spectral argument provides (presumably optimal) Sobolev regularity properties of the pathwise realization, see Flandoli et a1 14. We state here only the result for the 3D Brownian motion, to minimize the digression. Theorem 2.1. Let (Wt) be a 3-dimensional Brownian motion. T h e n the random 1-current S(p) defined by the Stratonouich integral above has a
pathwise realization S (w),with
S ( . ) E L2 (R, H-" (R3,R3)). for all s >
$.
148
It will be clear below that we are interested mainly in H-l currents. The Stratonovich integral (or the It6 one) does not have this property. Let us use the expressive notation (z) =
1'
6(z - Wt)0 dWt
for the random distribution such that ' 5 (cp) = s,'(cp(Wt), o d W t ) = [S ( w ) ] (9). If we want a random 1-current similar t o 6' but with a pathwise realization in H - ' , a natural idea is to mollify the 6 Dirac, just to the needed extent. Geometrically it means that in place of a single curve, namely a path of (Wt),we consider a sort of Brownian sausage, with a cross section that is not necessarily a ball. In place of set-theoretic sausage we prefer t o work with a smoothing based on a measure p. Here is the definition. Given a probability measure p on R3,consider the random current
or in more rigorous terms, the mapping
defined over all cp E D1, with values in Lo ( 0 ) .With the same arguments that yield the previous theorem we have: Theorem 2.2. Assume that the measure p has finite energy, in the following sense:
Then the random current cp H 5 (cp) just defined has a pathwise realization
< ( w ) , with
< E L2 (0,H-1
(R3,R3)) .
Remark 2.1. If A is a compact set in R3 with Hausdorff dimension > 1, then there exists at least one measure p supported on A (for instance the so called equilibrium measure of potential theory) which satisfies the previous condition. Therefore, if we want H-l samples, it is sufficient t o mollify the current just by means of a fractal cross section with Hausdorff dimension > 1.
<'
149
<
Remark 2.2. It is possible to show that the H-'-norm of is given (up to a multiplicative constant) by the following double stochastic integral in the Stratonovich sense:
where the "interaction" energy Hzyis given by 1
In Flandoli12, this double integral is rigorously defined and analyzed. It is proved that H,,p ( d z ) p ( d y ) has finite expectation. This provides a different proof of the last theorem above, not based on random currents. With such approach the previous hypothesis on p turns out to be necessary and sufficient. An interesting fact is that Hzycan be expressed as a double It6 stochastic integral plus the self-intersection local time of the Brownian motion (plus boundary terms). Another proof of the previous theorem can be found in Flandoli e t a1 15.
ss
2 . 2 . Back t o vortex filaments i n 3D fluids
The previous set-up and results are motivated by probabilistic models of vortex filament. We interpret the random distribution as a vorticity field of a fluid, concentrated in a tubular region around the curve (Wt),a region having a possibly fractal cross section p. The previous regularity property of implies that it defines a velocity field with finite kinetic energy. To explain this, consider a 3D fluid, in the whole space R3,with velocity field u(z)(we do not consider the time dependence here). The kinetic energy is
<
<
The vorticity field is defined as
< (z) = curl u(z). <
The relation between the regularities of u and is that u E L2 implies E H - ' , and given E H-' one can reconstruct (by Biot-Savard law) a velocity field u E L2. Therefore the requirement H ( u ) < 00 is equivalent to E H - l . So, up to now we have defined random vorticity fields, concentrated on narrow sets, such that the associated velocity fields have finite kinetic energy. The law po of on H-' is the image law of the Wiener measure
<
<
<
<
150
in R3. We may, as a first approximation, consider po itself as a possible statistical ensemble of vortex structures. More natural is to introduce the Gibbs measures PO ( d t ) = z i le-OH(u)po (&)
.
where u is recovered from E by the Biot-Savard law. In Flandoli et al l5 it is proved that pp is well defined for all 0greater than some 0 0 < 0, hence also for some megative inverse temperature (and it is also proved that e-@H(u) is not po-exponentially integrable for sufficiently large negative 0).The measures p~pare similar to those introduced by Chorin on the lattice. In that case po was the law of the self-avoiding walk, but also here there is, hidden in H(u),the presence of the self-intersection local time, see a remark above. We are not sure that the measures p~pare the best candidate to describe the statistics of vortex structures in turbulent fluids. Variants of them could be more interesting, as a work in progress indicate us, where many different vortex structures are taken into account simultaneously. Therefore the research on such measures is still a t the beginning. In spite of our ignorance about them, we indicate in the sequel an hypothetical path to relate them to the Navier-Stokes dynamics. 3. 3D stochastic Navier-Stokes equation: weak stationary
solutions
As we have remarked above, the statistical description of turbulence is based on certain relevant expected values, like the energy spectrum, the structure function, or the mean dissipation rate. A sound mathematical basis for them should be to take the expectation of certain observables with respect to a suitable measure p on the configuration space of the fluid (the space of all relevant velocity fields, or vorticity fields); for instance, E(k)
=
(Jsc,,10(k)12dk) P
. Let us restrict our attention t o persistent
turbulence (in contrast to decaying turbulence), which is a stationary long time phenomena. Under such a viewpoint, p should be an invariant measure for the Navier-Stokes dynamics. The final aim is to have quantitative informations on mean quantities related to turbulence, but for the time being let us comment on the preliminary question of the rigorous facts known about existence, uniqueness and ergodicity of invariant measures for the Navier-Stokes equations. The picture is different depending on the space dimension d = 2 or 3 and on the deterministic or stochastic nature of the equation.
151
1. For the deterministic 2D Navier-Stokes equations one can prove the existence of an invariant measure p, supported by the compact global attractor. The proof is a straightforward application of the existence KrylovBogoliubov theorem for invariant measures of continuous flows on compact metric spaces, along with the existence of the compact global attractor (see for instance Constantin et a1 '). Uniqueness of p is certainly not a general property and especially it is not expected at high Reynolds numbers. For instance, many flows with an unstable stationary solution are known; in such a case the delta Dirac at the stationary solution is an invariant measure, but another invariant measure certainly exists. At high Reynolds numbers one could even expect to have infinitely many invariant measures, as suggested by the Ruelle-Sinai-Bowen (RSB) theory. The first question is then how to identify the physical measure p which gives us the mean values of interest for turbulence. Again the RSB theory provides fundamental paradigms in this direction, but we have t o remind that it is applicable, a t present and in spite of great recent extensions to partially hyperbolic systems, only to rather artificial dynamical systems quite far from the Navier-Stokes equations. 2. For many classes of stochastic 2D Navier-Stokes equations one can prove the existence of an invariant measure p, and under several different assumptions on the noise also the uniqueness and ergodicity of p. This is one of the most notable achievements of the recent probabilistic efforts in fluid dynamics. 3. For both the deterministic and stochastic 3D Navier-Stokes equations one can prove the existence of a shift-invariant measure ji on the path space of solutions to the equations. In other words, there exists a stochastic process ( u (t)),Lothat is a solution of the 3D Navier-Stokes equations and is also a stationary process, hence its law ji in the path space is shiftinvariant. We shall state in a moment a rigorous theorem of this kind. The law p of u ( t ) is then independent of t and it can be considered as a measure on the space of configuration that may represent the stationary regime. Unfortunately, the lack of well-posedness of the 3D Navier-Stokes equations does not allow us to prove uniqueness and ergodicity of p under stochastic perturbations; but this seems to be a technical aspect, perhaps transient in the history of this subject (see for instance the irreducibility property proved in Flandoli"). On the contrary, the lack of uniqueness of p in the deterministic case is a true fact for many flows and has a fundamental origin. The results just quoted appeared in a long series of works by many authors, like Foias, Prodi, Temam and many others in the deterministic
152
case, and several works in the stochastic case, among which we just quote Kuksin et a1 21 and subsequent works, Vishik et a1 27. We complete this section with the precise statement of a rigorous result on point 3 above. In a sufficiently regular domain D c R3, consider the SPDE of NavierStokes type
* and references therein, Ferrariog, Flandoli et a1 17,
d u + [(u. V) u
+ Vp] dt = [VAU+ f]d t + G ( u )d W divu = 0,
UI~D
=0
where u , p , f , W are functions of space x E D , time t 2 0 (or sometimes E R), and the random element w E 0, where (0,A, P ) is an underline probability space. The field p is scalar and has the meaning of pressure, u is a 3D vector field with the meaning of velocity, f is a given 3D force field, W is a cylindrical Wiener process in a suitable Hilbert space, so G (u) d W is a random perturbation of the classical incompressible, Newtonian, constantdensity Navier-Stokes equations. Denote by H the function space
t
H
=
{Q : D
3
3
R31Q E [L2 ( D ) ] , divQ = 0 , Q .nlaD = 0
1
where n is the outer normal to i3D
V
=
{Q
E
[H' ( D ) I 3IdivQ = 0 , Q l a ~= O}.
Let { e i } be a complete orthonormal system in H and let { p i } be a sequence of independent standard Brownian motions on (0,A, {Ft}, P ) , where F = { F t }is a given filtration. Formally we set W ( t , x )= ei (.)pi ( t ) (this series converges only in suitable distributional spaces). Let G : H L2 (H) be a continuous mapping, where L2 (H) denotes the space of HilbertSchmidt operators on H. Even if W ( t , . )is not an element of HI the stochastic integral G (u ( s ) )d W (s) is well defined, for instance when u is an F-adapted process with paths in L" (0, T ;H ) . The following theorem of existence of martingale weak solutions has been proved by Flandoli et a1 l 3 in a great generality (even in some cases when G is defined only on V , so it may depend on the space derivatives of u),but close results may be found also in works of Schamlfuss, Capinski and Cutland, among others.
Ci
--f
Theorem 3.1. Assume that uo E H , f E V ' , and G : H + L 2 ( H ) is continuous with linear growth. Then there exists a stochastic basis (R, A, {Ft}, P ) , a sequence of independent standard Brownian motions
153
{pi} o n it, and a weakly continuous adapted process u in H , satisfying the stochastic Navier-Stokes equations as a n identity in V', with the property
f o r all T
2 0. If in addition IIG (
2
m 2 ( H ) 5 A0
1.;lI
+c
f o r all x E V and f o r a suficiently small A0 2 0 , t h e n there exists a stationary process u with the previous properties. For the sequel of our discussion we take a stationary solution u given by the second part of the previous theorem. Then the measure
p = law of u ( t ) is independent of time and therefore it is a candidate to describe the long time statistics of the fluid. In principle we do not know whether the stochastic Navier-Stokes equations define a Markov process, so we cannot speak of invariant measures in the usual sense, but p is clearly a substitute of such a concept (there are open paths to give a formal definition, like the coiicept of infinitesimally invariant measure, or the possibility to prove the existence of a Markov selection, that we believe to exist).
Remark 3.1. If u is a stationary measure then E IIu (t)11; is constant, so
(t)[lt]
An inequality of the form E S U ~ ~ ~IIu[ ~ , ~
[
4. The viewpoint of random dynamical systems Having in mind the search for quantitative properties of invariant measures of the Navier-Stokes equations, let us comment on the directions opened by the results of the previous section.
RSB theory. In the deterministic case, 2D for sake of rigor, p is concentrated on a compact set of configuration space, presumably a rather complicate geometrical object at high Reynolds numbers (there exists lower
154
bounds on the Hausdorff dimension of the attractor that show that the dimension diverges with the Reynolds number, see Liu22). Therefore we do not expect p to have a simple form, like a Gibbs measure. The paradigm arising from the RSB theory, however, open the door to a quantitative analysis, even if very difficult. The picture that emerges in the RSB theory is that a typical trajectory on the attractor crosses continuously local unstable manifolds Wp (such unstable manifolds are sets of points close to each other and approaching each other exponentially in the reversed motion). The measure p conditioned to W z is a Gibbs measure with energy proportional to a certain logarithmic determinant of the flow, so a quantity that in principle one can try to analyze to get quantitative informations. Hence the statistics of plwp reflect into statistics of the flow. The latter sentences require careful analysis since one has to mix up in a rather complex way the measures p[wp for different manifolds Wp to get statistical properties of a trajectory. Rigorously speaking, the point is still unclear. However, localization on the attractor (which is related to conditioning to W:) seems t o be compatible with scaling properties of p : one can presume that universal scaling properties do not depend so much on the local piece of attractor we observe, while more large scale properties (depending for instance on the particular geometry of the boundary) may vary in essential way over the attractor. Fokker-Planck equation. In the stochastic case, again 2D, we think that the unique invariant measure p is a sort of diffused regularization of the invariant measure p d e t of the deterministic system. Again for the models rigorously covered by the RSB theory, the measures ps of suitable random perturbations of order E converge to p d e t as E -+ 0. The additional regularity of the measure p has the good consequence that it satisfies certain elliptic infinite dimensional equations of Fokker-Planck type, so in principle there could be a way to obtain quantitative results from these equations. However, at present, really promising results in this direction are not known, especially as far as scaling properties of local quantities are concerned. This approach seems to be more promising to get large scale informations by suitable finite dimensional or large eddies approximations. An argument in favor of the SRB approach instead of the Fokker-Planck one is the following. In the section on vortex filaments we underlined the transient features of such vortex structures. The restrictions plw2 may capture features that change in time, while it looks less reasonable to see them directly from p using global tools like the Fokker-Planck equation.
155
RSB theory for random dynamical systems. In view of the previous facts and the additional ergodic properties of stochastic systems, we think that the most promising direction at present is an approach based simultaneously on the RSB paradigm and the invariant measure of the stochastic system. We remarked above that in the deterministic case there could be very many invariant measures and the physical one with the good RSB properties has to be identified. The viewpoint of random dynamical system (see Arnold') comes to help us. In the stochastic case it is still possible to introduce concepts intimately related to the geometry of configuration space as in the deterministic case, by means of the theory of random dynamical systems. In such a framework there exists a concept of random attractor and of random invariant measure p,, whose expected values are the classical invariant measures p . At the level of p, it seems possible to develop the concepts of the RSB theory. The lack of uniqueness of invariant measures suffered by the deterministic models is met again here: even if p is unique, it is not clear that p, is unique. But there is a theorem asserting that under a condition of ergodicity of the 2-point motion, there is a constructive way to identify a unique p,, with some properties similar t o those of the RSB theory. 4.1. Random attractors Consider in this section the case of additive noise:
G ( u )= G constant. Extensions of the following facts to multiplicative noise are of great interest, but only a few results have been proved until now. When the noise is additive it is possible to study the stochastic equation path by path, as a deterministic equation with a distributional forcing term G F . See the details in Flandoli & Schmalfu~s'~. For P-a.e. w E R , considered as given, the following fact can be proved: for every uo E H there exists a weak solution u = u(w),namely a weakly continuous function from [0, co)t o H , with
that satisfies the (deterministic) Navier-Stokes equation
au aW -+(u.V)U+V~=VAU+~+Gat at in the distributional sense and the initial condition u(0) = U O . We do not know whether this solution is unique, as in the deterministic case.
156
Denote by P ( H ) the family of all subsets of H . Consider as R the two-sided Wiener space R = Co (R, R ) N ,with the product a-algebra and product Wiener measure (on a single component CO(R, R), the two-sided Wiener measure is the measure of a process ,Ll ( t ) ,t E R,such that { p ( t ) } t-> o and {,Ll(-t)},,o are two independent Br0wnia.n motions). Consider on R the shift &, t E R,defined as (&w) (s) = w (t
+ s)
-
iJ ( t ) .
The previous existence result defines a multivalued random dynamical system, namely a family of mappings cp(t,w) : H + P ( H )
with t
2 0 and w
E
R,such that cp ( t ,w ) = cp ( t - s, Qsw) 0 cp (s, w )
(as composition of multivalued maps).
Remark 4.1. One can also associate a random dynamical system by lifting the dynamics in the path space L 2 (0,oo;H ) . This approach, introduced by Sell in the deterministic case, has been developed also in the stochastic one, see Cutland8, Flandoli et a1 19. Remark 4.2. The map cp ( t ,w ) does not have good continuity properties (similarly t o the lack of uniqueness). Just the following very weak form of continuity can be proved: b'x, E HI b'y, E cp ( t ,w ) x,, 3 { n k } , z E H , y E cp ( t ,w ) x such that x,, x and ynk y in H .
-
-
Remark 4.3. In 2-dimensions the map cp ( t ,w ) is single valued and continuous. The following definition has been given in Crauel et a1 '.
Definition 4.1. A random set A ( w ) is a compact global attractor if 1) it is compact and non empty, 2) cp ( t ,w ) A ( w ) = A ( O N ), 3) for every bounded set B C H we have d (cp ( t ,1 9 - t ~ B ) , A ( w ) ) + 0 as t + +m.
The following theorem is proved in the series of papers and the part on the weakly compact attractor in H by standard analysis is still a work in progress. See Cutland', Flandoli et a1 l 9 and references therein. One of the
157
claims require the following assumption, which represents one of the main open problems in the theory of the Navier-Stokes equations: t'uo E V there exists a global solution with u(.,w) E C([O,oo);V),P-a.s.
(29)
Theorem 4.1. For the 3 0 stochastic Navier-Stokes equations (with additive noise) there exists both a weakly compact global attractor in H for the multivalued random dynamical system and compact global attractor for the shift in the path space. Under the assumption (29), the flow is single valued and there exists a compact global attractor in H . In 2-dimensions the compact global attractor exists and (at least for certain noise) has finite Hausdorff dimension. 4.2. R a n d o m invariant measures
In this subsection we describe a few general facts for random dynamical systems, so it is not assumed that the dynamics come form the NavierStokes equations. The main facts are taken from works of Crauel'. Let H be a Polish space and let cp ( t ,w) be a random dynamical system on it (see Arnold'). Let P r ( H ) be the set of all Bore1 probability measures on H and cb ( H ) be the space of all bounded continuous functions on H . A random measure w H p, from R to P r ( H ) is called invariant for cp ( t ,w) if
Recall on the other side that a probability measure p E P r ( H ) is invariant for the Markov semigroup if P (f)=
(Ptf)
where Ptf (x)= E [f('p ( t ,.) x)]. Denote by F
A(w). Theorem 4.3. If in addition A (.) is F
158
Theorem 4.4. For any .?'
is invariant for the Marlcov semigroup. These theorems give us a strategy t o construct Markov invariant measures (but usually they may be constructed in easier ways). More than this, they provide a richer structure for the Markov invariant measures.
4.3. RSB properties of the random invariant measures The following result is a version of known facts proved in a series of works by Kifer, Baxendale-Stroock, Ledrappier-Young, Le Jan and others, see Dolgopyat et a1 lo. Its says that under the ergodicity of the 2-point motion there is a random invariant measure with some RSB properties. We restrict ourselves to discrete times for sake of simplicity. Let 'p ( n ,w),n E N,be a continuous random dynamical system on a Polish space H , having a compact global attractor A (w).Denote by On, as above, the underlying flow. Assume that cp ( n ,w ) generates a discrete time Markov process, with transition operator Pn. Assume in addition that 'p (n, w ) and 'p (k,O-kw) are independent, as it happens for systems generated by stochastic equations driven by white noise. Denote by P?' the transition operator of the 2-point process:
for all g E c b ( H x H ) . Let C C P r ( H ) be a set of measures closed by the action of p (n, w).
Proposition 4.1. Assume that P, has an invariant measure p E C and that the %point motion is exponentially mixing on C , in the sense that there is a constant X > 0 such that for every ul , u2 E C and every g E cb ( H x H ) one has
for some constant C > 0 depending only on g . Then there exists a (unique) random invariant measure p,, supported by A ( w ) , such that for every u E C, every f E C b ( H ) , every A' < X and P-a.e. w E R one has
159
for some random variable C’ ( w ) > 0. Moreover,
P-a.s., for all f E C b ( H ) . The application of this result to stochastic equations of Navier-Stokes type is still an open problem, but it is reasonable to expect positive results in the next future. General sufficient conditions on the coefficients of ordinary stochastic differential equations on compact manifolds to have the ergodicity of the two-point motion are known and they are generic in a suitable sense; see Dolgopyat et al lo. They are based on Hormander type conditions. The first step is t o try to understand these conditions for finite dimensional approximations of the 3D Navier-Stokes equations, the investigation of which is now a t a good stage, see R ~ m i t o Extension ~~. to the infinite dimensional full 3D Navier-Stokes equations, or at least to the well-posed 2D case, is another more open step. 5 . Conclusions
We arised the question whether the physical invariant measure p of the Navier-Stokes equations, conditioned to the unstable manifolds W:, is ‘ related to the statistics of vortex structures, like the Gibbs measure of vortex filaments. We believe that a statistical analysis of some typical instability of fluid flows could throw some light. Up t o now, just a few objects of such a story are known, like invariant measures for the Navier-Stokes equations, random attractors, random invariant measures possibly with some RSB properties, and some ensembles of vortex filaments.
References 1. L. Arnold, Random Dynamical Systems, Springer, Berlin 1998. 2. D. Bonn, Y. Couder, P. H. J. van Damm, S. Douady, Phys. Rev. E. 47, 28
(1993). 3. D. Boyer, J. C. Elicer-Cortbs, J. Phys. A : Math. Gen. 33,6859 (2000). 4. A. Chorin, Vorticity and Turbulence, Springer-Verlag, 1994. 5. P. Constantin, C. Foias, R. Temam, Attractors Representing Turbulent Flows,
Memoirs Amer. Math. SOC.314, Providence 1985. 6. H. Crauel, Random Probability Measures on Polish Spaces, Habilitations-
schrift, Universitat Bremen, 1995. 7. H. Crauel, F. Flandoli, Prvbab. Theory Rel. Fields 100, 365 (1994). 8. N. J. Cutland, these proceedings.
160
9. B. Ferrario, Stochastics and Stoch. Reports 60, 271 (1997). 10. D. Dolgopyat, V. Kaloshin, L. Koralov, Sample path properties of the stochastic flows, preprint 2001. 11. F. Flandoli, J . Funct. Anal. 149,160 (1997). 12. F. Flandoli, A n n . I. H. Poincari, P . Ei S. 38, 207 (2002). 13. F Flandoli, D. Gatarek, Probab. Theory and Rel. Fields 102,367 (1995). 14. F. Flandoli, M. Giaquinta, M. Gubinelli, V. M. Tortorelli, On a relation between stochastic integration and geometric measure theory, preprint 2002. 15. F. Flandoli, M. Gubinelli, Probab. Theory Rel. Fields 122,317 (2002). 16. F. Flandoli, M. Gubinelli, Random currents and probabilistic models of vortex filaments, preprint. 17. F. Flandoli, B. Maslowski, Comm. Math. Phyls. 171,119 (1995). 18. F. Flandoli, M. Romito, Trans. Amer. Math. SOC.354,2207 (2002). 19. F. Flandoli, B. Schmalfuss, J . of Dynamics and D i f f . Eq. 11,355 (1999). 20. U. Frisch, Turbulence, Cambridge Univ. Press, Cambridge 1998. 21. S. Kuksin, A. Shirikyan, Comm. Math. Phys. 213,291 (2000). 22. V. X. Liu, Comm. Math. Phys. 147,217 (1992). 23. L. Onsager, Nuovo Cimento 6, 279 (1949). 24. D. S. Pradeep, F. Hussain, J . Fluid Mech. 447,247 (2001). 25. M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, preprint 2002. 26. Z.-S. She, E. Leveque, Phys. Rev. Letters 72,336 (1994). 27. M. I. Vishik, A . V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer, Dordrecht, 1980.
SOME PROPERTIES OF BURGERS TURBULENCE WITH WHITE NOISE INITIAL CONDITIONS
CHRISTOPHE GIRAUD Laboratoire J.A.Dieudonne U M R CNRS 6621 Universite d e Nice Sophia-Antipolis Parc Valrose 06108 Nice Cedex 2, F R A N C E E-mail: [email protected] This paper intends t o review t h e main properties of t h e solutions of Burgers equation with random initial conditionsof white noise type. These properties are closely related t o those of the convex hull of a Brownian motion with parabolic drift. A special attention is given t o t h e latter.
1. Introduction
This text aims at surveying some key properties of the solutions of the one-dimensional (inviscid) Burgers equation (1)
&u+ud,u=O
with initial condition of white noise” type. Burgers introduced this equation around 1940 in its multidimensional form, &u u . Du = 0, as a toy model for hydrodynamic turbulence. It is known nowadays this far from accurate; see Kraichnan’l for a discussion on the similarities and the differences with Navier-Stokes equation. Yet, Burgers equation appears in many fields of mathematical physics, such as the formation of the large scale structures of the universe, or the dynamics of growing surfaces, see e.g. Woyc~ynski’~. The study of the solution of Burgers equation (1)with white noise initial condition takes place in the field of the analysis of solutions of PDE’s with random initial data. If we think to the phenomenon of turbulence, it seems interesting to exhibit the statistical properties of the solutions of some PDE of fluid mechanics, with random and chaotic initial conditions. Such studies also appear in astrophysics, when one considers the formation of the structures of the universe. Solutions of Burgers equation with random
+
aA white noise is the derivative, in the sense of distribution, of a Brownian motion. 161
162
Gaussian initial data seem to be in this case of particular interest] see Vergassolla et a1 26 for an up-to-date survey. Roughly, the analysis of Burgers turbulence may be viewed as a first step for depicting the solutions of more sophisticated PDE’s with random initial data. The choice of white noise as initial condition stems from the fact that it appears as a natural model for chaos. Some others initial conditions have yet also been considered. We refer t o Bertoin for the analysis of the Brownian case5 and a survey on the stable noise case6, and to Leonenko” and WoyczynskiZ7for other cases. The white noise initial data also arise naturaIly in statistical physics. Consider a time t = 0 particles of mass 1 spread on a regular lattice, say Z, with random initial velocities independent and identically distributed (i.i.d.) with centered law of finite variance. Next, let the system evolve according to the dynamics of free sticky particles: between collisions particles move at constant speed, and when some of them meet, they merge into a single particle] whose mass and momentum are given by the sum of the masses and momenta of the particles involved into the collision. Then, the velocity field of the hydrodynamic limit of such a system of ballistic aggregation is a solution to Burgers equation with white noise initial condition; see l2 and also next section for further explanations. Investigating solutions of Burgers equation with random initial data can lead to interesting problems in probability theory. Indeed, according to the celebrated Hopf-Cole formula, the solution u ( . , t ) of (1) at time t can be expressed in terms of the convex hull of the path 1 2 H u(5,O)d z -2’.
LZ
+ 2t
In the case of a white noise initial condition u(.,0), the analysis of u thus requires a deep analysis of the convex hull of a Brownian motion with parabolic drift, which is mainly based on the work of Groeneboomlg. See Section 3 for a sketch of this analysis. There are also some interesting connections with the phenomenon of coalescence and fragmentation] see Bertoin5. The rest of the paper intends t o review the main properties of the solutions of Burgers equation (1) with initial condition of white noise type. Section 2 recalls necessary background on Burgers equation (with deterministic initial condition). In Section 3, various results on the convex hull of a Brownian motion with parabolic drift are collected. Even if at first sight they seem to have little to do with Burgers turbulence] they are the key for the understanding of the proofs of the next sections. In Section 4 the main properties of the solution of (1) with white noise initial condition are depicted. A special attention is given to its time-evolution. Some other
163
types of white noise initial condition are presented in Section 5. Section 6 concludes with few open problems. 2. Some background on Burgers equation The purpose of this section is to present some standard features on solutions of Burgers equation (1). We refer to for proofs. Even for very smooth initial conditions, solutions may develop shocks (discontinuities) at finite time. We then lose the existence of strong solutions, as well as the uniqueness of weak solutions. We will focus henceforth on a special (weak) solution of (l),the so-called entropy solution, since it is the physically meaningful1 solution of (l),see g. This special solution can be obtained in adding a vanishing viscosity term to equation (1). More precisely, the viscid equation 11j12,20
dtu+udd33u=&d~ZU has a unique solution u, which converges as E + 0, except maybe on a set of Lebesgue measure 0, to the entropy solution u of (1). Provided that the so-called initial potential W ( z ):= u(z,0) dz fulfills the condition
W ( z )= o ( 2 ) as
121 + 00,
(2)
it is remarkable that the (entropy) solution u(., t ) a t time t can be expressed in terms of the convex hull 7-It of 1 2t Indeed, write a ( z ,t) for the right-most location of the minimum of
z
z
I--+
H
W ( z )+ -2.
W ( z )+ -1( z 2t
2
- x)
.
Then, on the one hand a(x,t) coincides with the right-continuous inverse of t times the derivative of the convex hull 'lit. On the other hand, a versionb of the entropy solution u of (1) is given by the Hopf-Cole formula u(2, t)=
x - a ( z ,t) t '
see 11$20, Notice already that the discontinuities of x ++ u(x,t ) come from the discontinuities of z H u ( z ,t ) . Since z H a(z,t)is right-continuous and bA weak solution is only defined up t o a set of Lebesgue measure 0, we can thus only speak of a version of it.
164
increasing, they are of the first kind and always negative (this is precisely the entropy condition). As mentioned before, we can interpret the entropy solution in terms of a system of ballistic aggregation. Consider a t time t = 0, infinitesimal particles spread on the real line according to the uniform density p(dz, 0) = d z , with velocities given by the velocity field u,(., 0). Then, let the system evolve according to the dynamics of free sticky particles described in the introduction. At time t , the velocity field of the system fits with (a version of) the entropy solution u(., t ) with initial condition u(., 0). Moreover, the function a ( z ,t ) defined above represents the right-most initial location of the particles lying in ] - 00, z] at time t. In other words, the density of mass in the system is given at time t by the Stieltjes measure
d l z , Yl, t ) = 4 Y , t ) - 4 x 1 t ) . Therefore, the jumps of z H a(z,t ) ,which correspond to the shocks of z H u(z,t ) , also correspond to the macroscopic clusters of particles (clusters of positive mass) present in the system at time t . Actually, a jump of a(., t ) at z corresponds exactly to a macroscopic cluster located in z, whose mass is
given by a ( z ,t ) - a ( z - , t ) ,where the notation a ( z - , t ) refers t o the left limit of a(., t ) a t z. The velocity V of this cluster is enforced by the conservation of momentum a(x,t) 2z - a ( z ,t ) - a ( z - , t ) u ( z ,0) dz =
2t V = 451 t ) -I4 z - >t ) aS( z - , t ) In the special case where z H a ( z ,t ) is a step function, we say that the shock structure is discrete at time t. The path z H u ( z , t )is then shaped as a toothpath made of pieces of line of slope l / t separated by negative
jumps (shocks). In terms of ballistic aggregation, a discrete shock structure corresponds to a state of the system where all particles have clumped into macroscopic clusters, whose locations form a discrete sequence on the real line. From a geometrical point of view, the shock structure is discrete if and only if the convex hull ?it of z H W ( z ) &z2 is piecewise linear. It is convenient in this case to introduce the so-called &-parabolic hull Pt of the initial potential W , defined by
+
When the convex hull ?it is piecewise linear, the parabolic hull Pt is made of pieces of parabola. Indeed, to a linear piece of X t with slope X / t , say ( z H 5 X z k ; a I z 5 b ) , corresponds a piece of parabola of Pt
+
165
with leading coefficient -$ and vertex of abscissa X . A moment of thought then shows that there is a one-to-one correspondence between the (pieces of) parabolas of 'Pt and the macroscopic clusters present in the system of ballistic aggregation a t time t. Indeed, to a parabola of ?t corresponds a cluster whose location X is given by the abscissa of the vertex of the parabola. Consider the two extremal contact points between this parabola and the initial potential W . Then, the distance between the abscissae of these contact points gives the mass of the cluster, whereas the slope of the segment linking these two points coincides with its velocity, see Figure 1. The state of the system is thus completely determined by Pt.
Figure 1. Geometrical interpretation of a shock
Finally, we emphasize that the above analysis still makes sense when the initial condition u(., 0) is not a real function, but is only the derivative in the sense of Schwartz of an initial potential W fulfilling condition (2). The solution u(., t ) is then a real function at any time t > 0 and when t -+ 0+, it converges in the sense of Schwartz to u(., 0), which is still said to be the initial condition. The white noise initial condition is to be understood in this sense.
3. Parabolic hull of a Brownian motion According to the work of Groeneboom'' (see also Pitman23), the convex hull of a Brownian motion W is 8.5. piecewise linear. A standard applica-
166
tion of Girsanov Theorem shows that this property still holds for the convex hull of a Brownian motion with parabolic drift, see Groeneboomlg and also Avellaneda & E4.
Theorem 3.1. The convex hull 7tt of a (two-sided) Brownian motion with parabolic drift (Wz &t2;t E R) is piecewise linear with probability one.
+
Recall that when the convex hull 7 t t is piecewise linear, the &-parabolic hull of W is made of pieces of parabola. We can index these pieces of parabola by Z, with indices increasing from left to right and the convention that parabola number 1 is the first parabola whose vertex is located at the right of 0. We write X, for the abscissa of the vertex of the piece of nt h parabola and also Mn-l and M, for the abscissae of its end-points; see Figure 1. One may notice that, in the notation of the previous section,
M,
= .(X,,t).
Mn)nE~. The parabolic hull Pt is fully determined by the sequence (X,, A characterization of the distribution of this sequence can be easily derived from the work of Groeneboomlg on Brownian motions with parabolic drift. It involves the Laplace transform C(X) of the integral of a Brownian excursion e of duration 1. According to Groeneboom's formula (see l9 Lemma 4.2.(iii))
n=l =
IE (exp
(-A
Jiu'
e, d s ) )
for X > 0, where 0 > -w1 > -w2 > . . . denotes the zeros of the Airy function A i (see on p 446). We also introduce, following Groeneboom's notations, the function g : R + Rf defined by its Fourier transform
Theorem 3.2. The sequences ((0, Mo), ( X n ,Mn)n>l} and ((0, Mo), (X-n+l, M-n)n>l} are two Marlcov chains, independent conditionally on Mo , with transitions given by
167
(an-i - z),
3
') dz,dm,.
- (an-l - zn-i)
6t2
(5)
Moreover, the law of MO is given by 1 P(Mo E da) = -92 5 / 3 t 2 / 3 ( - ( 2 t ) - 2 / 3 a ) g ((2t)-2/3a da.
1
This result has been recently recovered by F'rachebourg and Martin14. It is known that the "excursions" of the Brownian motion above its convex hull are distributed, conditionally on its convex hull, as independent Brownian excursions, see Groeneboom" and Pitman23. The next theorem states a similar path decomposition of the Brownian motion conditionally on its parabolic hull, see l5 for proof. We write elrn] for a Brownian excursion of duration m and
{
a(m) = min h e i r n I ; z
10,
m/} ~ ( m=) right-most location of this minimum. E
Theorem 3.3. The "excursions" of the Brownian motion above its parabolic hull Pt
€("I = (W(Mn-l
+ z) - Pt(M,-1 + z); 0 <_ z <_ Mn
-
Mn-
I)
are independent conditionally on Pt, with as conditional law, the law v(mn,t) of
(
2
H
1 eirnnl - 2tz(mn - X ) I a(mn) 2 I/t
)
where m, = Mn - M,-1.
Remark: A straightforward application of Girsanov Theorem shows that the law v ( m ,t ) is absolutely continuous with respect t o the law P[rn]of dml. Actually,
(- $ STeiml d z ) IE (exp (-$ S r eiml d z ) ) exp
d v ( m ,t ) =
dPIrn]
The law of the variables u(m) and q ( m ) plays a key role in the analysis of Burgers turbulence with white noise initial data. It is specified in the next theorem, in terms of the function C defined above. See l6 for proof.
168
Theorem 3.4. The scaling property of Brownian excursions entails the identity in law ( a ( m )rl(m)) , kW( m - 3 / 2 a ( 1 ) ,
.
For any a > 0 and 0 < x < 1, the probability density function of ( a ( l ) , ~ ( lis) )given by e-a2/24
P(a(1)E da, ~ ( 1E) dx) =
diG$Tj
C (ax3/') C ( a ( 1 - x ) ~ /dadx. ~ )
4. Burgers turbulence with white noise initial velocity
In this section, we turn our attention t o the solutions of Burgers equation ( 1 ) with initial condition u(.,O) distributed as a white noise. In other words, we consider an initial potential (W,; 5 E R) distributed as a twosided Brownian motion. We first describe the solution a t a fixed time t > 0, and then focus on its time-evolution. 4.1. S t a t e at a fixed t i m e t
>0
According to Theorem 3.1, when W is distributed as a Brownian motion, the convex hull of the path x H W, $x2 is piecewise linear with probability one. As a consequence (see Section 2), when u(.,O) is a white noise, the shock structure is discrete a.s. We recall that in this case, the solution x H u ( x , t ) is a toothpath, fully determined by the sequence ( ( X n , M n ) ;n E Z) described in Theorem 3.2. Indeed, X , gives the location of the nth shock a t the right of the origin, and ( M , - M,-I)/t the strength of this shock. In terms of ballistic aggregation, the state of the system is the following. All particles have a.s. clumped into macroscopic clusters located at (X,; n E Z), with masses and velocities given by (m, = M, - M,-l; n E Z) and
+
2Xn (vn =
-
M , - Mn-l 2t
Besides, it has to be mentioned that the scaling property of the white noise propagates t o the turbulence and induces the identity in law (see e.g. 4 ) ,
( u ( x t, ) ;x E R) 'EW(t-'/3u xt-2/3 1
(
1 ) ;
xE
R
).
169
4.2. T i m e evolution of the turbulence The previous section gives a complete description of the state of the turbulence at a fixed time t > 0. The natural question is now t o understand its time evolution. I t will be convenient in this view to use the ballistic interpretation of the turbulence. As time runs, the clusters present in the system aggregate according to the dynamics of sticky particles. This clustering is deterministic, because so are the dynamics. Clearly it induces a loss of information in the sense that we cannot recover the state of the system at a time tl from the state of the system at a time t 2 > t l . Suppose now that time runs backwards. Then, clusters dislocate and due to the loss of information, dislocations occur randomly. If we do understand how a cluster breaks into pieces in backwards times, then we will understand how it did aggregate in forwards times. Roughly, in this subsection we will answer the question: what does the genealogical tree of a given cluster look like?
Figure 2.
Genealogical tree of a Cluster
Henceforth, we focus on the fragmentation of the clusters in backwards times. The next theorem specifies the parameters on which the fragmentation of a cluster depends.
Theorem 4.1. Conditionally o n the state of the system at time t , each cluster present at time t breaks into pieces independently of the others, and according t o a conditional law only depending on its mass and t i m e t . Physically, the independence of the fragmentation of a cluster from its location and velocity may be viewed as a consequence of the invariance of the system under translation and Galilean transformations. The fact that it does not depend of the other clusters may be understood as follow. Consider at time 0 two (infinitesimal) particles, which belong at time t to two
170
different clusters. These two particles cannot interact up t o time t , else they would stick and belong t o the same cluster. Therefore, the particles which made up a cluster a t time t cannot interact before time t with the other particles. Since, in addition, the initial velocities of the particles are uncorrelated, the aggregation processes of the clusters are expected to be independent. Proof: We only sketch the proof of Theorem 4.1, and refer to l5 for details. The main point is to translate the fragmentation of the clusters in terms of the parabolic hull of the initial potential W . Recall there is a one-toone correspondence between the clusters present a t time t in the system and the (pieces of) parabolas of the ¶bolic hull of the initial potential. Consider a given cluster a t time t and its corresponding parabola with leading coefficient At time s < t , its corresponding parabola of the &-parabolic hull of W is stretched in the vertical direction, since its leading coefficient is larger. Let time s decrease from t to 0. The parabola corresponding t o the cluster gets more and more stretched, up to a time t* < t where it enters into contact with the initial potential W . This time t* corresponds to the time a t which the cluster splits into two clusters. Let time s decrease further. We now have two parabolas corresponding t o the two clusters. They are stretched in the vertical direction, up to the moment where one of them touches W a t a new point, and also splits into two new parabolas, giving at all three parabolas/clusters. And so on.
-A.
-&
2->-
Figure 3.
Time t' of splitting.
A moment of thought thus shows that the fragmentation of a given cluster a t time t only depends on the "excursion" E of the initial potential W above the parabola corresponding to the cluster. When W is distributed as a Brownian motion, it follows from Theorem 3.3 that conditionally on the state of the system at time t , each cluster breaks into pieces independently of
171
the others. Moreover, since the conditional law of & given Pt only depends on time t and the mass m of the cluster, the fragmentation of the cluster only depends on m and t , and not on its velocity or location. According to the previous theorem, we can focus on a single cluster of mass m at time t. We now turn our attention to its first splitting.
Theorem 4.2. With probability one a cluster splits into exactly two clusters at its first splitting. The law of the time t* of the splitting of a cluster of mass m at time t and of the mass m* of the left-most cluster arising from this splitting is given b y
P(t* E ds,m* E d m l )
for ( s ,m i ) E ] O , t [ x ] O , m [with , the notation bY (3). Moreover, we have for 0 < s < t
m2
= m - ml and C defined
We refer to l 5 for numerical illustrations of these laws. Proof: We write as before & for the "excursion" of the initial potential W above the parabola corresponding to the cluster at time t . Recall from the proof of the previous theorem that the time t* corresponds to the time a t which the parabola enters into contact with the initial potential a t a new point. When the initial potential is distributed as a Brownian motion, the cluster splits a s into two clusters, because the parabola enters as. into contact with the Brownian motion at a single new point, see l5 for proof. The location of this contact point gives the distribution of mass between the two new clusters. Indeed, it should be plain from the mechanism described above that l / t * and m* correspond to the maximum and the location of the maximum of
When W is distributed as a Brownian motion, the conditional law of & given Pt is u(m,t ) . Therefore, l / t * and m* are distributed as the variables a ( m ) and v(m)conditioned by {a(m)2 l / t } . Formulaes ( 6 ) and (7) follow thus from Theorem 3.3. I
172
The previous result depicts the first splitting. Combined with a Markov property at the times of fragmentation (see 1 5 ) , it yields a complete description of the fragmentation of a cluster. This description can be formulated as follows. We write ml, . . . , m k for the masses of the clusters resulting at time s = t - r of the fragmentation of a cluster of mass m at time t. The mass ml refers to the mass of the left-most cluster, the mass mk to the one of the right-most cluster. We write also
Theorem 4.3. The process ( r H M("it)(r);0 < r < t ) is a pure-jump (inhomogeneous) strong Markov process, with rate of jump at time r
1
M ( m ) t ) (+r h) = ( m l , .. . , mi,l,mi,2,. . . , m k ) M(mit)(r)=
with the function C defined b y (3) and
A2
( m l ,. . .,mi,. . . ,m k )
= mi - XI.
We refer t o l5 for the proof of the Markov property and l6 for the computation of the rate of jump. We end this section with a remark about the dynamics of fragmentation. The property stated in Theorem 4.1 bears the same flavor as the so-called fragmentation property considered by Aldous', PitmanZ4and Bertoin7. Nevertheless, the fragmentation process r ++ M("lt)(r)we study here is not homogeneous in time and therefore differs from those considered by Aldous et al. Besides, a cluster of mass m at time t statistically breaks into pieces in the same way as a cluster of mass mt-'l3 at time 1. This permits us to associate a time homogeneous Markov process to r H M ( m , t ) ( r ) . Indeed, the process := t-2/3e2"/3M(m,t)(te-s), s E Rf
fi(Wt)(')
is a time homogeneous strong Markov process, whose dynamic can be depicted as follows. Each cluster making up M("st) grows deterministically as s H e2s/3 and also splits randomly, independently of the others, according to the fragmentation rate
F(A1,x
c (A;y) c ((A - X 1 ) 3 / 2 )
A3/2 -
A,)
X
= J87rA1
(A - A,)
C(A39
173
5. Burgers turbulence with some other initial velocities of white noise type In this section, we consider other initial conditions of white noise type for equation (1). We outline in Section 5.1 the main properties of the solution of Burgers equation (1) with as initial condition u(., 0), a white noise on Rf and 0 on R-.In Section 5.2, we depict the case where u(., 0) is a periodic white noise. We omit the proofs.
5.1. The one-sided white noise case In this subsection, we deal with the initial condition U(.,O)
=
on ] - m,O] white noise on 10, m[ .
In terms of ballistic aggregation, such an initial condition arises a t the hydrodynamic limit of the following system. At time t = 0 the sticky particles are spread uniformly on Z; those on the right of the origin receive random i.i.d. velocities (with finite variance), whereas those on the left of the origin stay a t rest. The phenomenon of main interest here is the propagation to the left of the chaos initially located on the right of 0. The solution z ++ u(x,t ) has a shock front, which travels to the left as time t runs. At the left of this shock front u ( . , t ) equals 0, whereas a t its right z H u(x,t)is a s . a toothpath, made of pieces of line of slope l / t separated by a discrete sequence of shocks, see Figure 4. The location X , and the strength M n / t of the nth shock at the right of the shock front form a Markov chain, with transitions given by (5). We write henceforth xt and Mt for the location and t times the strength of the shock front.
Figure 4. Shape of x
H
u ( x ,t ) .
It is convenient to use the ballistic description of u ( . , t ) . There exists a so-called front cluster, travelling to the left, on the left of which
174
there are infinitesimal particles a t rest. On its right, all particles have clumped into macroscopic clusters, whose locations and masses are given . location and the mass of the front cluster correspond by (XnlM n ) n E ~The t o xt and Mt.
Figure 5 . Shape of the system of sticky particles.
The first property to mention about the shock front is the time-scaling identity in law
(xt,M t )
’aw(t2l3x1,t2l3M1).
This property originates from the scaling property of the white noise and permits t o focus on time t = 1. The second property to be noticed, is that the shock front is completely described a t time t = 1 by the variables z1 and M I . Indeed, according to the conservation of mass and momentum the velocity & of the shock front is given by V1 = - ; M I . This equality can be extended at any time t > 0 by
It is an easy task t o derive from the work of Groeneboomlg the law of ( X I , M I ) ,in terms of the function g defined by (4) and the function h(m,.) : R+ 4 R+ defined by the series
h(m,x) = 2 l l 3
O0
n=l
A i (2’I3m - wn) exp Ai’ (-tun)
(
-21/”zWn)
where, as before, 0 > -w1 > -w2 > ... represent the zeros of the Airy function A i ranked in decreasing order. See l7 for proof and also the law of x1 alone.
Theorem 5.1. In the above notation, the law of ( x 1 , M l ) is given by
for M ,x
> 0.
175
We now turn our attention to the time-evolution of the shock front. It is conspicuous from the ballistic description of the system, that the dynamics of the shock front are governed by two phenomena. First its movement to the left is continuously slowed down by the infinitesimal particles a t rest on its left. Second, macroscopic clusters on its right sometimes catch it and then increase sharply its velocity. We are mainly interested by the evolution of the location xt of the shock front. The identity (8) suggest that xt behave roughly as t H -t2l3. But we stress that the identity (8) is only true for a fixed time t > 0 and therefore does not give the time-evolution of t H xi. The identity (9) implies the equality
so that the evolution of the shock front can be fully expressed in terms of the process t H Mt,which is characterized in the following theorem.
Theorem 5.2. T h e process t H G t := t-'l2 Mt i s a pure-jump inhomogeneous and increasing Markov process, with rate of j u m p kt+h - MfE
for any M , m, t
d m I kt = M>
> 0.
We can also give the asymptotic behaviour o f t time t Proposition 5.1. W h e n t i m e t tends t o 0 or one the asymptotics
03,
H
xt for small and large
we have with probability
Some other aspects of the solution u(., t ) have also been investigated. The main contributions are perhaps the description of the flux of particles crossing a given point and the study of the different scaling regimes of the solution by Frachebourg, Jacquemet and Martin13, see also '. Besides, it can be noticed that the genealogy of a macroscopic cluster present in the system, is statistically the same as the genealogy considered in Section 4. Finally, we mention the work of Tribe & ZaboronskiZ5 and also of Frachebourg et
176
a l l 3 in the case where the initial condition is given by a white noise on a finite interval, and 0 elsewhere.
5.2. The periodic white noise case We focus henceforth on the solution of Burgers equation (1) with initial condition u(.,O) distributed as a periodic white noise. In other words, we consider the case where the initial potential W is 1-periodic and is distributed on [0,1] as a Brownian bridge of duration 1. Since the solution x H u(x,t ) is also 1-periodic at any time t > 0, we can focus on a period. It is convenient for investigating such a solution to use the ballistic description of J: w u(z,t ) . The system of sticky particles associated to u(., t) is 1-periodic and can therefore be thought of as a circular system, corresponding to the hydrodynamic limit of the following system. Consider at time t = 0, N particles uniformly spread on the unit circle, with random N angular velocities ( W ~ ) I , Ni.i.d., of finite variance and fulfilling wi = 0. Then, let the system evolve according to the next dynamic. Between collisions the particles evolve on the circle with constant angular velocities and when some particles meet, they merge into a new particle with conservation of mass and momenta. As before, the shock structure of u(., t ) is discrete a s . at any time t > 0. From a circular point of view it means that all particles have clumped into a finite number of macroscopic clusters. Moreover, it can be shown that when time t tends to 03 there remains a s . a single cluster of mass 1 and velocity 0. Its location follows the uniform law on the circle. The genealogy of this final cluster is distributed according to the law of the genealogy of a cluster of mass 1 at time t in Section 4, in the limit t -+ 03. This permits to compute the probability density of a given state in terms of the function C defined by (3). Indeed, the probability density to have a t time t exactly N clusters of mass ml, . . . , m N (fulfilling ml . . . m N = 1) located at 81 < ‘ < 8~ equals
+ +
where 4 is a completely determined ” polynomial-like” function of (mi, & ) l , N , see l6 Section 4 Proposition 1. Since the formula of 4 is somewhat complicated, we refer to l6 for its very definition.
177
6. Some open problems
To conclude we mention some open problems. Many questions on the onedimensional Burgers turbulence remain open. For example, concerning the periodic case, it would be interesting t o obtain a simple formula for the law of the number N of clusters present at time t . For more general initial conditions, we may wonder whether it is possible t o extend some of the above results (see for a discussion in the stable noise case)? Yet, going in higher dimensions appears now as the most challenging problem in Burgers turbulence, see Vergassola et al.26 for motivations and simulations. Besides, for a better understanding of the phenomenon of turbulence, it would be intersting t o exhibit some statistical properties of the solution of PDE’s of fluid mechanics (especially of Navier-Stokes equation), with random initial conditions.
References 1. M. Abramowitz, I.A. Stegun: Handbook of mathematical functions. Washing-
ton: Nat. Bur. Stand. 1964 2. D. Aldous: Deterministic and stochastic models for coalescence (aggregation, coagulation): review of the mean-field theory f o r probabilists. Bernouilli 5 (1999), pp 3-48. 3. M. Avellaneda: Statistical properties of shocks in Burgers turbulence II: tail probabilities for velocities, shock-strengths and rarefaction intervals. Comm. Math. Phys. 169 (1995), pp 45-59. 4. M. Avellaneda and W. E: Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys. 172 (1995) pp 13-38 5. J. Bertoin, Clustering statistics for sticky particles with Brownian initial velocity. J. Math. Pures Appl. 79 no 2 (2000), pp 173-194. 6. J. Bertoin, Some properties of Burgers turbulence with white or stable noise initial data. In LBvy Processes : Theory and Applications. Eds BarndorffNielsen, Mikosh et Resnick. Birkhuser (2001). 7. J. Bertoin Homogeneous fragmentation processes. Probab. Theory Related Fields 121 (2001), no. 3, pp 301-318 8. J. Bertoin, C. Giraud, Y. Isozaki: Statistics of a flux in Burgers turbulence with one-sided Brownian initial data. Cornrn. Math. Phys. 224 (200l), pp 551-564 9. Y. Brenier, E. Grenier: Sticky particles and scalar conservation laws SIAM J. Numer. Anal. 35 No 6 (1998), pp 2317-2328. 10. J.M. Burgers: The nonlinear diffusion equation. Dordrecht, Reidel 1974 11. J.D. Cole: On a quasi linear parabolic equation occuring in aerodynamics. Quart. Appl. Math. 9 (1951), pp 225-236 12. W. E, Ya.G. Rykov, Ya.G. Sinai: Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys. 177 (1996), pp 349-380
178
13. L. Frachebourg, V. Jacquemet, Ph.A. Martin: Inhomogeneous ballistic aggregation. J. Statist. Phys. 105 (2001), no. 5-6, pp 745-769 14. L. Frachebourg, Ph.A. Martin: Exact statistical properties of the Burgers equation. J. Fluid. Mech. 417 (2000), pp 323-349 15. C. Giraud: Genealogy of shocks in Burgers turbulence with white noise initial velocity. Comm. Math. Phys 223 (2001), p. 67-86. 16. C. Giraud: Statistics of the convex hull of Brownian excursion with parabolic drift. preprint (2002). 17. C. Giraud: On a shock front in Burgers turbulence. preprint 2002. 18. P. Groeneboom: The concave majorant of Brownian motion. Ann. Probab. 11 no 4 (1983), pp 1016-1027 19. P. Groeneboom: Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 (1989), pp 79-109 20. E. Hopf: The partial differential equation ut uuz = pZz. Comm. Pure Appl. Math. 3 (1950), pp 201-230 21. R.H. Kraichnan Lagrangian history statistical theory for Burgers’ equation. Phys. Fluids 11 (1968), pp 265-277 22. N. Leonenko: Limit theorems for random fields with singular spectrum. Kluwer, 1999 23. J . Pitman: Remarks on the Ccnvex minorant of Brownian motion., Seminar on Stochastic Processes (1982), Birkhauser, Boston. 24. J . Pitman, Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, pp 1870-1902 25. R. Tribe, 0. Zaboronski: On the large time asymptotics of decaying Burgers turbulence. Comm. Math. Phys. 212 (2000), pp 415-436 26. M. Vergassola, B. Dubrulle, U. Frisch and A. Noullez: Burgers’ equation, devil’s staircases and the mass distribution for large-scale structures. Astron. Astrophys. 289 (1994), pp 325-256 27. W.A. Woyczyhski: Gottingen lectures on Burgers-KPZ turbulence. Lecture Notes in Math. 1700, Springer 1998. 28. Ya.B. Zeldovich: Gravitational instability : an approximate theory for large density perturbations. Astron. Astrophys. 5 (1970), pp 84-89.
+
DETERMINISTIC VISCOUS HYDRODYNAMICS VIA STOCHASTIC PROCESSES ON GROUPS OF DIFFEOMORPHISMS
Y. E. GLIKLIKH Mathematics Faculty Voronezh State University Universitetskaya p l . , 1 394006 Voronezh, Russia E-mail: [email protected]
The flow of viscous incompressible fluid on an n-dimensional flat torus is presented a s the expectation of a certain stochastic process on the group of diffeomorphisms of the torus. The above-mentioned process is governed by a stochastic analogue of the second Newton’s law subjected to the mechanical constraint that garantees incompressibility. The diffusion term of the process is connected with viscosity coefficient of the fluid. The constraint is given in invariant geometric terms, the Newton’s law is formulated in terms of Nelson’s mean backward derivatives. The Navier-Stokes equation is derived as an Euler type equation in “algebra” of the group. The construction is translated into the finite-dimensional language of processes on the torus (as far as it is possible). Relations with some other stochastic approaches to viscous hydrodynamics is discussed.
1. Preliminaries and Introduction
The paper is devoted to the approach to hydrodynamics in terms of geometry of groups of diffeomorphisms, suggested for perfect fluids by Arnold and Ebin and Marsden ‘. In previous papers by the author it was found that the adequate description of viscous fluids in this language requires involving stochastic processes (see, e.g., and s). In particular, the second Newton’s law on the groups of diffeomorphisms, used in the case of perfect fluids, is replaced by its special stochastic analogue in terms of Nelson’s mean derivatives. Here we engage some additional geometric machinery that provides clear finite-dimensional interpretation of the construction. Consider a stochastic process [ ( t ) in Rn, t E [O,Z], given on a certain probability space (0,F ,P) and such that [ ( t )is an L1-random variable for all t. The ”present” (”now”) for [ ( t )is the least complete cT-subalgebra Nf of 3 that includes preimages of Bore1 set of R” under the map [ ( t ) : R + 179
180
R". We denote by E i the conditional expectation with rcspcct to Nf. Below we shall most often deal with the diffusion processes of the form
in R" and flat torus Inas well as natural analogues of such processes on groups (infinite-dimensional manifulds) of diffeornorphisms. In (1) w ( t )is a Wiener process, adapted to ( ( t ) ,a ( t ,x) is a vector field and 0 > 0 is a real constant. Following Nelson (see, e.g., - 11) we give the next:
Definition 1.1. (i) The forward mean derivative D l ( t ) of process e ( t ) a t t is the L1-random variable of the form
where the limit is supposed to exist in L1(R, F, P ) and At 4 f O means that A t 0 and A t > 0. (ii) The backward mean derivative D,<(t) of ( ( t ) at t is L 1 - I a ~ ~ d o ~ ~ ~ variable ---f
where (as well as in (i)) the limit is supposed to exist in L1(R, F, P ) and At +O means the same as in (i). ---f
Notice that generally speaking D [ ( t ) # D*((t) (but, if [ ( t ) a.s. has smooth sample trajectories, those derivatives evidently coincide). jFrom the properties of conditional expectation it follows that D ( ( t ) and D * ( ( t ) can be represented as compositions of l ( t ) and Bore1 measurable vector fields
on R" (following Parthasarathy we call them the regressions): D ( ( t ) Y o @l ,( t ) )and D*r(t)= Y,O(t,l ( t ) ) .
=
Lemma 1.1. For a process of type (1) D [ ( t ) = a ( t ,( ( t ) )and so Y o ( tx) , = 4tl
x).
181
See details of proof, e.g. in and '. Mean derivatives of Definition 1.1 are particular cases of the notions determined as follows. Let x ( t ) and y ( t ) be L1-stochastic processes in F defined on ( R , 3 , P). Introduce y-forward derivative of x ( t ) by the formula
x(t D Y x ( t )= lim E,Y( At-+O
+ At)
-
x(t)
At
1
(5)
and y-backward derivative of x ( t ) by the formula
x ( t ) - x ( t - At) D,Yx(t)= lim E,Y( 1
at
At++O
where, of course, the limits are assumed to exist in L1(R, 3,P). Let Z ( t ,x) be C2-smooth vector field on R".
Definition 1.2. L1-limits of the form
are called forward and backward, respectively, mean derivatives of Z along
I(.)a t time instant t . Certainly D Z ( t , J ( t ) )and D * Z ( t , [ ( t ) )can be represented in terms of corresponding regressions, defined analogously to (4). If it does not yield a confusion, we shall denote those regressions by D Z and D , Z .
Lemma 1.2. For process ( 1 ) an R" the following formulae take place: DZ
=
a -2 at a
D*Z = - Z + at
o2 + (YO.0)Z+ -v2z, 2
(9)
o2
(Y*".V ) 2 - - P Z , 2
(A,
where V = ..., &), V2 i s the Laplacian, the dot denotes the scalar product in Rn and the vector fields Y o ( t x) , an,d Y f ( t ,x) are introduced in
(4). The main idea of description of viscous hydrodynamics in the language of mean derivatives is as follows. For the sake of convenience we deal with fluids moving in a flat ndimensional torus I".It is the quotient space of €2" with respect the integral lattice where the Riemannian metric is inherited from Rn. Consider
182
+ 1) on I".
the vector space V e c t ( s )of all Sobolev HS-vector fields (s > Introduce the L2-scalar product in V e c t ( s )by the formula
(X, Y )=
1
< X(X), Y ( x )> CL(dx)
(11)
I n
where < ., . > is the Riemannian metric on 7"and p is the form of Riemannian volume (here it is the ordinary Lebesgue measure on 7"). Denote by p the subspace of V e c d s ) consisting of all divergence-free vector fields. Then consider the projector
P : v e c d s ) -+ p
(12)
orthogonal with respect to (11). Notice that from Hodge decomposition it follows that the kernel of P is the subspace consisting of all gradients. Thus for any Y E Vect(s)we have
P ( Y )= Y
- gradp
(13)
where p is a certain HS+l function on I" that is unique to within the constants for given Y . Let a random flow [ ( t ) be given on a flat n-dimensional torus 7".Suppose that it is a general solution of a stochastic differential equation of the type
d J ( t )= a ( s , J ( s ) ) d s
+ udw(t)
(14)
where u > 0 is a real constant. Let o , c ( t ) = u ( t , E ( t ) ) where , u ( t , z )is a divergence-free vector field on I", C1-smooth in t and C2-smooth in m E 7".Suppose that [ ( t ,x) satisfies the relation
(15)
PD*D*J(t)= F ( t l< ( t ) ) ,
where F ( t ,x) is a divergence-free vector field on 7".Taking into account formulae (10) and (13), we obtain
d P D * D * [ ( tx) , = P(-u at
a + (ulV)u at
= -u
+ (21,V)u- -0%) 2 U2
U2
-
-V2u - gradp.
2 Thus (15) means that the divergence-free vector field u ( t l x )satisfies the relation
a + (u,0 ) u at
-u
U2
-
-V2u 2
-
gradp
= F,
that is the Navier-Stokes equation with viscosity
F ( t ,XI.
$ and
external force
183
We interpret (15) as a stochastic analogue of the second Newton's law on the group of Sobolev diffeomorphisms D ' ( 7 " ) of the torus, subjected to a certain mechanical constraint expressed in geometrically invariant form. In spite of the fact that the constraint is holonomic (i.e., integrable), we do not restrict the consideration to its integral manifolds. This allows us to apply both finite and infinite-dimensional language to the investigation more easily. Involving constraints is a new point of our presentation.
2. Basic notion from the geometry of groups of
diffeomorphisms Consider a flat n-dimensional torus I" as in 51. The tangent bundle t o = 7"x R" and so any tangent space to T I " admits the decomposition T(,,x)TIn= R" x R" where the first multiplier, called horizontal (denote it by H(,,x)), is tangent t o I" and the second one, called vertical (denote it by V,,,,)), is tangent to R". The family of subspaces H(,,x) in all tangent space T(,,x)TI" is a flat connection on the torus. Introduce the Riemannian metric < ., . > on I" such that given X , Y E T m I n the value < XIY > is their ordinary scalar product in R". This metric is called flat and I" with this metric is called the flat torus. Everywhere below we deal with the flat torus. Notice that both 'H(z,x)and V(,,X) are isomorphic to Tm7" (here all three spaces are canonically isomorphic to R", see above). Thus we can send any vector X E Tm7" into 'H(,,x) and into V ( , , X ) . The former is called the horizontal lift of X and denoted by XT while the latter is called the vertical lift of X and denoted by X 1 . The same notations will be in use for the groups of diffeomorphisms below. Thus there is a natural map K : T T I " 4 T I " that sends the vector Y E T(,,x)TIn into the second factor in T(,,x)TIn = R" x R", i.e., K : T(,,x)TI" = 'H(,,x) x V(,,x) + V(,,X) = R" = T m I n . This map is called the connector. The connection 'H is its kernel. At any point ( 5 ,X ) E T I " consider the vector 2 ( , , ~ that ) belongs to ?t(,,x) and satisfies the relation T7r2(,,x) = X , E T m I n where 7r : T I " 4 I"is the natural projection and TT : T T I " -+ T I " is its tangent map. For the flat torus, taking into account the above decomposition of the second tangent space, the vector 2(,,x) is described as 2(,,x) = ( X , O ) E 7i(,,x) x V(,,X).The vector field 2 on T I " is called the geodesic spray of the connection. Consider the set D " ( 7 " ) of all diffeomorphisms of 7"belonging to the Sobolev space H S , s > $n 1. Recall that for s > $n 1 the maps from
I" is trivial: T I "
+
+
184
H" are C1-smooth. There is a structure of a smooth (and separable) Hilbert manifold on D"(7")as well as the natural group structures with the composition involved as multiplication. A detailed description of the structures and their interconnections can be found in 6. Note that at the unit element e = id the tangent space T e D s ( 7 " )= Vecd") (see above). As above denote by ,Ll its subspace consisting of all divergent-free vector fields on 7"belonging to H". The space T f D " ( ( I " ) ,f E D S ( I n ) ,consists of the maps Y : I" + T M such that .rrY(z)= f(z). Obviously for any Y E T f D 5 ( I n )there exists unique X E TeD"('Tn) such that Y = X o f . In any T f D " ( I " ) we can define the L2-scalar product in analogy with (11) by the formula
(X, Y ) f=
1
I"
< X ( z ) ,Y(,)
>f(.)
ll(dz).
(18)
The family of these scalar products form the weak Riemannian metric on D s ( I n ) (it generates the topology, weaker than H ' ) . The right-hand translation R f : D"(I") + D"(I"), Rf o 0 = 0 o f , 8 , f E V s ( I n )is , Cw-smooth and thus one may consider right-invariant vector fields on D s ( I n ) . Note that the tangent to right translation takes the form: T R f X = X o f for X E T D " ( I n ) . A right-invariant vector field X on D i ( 7 " ) generated by a vector X E T e D " ( l n )is C'-smooth iff the vector field X on 7"is Hs+'--smooth This fact is a consequence of the so-called w-lemma (see 6 , and it is valid also for more complicated fields. For example, if a tensor (or any other) field on 7"is Cw-smooth, the corresponding right-invariant field on D s ( 7 " ) is Cw-smooth as well. One can easily check that the second tangent bundle T T D ' ( 7 " ) consists of H" maps from 7"to TT7" with additional properties that they are projected into maps from D S ( 7 " ) . Thus we can apply the connector K : TT7" -+ T I " of introduced above to obtain the connector on TTV"(7")by the formula
K : TTD"(7") 4 TD"(7").
(19)
The family of its kernels in second tangent spaces form the connection on D " ( I " ) ,denoted by 7-1. is described as follows: The geodesic spray 2 of
2(X)=2oX (20) for X E T V S ( I n )where , 2 is the geodesic spray of the connection 'Ft on
I" (see above). One can easily obtain from (20) the following statement:
185
2 is V s(In)-right-invariant and C"-smooth on TDs(7"). Introduce the subspace p f c TfDs('Tn)as TRfP. Thus we obtain the smooth subbundle p of T P ( 7 " )that will play the role of constraint below. Consider the map P : TDS(7")+ 6 determined for each f E D S ( 7 " )by the formula
Pf = T R f o P o T R f ' .
Be
where P = P, : V e c d " ) = T e D S ( I n )--f /3 = is the projection introduced in (12). It is obvious that P is D;(I")-right-invariant. There is an important and rather complicated result (see 6 , that P is C"-smooth. Construct the vector field S on the manifold by the formula
p
S ( X ) = T B ( 2 o X ) , X E p. Since P and 2 are P ( P ) - r i g h t - i n v a r i a n t and C"-smooth it evidently follows from (21) that so is S. Introduce the operators:
B :TI"
--f
(21)
on T D S ( 7 " ) ,
R",
the projection onto the second factor in 7"x R";
A ( z ) : R"
4
Tm7",
(22)
the converse to B linear isomorphism from Rn onto the tangent space to m E I", and
I"at
Qg(z) =
A ( g ( z ) )0 B
(23)
where g E D s ( I " ) , m E 7". For a vector Y E T f D s ( I " ) we get QgY = A ( g ( z ) )0 B ( Y ( z ) )E In particular, Q,Y E V e c t ( " ) . Notice T,DS(7") for any f E Ds(In). that for Y E the vector QeY may not belong to Be. The operation Q , is a formalization for D s ( I " ) of the usual finite-dimensional operation that allows one to consider the composition X o f of a vector X E V e c d s ) and diffeomorphism f E D s ( l n ) as a vector in V e c t ( s ) .It denotes the shift of a vector, applied at the point f(z),to the point z with respect t o global parallelism of the tangent bundle to torus. The map A has the following property. For the natural orthonormal frame b in R" we have an orthonormal frame A,(b) in T , P , the field of frames A(b) on T7" consists of frames inherited from the constant frame b. Thus for a fixed vector X E R" the vector field A ( X ) on I nis constant (i.e., it is obtained from the constant vector field X on R" and has constant coordinates with respect to A(b))and in particular A ( X ) is Coo-smooth and
pf
186
divergent-free since such is the constant vector field X on R". So, A may be considered as a map A : R" p = 0, c T e D s ( l n ) . Consider the map A : D'(7") x R" -+ T D S ( I n )such that A, : Rn 4 T , D S ( 7 " ) is equal to A, and for every g E D'(7") the map A, : R" -+ T g D S ( 7 " )is obtained from A, by means of the right-translation: -+
A,(X)
= TR,A,(X) =
( Ao g ) ( X ) .
Since A is Cm-srnooth, it follows from w-lemma that A is Cm-smooth jointly in X E R" and g E D'(7"). 3. Description of viscous hydrodynamics
For the sake of simplicity of presentation, in this section we suppose s > 2. This means that H S vector fields on 7"are a t least C2. We shall deal with It6 type equations on D'(7"). We refer the reader to 3 , and for global geometric-invariant constructions of such equations on manifolds suggested by Belopolskaya and Daletsky in terms of exponential are maps of connections (in particular, in and equations on Ds(ln) considered). Local presentation in charts of those equations are known as the Baxendale form of It6 equations. In, e.g., and it is shown that Lemma 1.1 is true for It8 equations in Belopol'skaya-Daletsky form and so this is an adequate machinery for working with mean derivatives. Since the connection on D'(7") is generated by the flat connection on the torus, the corresponding exponential map is like that on a linear space. So, without loss of generality we use the notations, usual for It6 equations in linear spaces. Below we consider a certain equation on the manifold in general form with respect to the exponential map of some special connection. Let a ( t , z) be a divergence-free H' vector field on 7".Denote by a(t, f) the corresponding right-invariant vector field on DS('Tn).The flow on I", generated by equation (14), is a solution of the equation
+
d l ( t ) = q t ,l ( t ) ) d t
+ cA(l(t))dw(t)
(25)
on D s(7"). Definition 3.1. If<(t) satisfies an equation of (25) type with some (maybe random) initial condition, we say that it is a process with diffusion term
aA.
t
Suppose that a process ( ( t ) with diffusion term a A is well-posed for > 0. Recall the well-known fact that the process
E [O,T]for some T
187
v ( t ) = [(T - t ) with inverse time direction has the same diffusion term aA but, generally speaking, different drift. The definition of mean derivatives for processes on V s(7") is analogous to that on R" and on 7".In order to distinguish the derivatives on Ds('Tn) and on Tnwe denote the former by D and D* while D and D , remain valid for I". The mechanical meaning of the subbundle fl is a constraint. According to the ideology of geometric description of constraints suggested by Vershik and Faddeev, we give the following
Definition 3.2. A stochastic process ( ( t ) is called forward admissible to the constraint fl if D [ ( t ) E & ( t ) a.s. for all t. A stochastic process [ ( t ) is called backward admissible to the constraint fl if D,[(t) E f l ~ (a~s .) for all t . A vector field X is called admissible, if X f E bf at any f E D'(7"). Notice that for a solution [ ( t ) of (25) we have D [ ( t ) = a ( t , [ ( t ) )(see Lemma 1.1).Thus this [ ( t )is forward admissible. Following general ideas of mechanics with constraints we can introduce the notions of covariant mean derivatives with respect to a constraint. Definition 3.3. For an admissible vector field X and forward admissible process [ ( t ) the expression P D X ( t ,[ ( t ) ) is called covariant forward mean derivative with respect to the constraint /3. For an admissible vector field X and backward admissible process [ ( t ) the expression P D , X ( t , [(t)) is called covariant backward mean derivative with respect to the constraint p. Let v(t) be a backward admissible process. Then, according to Definition 3.3, we can consider the covariant backward mean derivative PB,,D,[(t). Let F ( t , x) be a divergence-free Hs-vector field on In, i.e., it can be considered as a time-dependent vector F ( t ) E fie. Denote by p ( t ,f ) the right-invariant vector field on Vs(7") generated by F ( t ) .
Theorem 3.1. Let a process [ ( t ) on D'(7") has the diffusion term a A and let D * [ ( t )= u(t,[ ( t ) ) where G ( t l f ) is a right-invariant vector field on VS(7"), generated by a divergence-free HS-uector field u ( t , x ) on 7".If [ ( t ) satisfies the constraint Newton's law
u(t,x) on 7"satisfies Navier-Stokes equation (1 7).
188
The proof of Theorem 3.1 is reduced to the finite-dimensional arguments of 31. The divergence-free vector field u ( t ,x) on I"from Theorem 3.1, i.e., a , be obtained by right transtime-dependent vector in ,Be c T e D S ( I n )can lation of backward velocity a t e , and so the Navier-Stokes equation (17) plays the role of Euler equation in the "algebra" T , D S ( I n ) according t o general approach to Euler equations. The flow of u(t,x) on I " ,that is a curve on D s ( I n ) describing the motion of viscous incompressible fluid, may be considered as the expectation of the process [ ( t ) . So, we need to construct a backward admissible process on D s ( l n ) with diffusion term a A satisfying (26). It is a complicated problem to find a process with given backward mean derivatives. That is why we shall try to construct [ ( t ) by solving first a certain equation of (25) type and then changing the time direction in its solution. Let a process q ( t ) on D E ( I " ) be a solution of stochastic differential equation of (25) type with initial condition q(0) = e and let it exist for t from a certain non-random time interval [0,TI.Consider the process with inverse time direction [ ( t ) = q(T - t ) . Our aim now is to construct an equation for q such that (26) is fulfilled for [ ( t ) ,and D,[(t) = G ( t , E ( t ) ) where u(t,f ) is an admissible right-invariant vector field with initial condition u(0,e ) = uo E ,Be where uo = uo(x)is a divergence-free HS-vector field on I". Since the backward mean derivative for [ ( t ) is equal to the forward mean derivative for q(T - t ) with minus, we have Dq(t) = -D,[(T - t ) = -u(T - t , q ( t ) ) . Hence, taking into account Lemma 1.1 and the fact that T r S ( X ) = X and TTF'= 0, we can derive that ( ( t )will satisfy (26) if q(t) satisfies the equality
o*[(t)
+
d q ( t ) = -G(T - t , q ( t ) ) d t o A ( q ( t ) ) d w ( t ) . and the process
u(T- t , q ( t ) )in
DVG(T- t , q ( t ) )= -S(G(T
-
(27)
satisfies the equality
t ,~ ( t ) ) )F'(T - t ,G(T - t , ~ ( t ) ) (28) )
where F'(T - t , G(T - t , q ( t ) ) )is the vertical lift of F(T - t , G(T - t , q ( t ) ) ) . Denote by AT the horizontal lift of the field A onto T D ' ( 7 " ) . On there is a natural connection such that the projections of its geodesics onto D'(7") are geodesics of the connection 7? (see, e.g., '). Denote the exponential map of this connection by expT.
p
Theorem 3.2. If the process u(T - t ,q ( t ) )on in Belopols~aya-Daletskiif o r m T
p satisfies the It;
d u ( T - t , q ( t ) )= exp,(T-t,l)(t))(-S(iZ(T-t,q ( t ) ) ) d t - F ' ( t ,
equation
q(t)))dt
189
-
-T
+aA
- t , rl(t)))dw(t)),
(29)
the process q ( t ) and the right-invariant admissible vector field ii on V s( I " ) satisfy (27) and (28) and so ( ( t )= q ( T - t ) satisfies (26) and the divergencefree vector field u(t,x) on I" is a solution of (17). Theorem 3.2 follows from a statement of Lemma 1.1 type for equations in Belopolskaya-Daletskii form (see, e.g., 7, 8 ) . The next finite-dimensional interpretation makes the construction more clear. Notice that the process q ( t ) with initial condition q ( 0 ) = e on V8(In), that satisfies (27), is a random flow on I". Denote this flow by q ( t , x ) with q ( 0 , x ) = x. It is the general solution of It6 stochastic differential equation on I"
dq(t,X) = -u(T - t , q ( t ,x ) ) d t -tadw(t)
(30)
with divu(t, x) = 0, the finite-dimensional version of (27). By direct calculation of forward mean derivatives for the finite dimensional process q ( t ,x) we show that
Drl(t1.)
=
- 4 T - t177(4 x)),
P D D q ( t ,z) =
a
--v(T
at
-
t ,q(t,.))
f
(Gt , 77(t1z)), V)U(T- t , d t , x))-
U2
-V2u(T - t , q ( t ,x)) - gradp. 2 The latter equality is turned into (16) under the change of variables q ( t ,x) = [ ( T - t ) . Thus equation (29) guarantees that for the process q ( t ) ,satisfying (30), the relation P D D q ( t , x ) = F ( t , x ) holds. The same relation can be achieved also by another way. For a stochastic differential equation with respect t o a process ( ( t ) on Vs(In) denote by & ( s ) its solution with initial condition & ( t )= e. Consider the following system on V s(I"):
dq(t) = -G(T
-
+
t , q(t))dt a A ( q ( t ) ) d w ( t ) P t
where Qe is introduced in (23) and u0 = u(0) E Pe is the initial value for u(t),introduced above. Notice that the first equation of (31) is (27). Theorem 3.3. If the process q ( t ) and the vector u ( t ) satisfy (31), then u ( t ) , considered as a divergence-free vector field on I " ,satisfies (17).
190
Indeed, taking into account t h e routine stochastic presentation of solutions of PDE’s one can easily derive from t h e second equation of (3.3) t h a t
PDDq(t,x) = F ( t ,x). It should be pointed out t h a t system (31) is similar t o t h a t considered by Ya. Belopolskaya (see also 4). Equation (30) as a part of another system of stochastic differential equations, connected with Navier-Stokes equation, was considered also by B. Busnello (the problem was set u p by M. F’reidlin).
Acknowledgments T h e research is supported in part by Grant 99-00559 from INTAS, Grant UR.04.01.008 of the program Universities of Russia and by U.S. CRDF RF Ministry of Education Award VZ-010-0.
References 1. Arnol’d V. Sur la gkomktrie diffkrentielle des groupes de Lie de dimen-
sion infinie et ses applications a l’hydrodynamique des fluides parfaits. Ann.Inst.FourierT.16, N 1,319-361 (1966). 2. Belopolskaya Ya.1. Probabilistic presentation for solutions of boundary-value problems for hydrodynamical equations. Trudy POMI, V. 249, 71-102 (1997). 3. Belopolskaya Ya.1. and Dalecky, Yu.L. Stochastic processes and differential geometry. Kluwer Academic Publishers, Dordrecht 1989 4. Belopolskaya Ya.I., Gliklikh Yu.E. Diffusion processes on groups of diffeomorphisms and hydrodynamics of viscous incompressible fluid. Transactions of RANS, ser. MMMIC, V. 3, N. 2 , 27-35 (1999). 5. Busnello B. A probabilistic approach to the two-dimensional Navier-Stokes equation. The Annals of Probability, V. 27, No. 4, 1750-1780 (1999). 6. Ebin D.G. and Marsden J. Groups of diffeomorphisms and the motion of an incompressible fluid Annals of Math.,V.92, N 1, 102-163 (1970). 7. Gliklikh Yu.E. Ordinary and Stochastic Differential Geometry as a Tool for Mathematical Physics.- Dordrecht: Kluwer, 1996. 8. Gliklikh Yu.E. Global Analysis in Mathematical Physics. Geometric and Stochastic Methods.- N .Y.: Springer-Verlag , 1997. 9. Nelson, E. Derivation of the Schrodinger equation from Newtonian mechanics. Phys. Reviews, 150 (4), 1079-1085 (1966) 10. Nelson, E. Dynamical theory of Brownian motion.-Princeton: Princeton University Press, 1967 11. Nelson E. Quantum Fluctuations.-Princeton: Princeton University Press, 1985.
FURTHER CLASSES OF PSEUDO-DIFFERENTIAL OPERATORS APPLICABLE TO MODELLING IN FINANCE AND TURBULENCE
NIELS JACOB AND AUBREY TRUMAN
University of Wales, Swansea Department of Mathematics Singleton Park Swansea SA2 8 P P E-mail: N . [email protected] A . [email protected] 0. Barndorff-Nielsen and S. Levendorskii used some classical (SE6-) pseudodifferential operators t o construct Markov processes in order to model some situations in finance (and turbulence). In our note we describe various symbol classes consisting of non-classical but smooth symbols which lead to Markov processes and obey a symbolic calculus. In particular it is pointed out that in many cases it is possible to make parameters of the characteristic exponent of a LBvy process state-space dependent to get a corresponding Markov process generated by a psuedo-differential operator.
1. Introduction Since the pioneering work of E. EberleinlO on comparing solutions for finance market models obtained from models driven by diffusions with realworld data it is clear that jump processes yield much better models. The most widely used class of jump processes for modelling are L&y processes. We refer to the surveys of E. Eberleing and 0. Barndorff-Nielsen and N. Shephard4, respectively, and the references given therein. The fact that L6vy processes have stationary and independent increments implies a certain “translation-invariance” of the distribution corresponding to the underlying process. This fact excludes for example that a change of parameters occurs when a certain threshold (for prices for example) is reached. In the very original paper, 0. Barndorff-Nielsen and S. Levendorskii3 therefore started to model finance markets by using distributions involving parameters depending on the price, or by an abuse of the language of physics: price-homogeneous distributions were replaced by priceinhomogeneous distributions. Tracking back to the generator of the un191
192
derlying Markov process, this change necessitates the switch from constant coefficient operators to variable coefficient operators. In their paper, Barndorff-Nielsen and Levendorskii used the representation of the generator as a pseudo-differential operator following l6 where it was emphasised that pseudo-differential operators are canonical tools in the theory of Markov processes. They chose to work with classical pseudo-differential operators, i.e. with symbols q(x,<)in the class Sz6 with the additional assumption that for all z E R",q(z, .) : Rn -+ C is a continuous negative definite function to ensure that the generated semigroup is a Feller semigroup. Their model symbol is:
which gives for frozen coefficients, i.e. z = ZO, just a normal inverse Gaussian distribution with parameters ~ ( z o )~, ( z o )~, ( z oand ) P ( s 0 ) . The fact that modelling with normal inverse Gaussian distribution is rather successful and that Eq. (1) belongs to the Hormander class allowed them to emphasise the need for having smooth symbols in modelling finance markets. The purpose of this note is to show that there are large and rather general classes of smooth but non-classical symbols leading to pseudodifferential operators generating Feller semigroups. After we have discussed some basic facts on pseudo-differential operators and Markov processes, we will first discuss W. Hoh's l 1 > l 2symbol class and then the Weyl calculus approach due to F. Baldus Further we will have a short look a t subordination in the sense of Bochner as well as t o pseudo-differential operators of variable order of differentiation. Many results in finance have a counterpart in turbulence, compare for example Barndorff-Nielsen2 and Barndorff-Nielsen and Shephard4. In particular experimental observations show that the time derivative of a fluid's velocity field is not log normally distributed. It has instead a hyperbolic or normally inverse Gaussian type of distribution such as those arising for the above jump processes. Thus, the classes of pseudo-differential operators introduced here may also be helpful in modelling turbulence problems, e.g. by considering classical models of fluid dynamics in a random environment where the driving noise is a jump process like those described in what follows. Extensive results are already known for models involving Burgers equation when the driving noise is white noise. Here we have exact solutions for the Burgers velocity field in terms of a stochastic mechanics with additive white noise. The challenge is to replace this noise with a jump
'.
193
process and Burgers dynamics with Navier-Stokes dynamics so as to match the experimental observations (Barndorff-Nielsen2, Barndorff-Nielsen and Shephard4, Davies, Truman and Zhao', Truman and Z h a ~ ~ ~ ) . As explained in Barndorff-Nielsen and Levendorskii3 , or more precisely in Bogarchenko and L e ~ e n d o r s k i i ~the > ~ application >~, in finance follows by solving a generalised Black-Scholes (backward Kolmogorov) equation
&u(t,). - (A
+ q(z,Ox)) 4 6 ).
=0
(2)
with u(t,z) being the price of a contingent claim. But the calculi introduced by W. Hoh and F. Baldus allow to attack this equation for non-classical symbols in a straight forward way analogously to the classical case. Our report is intended to inform those who do modelling in finance and turbulence of the mathematical tools available. In particular it should be emphasised that it is often possible to pass from LBvy processes t o jump processes with a pseudo-differential operator as generator by making the parameters "location dependent". The authors are very grateful to 0. Barndorff-Nielsen and S. Levendorskii for discussions on their work. 2. Some basic f a c t s on Feller processes
We restrict ourselves to Feller processes ( X t ) t > O, PX)
with state space ( R".The fundamental quantity characterising the process is its symbol XERn
which reduces in the case of L6vy processes to the c h a r a c t e r i s t i c exponent $([) of the LBvy process, for details see Jacob" or the survey Jacob and Schillinglg. From Eq. (3) it is clear that for z E Rn fixed H q ( z , < ) must be a characteristic exponent, i.e. we have the LBvy-Khinchin representation
<
c n
q(z,
C) = 4.) + ib(z). E +
al(z)EkEl
l,k=l
and E H q ( z , [ ) is therefore a c o n t i n u o u s n e g a t i v e definite f u n c t i o n , i.e. has a LBvy-Khinchin representation. We call such symbols by a small abuse of language a n e g a t i v e definite symbol.
194
The basic observation is that (for reasonably nice processes) the generator of the semigroup Ttu(z)= Ex(u( X t ) )is given by the pseudo-differential operator -q(x, D ) u ( x )= - (27r-"
"S Wn
eiz'cq(z,<)ii(<)d<,
(5)
ii denoting the Fourier transform of u.For translation invariant operators, i.e. operators with constant coefficients (i.e. generators of LBvy processes) we find the well-known formulae
and
$I being the characteristic exponent of the L6vy process. Thus modelling a phenomenon with "varying parameters" simply means to pass from the generator -$I(D) (LBvy process case) with symbol -$(<) (characteristic exponent) to the generator - q ( x , D ) with symbol -q(x, <) where q(xo,<) N $(<) for all 20. Thus the fundamental problem is twofold, construct the process starting with q(x,<) and study the process (if constructed) by using q(x,<), in particular try to prove that close t o zo it behaves like a LeGy process with characteristic exponent q(z0,<). Note that so far no smoothness assumptions on (5, <) H q ( x ,<) were imposed. 3. Hoh's symbolic calculus The philosophy of the theory of pseudo-differential operators is to have a symbolic calculus which allows one to reduce operations on the level of operators to operations on the level of their symbols. Such a calculus needs some smoothness for (z, <) H q(z,I). Hormander's calculus requires C"smoothness and in addition some type of homogeneity of (the principal symbol of) q(z,<). W. Hoh11y'2 had developed a symbolic calculus for negative definite symbols without assuming homogeneity properties. We just describe his approach. Given a continuous negative definite function $I : Rn -+ R with LBvyKhinchin representation
$(<) =
1
Wn\{o)
(1 -cosY+(dY)
.
(8)
195
The L6vy measure v has to integrate y H 1 A IyI2, and the integrability properties of v determine the smoothness of $. To see this just differentiate in Eq. (8) formally under the integral sign to find for a multi-index Q
aa$(E) =
1 a; 1
(1 - cosy. E ) v (dy)
a-\{o) = ca
cos) (Y . E ) v (dy)
ya
7
Wn\to)
ca being f l . lyal v (dy) < 00, then a"$ exists by the dominated conThus, if wn\{o} j I n, vergence theorem. In fact, we find more. For Q = ~ j 1,I
s
IyI2 v (dy) it follows
or with Mz := Rn\to)
P""(E)I
I (2-'M2)+ ($ (014
and for ( a (2 2 we always find
la"$ (511 I Mlal with Mlal =
s
lylla' v (dy).
an\{0)
Finally we proved the following result due to W. Hoh": Lemma 3.1. If $ has t h e representation Eq. (8) and if Mla1 exists for 2 5 J Q J5 m.t h e n E C" (Rn) and
ppm1 I
CIaI
holds where p ( k ) = k A 2, k
(1+
+ )(I'
E No.
Z-P(lul)
>
IQI I m
1
(11)
196
<
Note that the continuous negative definite function H 1-cos a.<, a E R", shows that Eq. (11) is optimal. Given any continuous negative definite function with representation Eq. (8) we may always construct a continuous negative definite function $R E C" (R")satisfying Eq. (11) for all m E No. We just have to consider
$R(E)
=
J
(1 - C O S Y . <) v (dy) =
J
(1 - cosy
.
an\{o}
BR(O)\{O}
Now the way is open to construct a symbolic calculus related to a fixed continuous negative definite function $I. Let $I : Rn 4 R have the represent at ion
J
=
(1 - cosy.
(13)
BR(O)\{O}
with some L&y measure v on Rn\{O} and define
A(<) = (1+ $ ( E l ) + .
(14)
We consider Hoh's symbol class SF,' consisting of all p : R" x p E C" (R"x R"),satisfying
IaE"a:P(x,
0 1I ca,&m"-P(IaI)
.
R" + @, (15)
As worked out by W. Hoh11,12,it is possible to develop a complete symbolic calculus including the parametrix construction for "elliptic" elements for pseudo-differential operators p(x,D)with symbol p E S~~'. In particular, if in addition $ satisfies a growth condition from below,
NE)2 co IEI? , r > 0 and IEI large,
(16)
and if p ( z ,.) : R" -+ R is a continuous negative definite function for each x E R",then we have
Theorem 3.1. (W. Hoh) A s s u m e Eq. (16), p E 5':)' R i s a negative definite function, and
such that p(x,.) :
R"
P(Z,
02 fix2 (El
(17)
for large 1 1 1 and some 6 > 0. T h e n (-p(z, D ) , C r (EXn)) is closable in C, (R")and the closure generates a Feller semigroup.
Corollary 3.1. In case of Theorem 3.1 we find that p(x,5) i s the symbol of a Feller process.
.
197
Of course we still have the restriction that p ( . , .) must be a Cm-function with respect to x and (as observed in 3 ) . But there is a way t o overcome this restriction too. Suppose that p : R" x R" + R is a continuous function such that p ( z , . ) is negative definite with uniform bound
<
p ( z , <) 5 c (1
+ /(I2)
and L6vy-Khinchin representation
P(Z,O=
.i'
(1- C O S Y . 5 ) N ( G d Y )
.
(18)
an\{o) When we make a uniform decomposition P(X,C) =
.i'
(1-cosY.<)N(x,dY)+
= PR(x,
s
(1-cos(Y.t))"z,dY)
ak(o)
BR(O)\{O)
t)+ $ R ( x ,6)
>
it is often possible to identify ~ R ( z () , as a symbol in some class satisfying Eq. (17). Further, PR(z, D ) is a bounded operator which is an admissible perturbation of p ~ ( xD, ) in the sense that if - p ~ ( zD, ) generates a Feller semigroup, so will - p ( z , D ) = - ~ R ( z , 0 ) - I j ~ ( z0,) . For details we refer t o W. Hoh l1 and 12. In conclusion: Hoh's symbolic calculus works almost as Hormander's SE6-calculus and allows t o construct Feller semigroups leading to Feller processes with C"-symbols not belonging to the Hormander class. 4. Baldus' Weyl calculus approach
In this section we will briefly discuss results due to F. Baldus' who used the Weyl calculus to construct Feller semigroups. Unfortunately we need quite a lot of special notions to state the result. For a detailed discussion we refer to F. Baldus' and also to the original paper and monograph by L. HOrmander'sl4>15. Denote by 0 the standard symplectic form on Rn x R",i.e.
* ((z,0 (Y, 7 ) )= Y . < - z 7
and for a positive definite quadratic form y on
'
7
7
R" x Iw" we set
of In the following a metric on Rm simply means a family y = (yz)zEWm positive definite quadratic forms on W" which we may interpret as Rieman-
198
nian metric and denote it sometimes by y ( d z , d z ) . Given a metric y on we say that it splits if we have for each (y, 17) E R" x R"
R" x R",
Definition 4.1. A. A metric y on R" is called a slowly varying metric if there exists a constant cy such that for z , y E R" satisfying yz (z - y, z - y) 5 1it follows that c-7
holds for all z E R". B. Let y be a slowly varying metric on R". A function M : R" 4 R+ is called y-slowly varying if there is a constant CM such that for all z, y E R" with y,(z - y, z - y) 5 we have
&
Next we have to introduce the notion of a Hormander metric and that of (sub-) admissible weight functions.
Definition 4.2. A. A slowly varying metric y on R" x R" is called a Hormander metric if there exist constants cy > 0 and N y E N such that for all (z, [) E R" x R" we have
B. Let y be a slowly varying metric on R" x R".We call M : Rn x R" + R+ a y-admissible weight function if M is y-slowly varying and satisfies with C M > 0 and N M E N
for all (z, <), (y, 7) E R" x R". Denoting for a metric y on
R" x R"
the function h, by
199
we have:
Definition 4.3. Given a Hormander metric y on Rn x R". A function M : R" x R" R+ is called a sub-y-admissible weight function if there exists a y-admissible weight function MO such that M 5 MO and for some m E N and c > 0 it follows that ---f
hTM0 5 CM .
(28)
6
If M and are both sub-y-admissible we call M an invertible sub-yadmissible weight function. For ( y , ~ E) Rn x R" and
ZL : Rn
d(,,,)+
x R"
@, we set
-+
E ) = ((Ill 77)
I
0)
V 2 n U (2,
I
where Van is the gradient in R" x R" and (., .) is the scalar product. Further we set for a metric y on R" x R" and Ic E No
J~l:"'q')
(z, 5)
(30)
Definition 4.4. Given a metric y on R" x R" and a weight function M : Iw" x R" 4 R+. The symbol class S ( M , y) consists of all functions q E C" (R" x R") which satisfy for all Ic E NO
Now we can introduce the operators associated with S ( M ,y).
Definition 4.5. Given q E S ( M , y ) . We define the associated Weylpseudo-differential operator q w ( x 1D ) : S(Rn) -+ S'(Rn) by
qw (Z] D)U ( Z ) = (27r)-"
(F, <)
J' ei(z-Y).Eq
u(y)dyd[ .
(32)
P"W n
The set of all operators qw(zlD ) with symbol q E S(Mly) is denoted by Q
( M ,7).
Example 4.1. For 0 5 6 5 p 5 1,6 < 1, a Hormander metric is given by
200
Taking in addition the weight function M ( z ,S )
=
(1
+
S ( M , y ) = SE6 .
m -
we find
(34)
Note that the Weyl-pseudo-differential operators q w ( z ,0)we are interested in can always be transformed into the "usual" form q(z, D ) u ( z )= (27r)-?
] eiz.5q(zl[)ii(J)d< .
(35)
IWn
Let us denote by B ( L 2(R"))-' the set of all bounded linear operators A : L 2 ( R n ) -+ L 2 ( R n )which have a bounded inverse, and denote by 9 ( M ,y)-l the set of all q w ( z ,D)E 9 (MIy) with inverse in 9 ( M ,y). Now we can state the result of F. Baldus
':
Theorem 4.1. Let y be a Hormander metric o n R" x Rn which splits and assume
Q(I,~)~B(L~(w= ~ s) () i-, 'y ) .
(36)
Further let M be a n invertible sub-y-admissible weight function and m 5 1 a n arbitrary sub-y-admissible weight function satisfying with some k E N and C M > 0
where h, i s given by Eq. (27). If q E S ( M ,y) satisfies
for all k E No, as well as
IX + 4(z,El
for all (z,c) E
+ cql 1
54
(A
+ M ( z ,<)I
(39)
R" x R",X 2 A, 2 0 and cq,Cq 2 0 , and
E H q ( z ,I ) is
a negative definite function,
(40)
then the operator -q(z, D ) : C r (Rn) 4 C , (R") is a densely defined operator o n C, (R")which extends to a generator of a Feller semigroup, hence q(x,<) i s the symbol of a Feller process.
Example 4.2. A. Elliptic elements p E Sz6 such that for all z E R" the function 5 H p(x,<) is negative definite are included in Theorem 4.1.
20 1
B. The class Sr,' considered by W. Hoh, see Section 2 is included when working with the metric
Note however that certain extensions of Hoh's results, i.e. the perturbation theory, is not covered. C . Symbols of mixed homogeneity are partly included.
5. Relations to subordination in the sense of Bochner and operators of variable order of differentiability Subordination in the sense of Bochner is a procedure t o construct a new stochastic process out of a given one by a random time change. Most importantly, i t has a nice analytic counterpart. A non-technical outline is given in Jacob", Chapter 5, and in Jacob and Schillinglg, Section 4. In this section we sketch only very briefly how to get using subordination in the sense of Bochner further pseudo-differential operators generating (Feller) processes. By definition a Bernstein function f : ( 0 , ~ + ) R is a function with represent ation
f ( z )= a
+ bz +
i:
(1 - e-zt) p (dt)
(42)
where a , b 2 0 and p integrates t H 1A t , t > 0. To every Bernstein function f there is associated a one-sided L6vy process (St)t>O called a subordinator the paths of which are almost surely monotone increasing. If ( X t ) t 2 0is any Markov process and (St)t20is an independent subordinator, then rt := X s , is a new Markov process called the subordinated (in the sense of Bochner) process. For the case where ( X t ) t > ois a Lkvy process with characteristic exponent then the characteristic exponent of ( X S , ) ~ is > ~f o $. It is a fact that f o $ is always a continuous negative definite function for f being a Bernstein function and $ being a continuous negative definite function. Now suppose that q : Rn x R" + R is a symbol of a generator of a Feller process. In particular q(z,.) : R" 4 R is a continuous negative definite function. It follows that for every Bernstein function f the function f o q : Rn x RT2+ R,(z, I ) H f ( q ( x , E ) )is negative definite too. Hence we may try t o construct a Feller process starting with the symbol f ( q (z, 0). Clearly, if q ( z , J ) is independent of z then we just get the subordinated L6vy process with characteristic exponent f ($ (t))= f ( q (z0,t)) for some, hence all z o E R".However, if -q(z, D ) generates a Feller process (Xt)t,O -
+
202
and q(xl<) depends on x , then the subordinated process X,f := Xs, and the process yt with symbol f ( q (zl6)) are clearly distinct! The symbolic calculi introduced in Sections 2 and 3 may be used to relate X s , and Yt. We refer to Jacob and Schilling18 for a first simple approach and t o F. Baldus', Section 6.5. What we may expect (and what holds true in many situations) is that the generator -f (-4 (x,D))of X [ and the operator - (f o q) (x,D) differ only by a "low order" term which will follow a reasonable asymptotics and vice in terms of that of (K)t>O of the transition function of ( X ! ) t2o versa. For modelling purposes maybe a different] but very related concept is more important. Stable, especially symmetric stable processes, are very often used for modelling] in finance and turbulence, but also in other problems. We may interpret the symmetric stable process with index 2a, 0 < Q < 1, as the process obtained from Brownian motion by subordinating with the one-sided L6vy process associated with the Bernstein function fa(s) = s". Indeed the characteristic exponent of the symmetric stable process of index 2a is given by $a (I =) 1 < 1 2 " which is just fa 1<12) and
(
<
of course H [
c)
<
This function has the property that if H q(xl E ) is a continuous negative definite function, so is H q(zl<)"(").Hence, it may lead t o a stochastic process with generator
<
In case of q(xl <) = 1<12 (or more generally q(xl <) = ~ i , ak,l(x)
203
In conclusion : stable-like processes have generalisations to processes generated by pseudo-differential operators of variable order of differentiation, a n d these classes of processes (or operators) are at our disposal when modelling. We feel that because of empirical merits such processes should come into their own i n t h e modelling of turbulence a n d financial markets. References 1. Baldus, F., S ( M ,g)-pseudo-differential calculus with spectral invariance on Rn and manifolds for Banach function spaces. Dissertation Universitat Mainz 2000, Logos Verlag, Berlin 2001. 2. Barndorff-Nielsen, 0. E., Probability and statistics: Selfdecomposability finance and turbulence. In : Probability Towards 2000 (eds. L. Accardi and C. C. Heyde), Springer Verlag, Berlin 1998, 47 - 57. 3. Barndorff-Nielsen, 0. E., and Levendorskii, S. Z., Feller processes of normal inverse Gaussian type. Quantitative Finance (to appear). 4. Barndorff-Nielsen, 0. E., and Shephard, N., Modelling by LBvy processes for financial econometrics. In: L6vy processes: Theory and applications (eds. 0. E. Barndorff-Nielsen, T. Mikosch, S. J . Resnick), Birkhauser Verlag, Boston 2001, 283 - 318. 5 . Bogarchenko, S. J., and Levendorskii, S. Z., Option pricing for truncated Le'vy processes. Intern. J. Theor. Appl. Finance (to appear). 6. Bogarchenko, S. J., and Levendorskii, S. Z., Perpetual American options under LBvy processes. Preprint 2000. 7. Bogarchenko, S. J., and Levendorskii, S. Z., Barrier options and touch-andout options under regular LBvy processes of exponential type. Preprint 2000. 8. Davies, I. M., Truman, A . , and Zhao, H. Z., Stochastic heat andSurgers equations and their singularities - geometrical and analytical properties (The fish and the butterfly, and why). Preprint 2001. 9. Eberlein, E., Application of generalised hyperbolic Le'vy motion to finance. In: LBvy processes: Theory and applications (eds. 0. E. Barndorff-Nielsen, T. Mikosch, S. J. Resnick), Birkhauser Verlag, Boston 2001, 319 - 336. 10. Eberlein, E., and Keller, U., Hyperbolic distributions in finance. Bernoulli 1 (1995), 281 - 299. 11. Hoh, W., Pseudo differential operators generating Markov processes. Habilitatsionschrift Universitait Bielefeld, 1998. 12. Hoh, W., A symbolic calculus for pseudo differential operators generating Feller semigroups. Osaka J. Math. 35 (1998), 789-820. 13. Hoh, W., Pseudo differential operators with negative definite symbols of variable order. Revista Mat. Iberoamericana 16 (2000), 219-241. 14. Hormander, L., The Weyl calculus of pseudo-differential operators. Comm. Pure Appl. Math. 32 (1979), 359 - 443. 15. Hormander, L., The analysis of linear partial differential operators, vol. 3, Springer Verlag, Berlin, 1985. 16. Jacob, N., Pseudo-differential operators and Markov processes. Akademie Verlag, Berlin, 1996, 17. Jacob, N., and Leopold, H. G., Pseudo-differential operators with variable
204
18.
19.
20. 21.
22.
23.
order of differentiation generating Feller semigroups. Integr. Equat. Oper. Th. 17 (1993), 544-553. Jacob, N., and Schilling, R. L., Subordination in the sense of Bochner - An approach through pseudo differential operators. Math. Nachr. 178 (1996), 199-231. Jacob, N., and Schilling, R. L., Le'vy-type processes and pseudo differential operators. In: LBvy processes: Theory and applications (eds. 0. E. BarndorffNielsen, T. Mikosch, S. J. Resnick), Birkhauser Verlag, Boston 2001, 139 168. Kikuchi, K . , and Negoro, A , , On Markov processes generated by pseudo differential operators of variable order. Osaka J. Math. 34 (1997), 319 - 335. Negoro, A . , Stable-like processes : Construction of the transition density and the behaviour of sample paths near t = 0. Osaka J . Math. 31 (1994), 189 214. Negoro, A . , and Tsuchiya, M., Stochastic processes and semigroups associated with degenerate L6vy generating operators. Stochastics and Stochastics Report 26 (1989), 29 - 61. Truman, A . , and Zhao, H.Z., Stochastic Burgers' equations and their semiclassical expansions. Commun. Math. Phys. 194 (1998), 231-248.
MATHEMATICAL ANALYSIS OF A STOCHASTIC DIFFERENTIAL EQUATION ARISING IN THE MICRO-MACRO MODELLING OF POLYMERIC FLUIDS.
BENJAMIN JOURDAIN, TONY LELIEVRE CERMICS, Ecole Nationale des Ponts et Chausse‘es, 6 & 8 Av. Blaise Pascal, 77455 Champs-sur-Marne, France. E-mail: {jourdain, lelievre}@cermics.enpc.f r
We analyze the properties of a stochastic differential equation (SDE) arising in the modeling of polymeric fluids. More precisely, we focus on the so-called FENE (Finite Extensible Nonlinear Elastic) model, for which the drift term in the SDE is singular.
1. Introduction The rheology of non-newtonian fluids is a very lively field of modern fluid mechanics. The challenge is to find a good relation linking within the fluid the stress tensor to the velocity field in order to reproduce the behavior of the fluid in some classical situations (shear flow, elongational flow) and to simulate it in some more complex cases. This relation may be complicated since the stress generally depends on the whole history of the velocity field. Many approaches consist in deriving this relation from the microscopic structure of the fluid. In some cases, it is possible to directly attack the full system coupling the evolution of these microscopic structures to the macroscopic quantities (such as velocity or pressure) : this is the so-called micro-macro approach. We are here interested in the modeling of polymeric fluids. More precisely, we consider dilute solutions of polymers, so that the chains of polymers (the ‘hicroscopic structures”) do not interact with each other. In order to describe the microscopic structure of this fluid, one can model a polymer by a chain of beads and rods (this is the Kramers model) or more simply by some beads linked by springs (see Figure 1). We consider here the simplest model consisting in two beads linked by one spring : this is the dumbbell model. In this model, the evolution of the end-to-end vector (which joins the two beads) is described by a SDE. We refer the interested 205
206
reader to Refs 10)1,2,6 for the general physical background of these models. This SDE is actually coupled to the Navier-Stokes equation through the expression of the stress tensor as an expectation value built from the end-to-end vector.
Figure 1.: A hierarchy of models : from Kramers chain (top) to dumbbell (bottom).
The spring force can be linear (Hookean dumbbell model) or explosive (Finite Extensible Nonlinear Elastic dumbbell model). In the following, we consider the start-up of a Couette flow of a polymeric fluid (see Figure 2) : the fluid is initially at rest, and for t > 0, the upper plate moves with a constant velocity. For a complete analysis (existence, uniqueness, convergence of a finite element method coupled with a Monte Carlo method) of this model in the Hookean dumbbell case, we refer to Ref. 8. This reference also contains a more detailed introduction to these types of models and the way to discretize the corresponding system of coupled PDE-SDE. We here complement the mathematical analysis of the FENE model presented in Ref. by focusing on the SDE modeling the evolution of the conformation of the polymers in the FENE case. It is proven in Ref. that a solution to the coupled micro-macro system uniquely exists under natural assumptions. Our concern in the present paper is in particular to investigate the role played by the finite extensibility coefficient b (see formulas (2) and (3) below) in the existence and uniqueness of solution of the SDE itself, the fluid velocity being considered known.
207
u=o
Figure 2.: Velocity profile in a shear flow of a dilute solution of polymers. Let us now introduce the equations we deal with. They read, in a nondimensional form :
where the parameter b > 0 measures the finite extensibility of the polymer. The space variable y varies in c? = ( 0 , l ) and t varies in the whole of R+. The random variables are defined on a filtered probability space (R,.FIFtllP). The random process (&, Wt) is a (.Ft)-twodimensional Brownian motion. We take Dirichlet boundary conditions on the velocity. The initial velocity is u(t = 0, .) = U O , and ( X O YO) , is a FOmeasurable random variable. We will suppose that ( X o ,Yo) is either such that P ( X i Y: > b) = 0 (Section 2) or such that P ( X i Y; 2 b) = 0 (Sections 3 and 4). We fix y in 0 , set g ( t ) = a,u(y,t) and suppose throughout this paper that we have at least the following regularity on g :
+
+
where R+ = [O,+w). We are then interested in solving for t 2 0 the following SDE, which is a rewriting of the SDE (3) of the initial coupled
208
system :
{
dXf =
[-i1-
dY,g = -1.2
1-
qg
(X?P+(Y,9)2 b
with initial condition ( X O YO). , Let us begin by recalling from Ref. we give to (5).
) dt+dWt,
(5)
the precise mathematical meaning
Definition 1.1. Let Xo = ( X o ,Yo)and Wt = (K,Wt). We shall say that a (Ft)-adapted process Xi = ( X a ,yt”) is a solution to (5) when : for lP-a.e. w , lft 2 0 ,
Remark 1.1. Because of the convention 1 - ! 2 = +m if )zI2= b, we deduce that a solution to (6) is such that the subset of R+ (0 5 u < 00, IX:12 = b} has lP-as. zero Lebesgue measure. The paper is organized as follows : in Section 2 , we prove the existence and uniqueness of the solution to (6) with values in B,where
B = B(0,h)= { (x,y), x2
+ y2 < b} .
The existence of such a solution is derived from results concerning mulWe then focus on the probability for this tivalued SDEs (see Refs solution to reach the boundary of B (see Section 3). When b < 2 and I P ( ( X O (<~ b) = 1, this probability is equal to one. This enables us to construct (for g = 0) a solution to (6) that leaves a.s. B. Hence, if b < 2 , uniqueness of solutions does not hold for solutions to (6) without the additional requirement to take values in B. When b 2 2 and again lP(IX0l2 < b) = 1, the probability to reach the boundary is equal to zero and trajectorial uniqueness holds. We exhibit the unique invariant probability measure of the SDE (6) with g = 0 (see Section 4). All these results on the SDE have an impact on the analysis and the understanding of the coupled SDE-PDE system (for which we refer to Ref. They show that the assumption b 2 2 adopted in Ref. to prove existence and uniqueness of solution t o the coupled system is in some sense “optimal”. 415).
209
2. Existence and uniqueness
In this section, we suppose that ( X OYo) , is such that IP (Xi+ Y: > b) = 0. Our aim is to prove the following : Proposition 2.1. Under assumption (4), for any b > 0 and for any initial condition ( X O YO) , such that IP (Xi+ Y$ > b) = 0 , there exists a unique solution to (6) with values in B.
We first prove the uniqueness statement (Section 2.1), then turn to the existence first when g E L y (Section 2.2) and finally when g E L:,c (Section 2.3). In the following, the point is to notice that the singular term in the drift derives from a convex potential II : R2 +]- 00, +m] : WX,Y)
=
"'1'> + y2 < b,
(1 - 2 if x2
(7)
otherwise.
We have : Vx E B , W I ( x ) =
+&.
Moreover, the function
II is a
b
continuous convex function with domain B. 2.1. Trajectorial uniqueness for solutions with values in B
Let us begin with the uniqueness.
Xi
Proposition 2.2. Let us suppose we have two solutions Xf and to ( 6 ) and such that IP-a.s., X: = X,". Then these two solutions are indistiguishable until one of the processes leaves B. In addition, if P ( 3 t 2 0IXfl2 = b ) = 0 , then X: and X: are indistiguishable. Proof :
(
Z t = X:
Let us consider r = inf{t 2 0 , ( ( X f ( 2V ( X ; l 2 )> b } and
- g, . By ItG's formula, we have
- X,
:
dlZl: = 2 2 t . d Z t , = -2(VrI(X,s) -
on(x;)).z,dt + 2g(t)(X,g- x,g)(qg
- P)dt,
where x.y denotes the scalar product of x and y E lR2. Using the fact that, since II is convex, for a,ny x and 2 E B , (VrI(x) VrI(%)).(x- 2 ) 2 0, we obtain, for any t 2 0 :
210
Using Gronwall Lemma and the fact that g E L~o,(R+),we have thus shown - 9 that P-a.s, 'dt 2 0, XfAT= X t A T .Therefore, on {T < m}, IX$12= b. We deduce that in case P(3t 2 0, IX:lz = b) = 0, T = 03 P-a.s. . 0 2.2. Existence in the case g E L r
In this section, we suppose : 9 E L" @+).
(8)
In order to prove an existence result, we will use a multivalued stochastic differential equation. In this section, we use the results of E. C6pa4 and E. Ckpa and D. Lepingle5. Since the function II is convex on the open set B , its subdifferential X I is a simple-valued maximal monotone operator on R2 with domain B : 8rI(X) =
{
{VII(z)} if z E B , 0 ifxfB.
Let us now consider the two-dimensional process X t solution of the following multivalued SDE :
+ BII(X:)d t 3 ( g ( t ) K g0), dt + d W t ,
Lo xo dXi
-
= (xo,yo)l
(9)
We first recall the precise meaning of a solution to (9). Definition 2.1. We shall say that a continuous (.Ft)-adapted process X; = ( X a ,Kg)with values in B is a solution to (9) if and only if X i = X Oand the process K : = Wt ~ ~ ( g ( s ) Y ~ds, O - )( X : - X i ) is a continuous process with finite variation such that : for any continuous (.Ft)-adapted process at with values in R2,for P-a.e. w , 'do 5 s 5 t < 03,
+
l
t
I I ( X t )du 5
l
t
II(a,) du
+
l
t
( X t - a,).dKt.
(10)
Remark 2.1. A condition equivalent to (10) is the following : for any continuous (&)-adapted process at with values in B , the measure on R+ :
( X t - a,). (dKt - VrI(aU) du) is P-a.s. nonnegative. Since (8) ensures that x = ( 5 , ~++ ) (g(t)y,O)is (uniformly in time) Lipschitz and with linear growth, according to E. C6pa4, we have : Proposition 2.3. Under the assumption (8), for any b ued SDE (9) has a unique strong solution.
> 0 , the multival-
211
We are now going to recover a solution to (6) from the solution of (9). More precisely, we follow the method of E. C6pa and D. L6pingle5 (see Lemmas 3.3, 3.4 and 3.6) in order to identify the process K f . We can thus show that for all 0 < t < co,we have :
)
IE ( l l d I I ( X t ) l du < 03, with convention ldII(z)I = +co if x @ B . As a consequence, for any 0 du
< t < co,IP-as.,
< 00
with convention
& = +co.
(11)
Moreover, the process K i is IP-a.s. absolutely continuous on (0 5 u < co,X t E B } , with density V I I ( X : ) so that dKt has the following form :
dK: = V I I ( X : ) du
+ dG:,
(12) where G g is a continuous boundary process with finite variation IGgl :
Finally, one can identify this process Gf : for all t 2 0,
5
where, for any x E d B, n ( x ) = is the unitary outward normal to B at the point x. Hence the process X : is solution of the following SDE with normal reflexion at the boundary of B :
d X i = - V I I ( X i ) dt
+ (g(t)Ytg,O)dt + d W t - l{x;Eas}n(X:)dlGglt.
It just remains to show that IGgl, = 0, for u 2 0, in order to recover (6). Notice in particular that by ( l l ) ,the property of integrability of the drift term in (6) holds for the solution X : of the multivalued SDE (9). Lemma 2.1. IGgl = 0.
Proof : We follow here again the ideas of E. Ckpa and D. L6pingle5 (see Lemma 3.8 p. 438) to prove that IGgl = 0. Let us consider Rf = b - IXfI’. By It6’s formula, dR: = -2X:.dXi - 2dt, = -2VII(Xf).Xf dt - 2g(t)X/Kgdt - 2dt - 2Xf.dWt
+ 2 lxgl2l dlGglt, 4
b2
= - dt - 2g(t)X/Kgdt - ( 2
R:
+ b) dt - 2Xf.dWt + 2&dlGglt,
(14)
212
the last equality using the fact that d J G g J = t l(X;EBB)dJGglt. We know that Rf is a continuous semimartingale with values in [0, b ] . We want to prove that dRf = l R ; > O d R f . Using Tanah's formula (see" p. 213),
where, for any a E [0, b ] , Lg denotes the local time in a of Rg. Using now the occupation times formula (see Ref. l1 p. 215), we know (using ( 1 1 ) ) that, for any fixed t > 0 :
Since a + Lg is a.s. cadlag (see" p. 216), we deduce that for any t IP-a.s., L: = 0. Using this in ( 1 5 ) , we obtain
> 0,
dRf = 1Rf>O d R f . Using this equality in ( 1 4 ) , we have : V t 2 0,
s"
(-z+
1 b2 1Rj=O ds 2g(s)X,SY,S d s ( 2 13)d s 2 X : . d W s 2& 0 Since, according to ( l l ) ,IP-a.s., ( 0 5 t < m,Rf = 0 ) has zero Lebesgue measure, the right hand side is null. We conclude by using dlGglt = lR:=odlGglt. 0
-
+ +
+
We have thus shown the following properties on the process X: : 0
for any 0 < t
< co,IP-as-.,
0
d X f = -VII(X:) d t
h 1-w + t
1
+ (g(t)Kg,O)dt
du
< co,
dWt.
We have thus built a solution Xf = ( X a , q g )to our initial problem ( 6 ) in case g 6 L"(R+). This result is not sufficient in our context since the energy estimates on the coupled system (1-3) yields less regularity on g (see Ref. 8 ) . 2.3. Existence in the case g E L$,(R+)
We now want to build a solution to ( 6 ) using the multivalued SDE (9), but with a weaker assumption on g, namely (4). In this case, the general results of existence on multivalued SDE do not apply immediately.
213
Therefore, we consider the following sequence of approximations of this problem :
+ aII(X:")d t 3 ( g n ( t ) q g " ,0) d t + d W t , x;"= xo, dX:"
where n E IN* and g n ( t ) = -nV ( n A g ( t ) ) .Since gn is bounded, the results of the previous section apply and we obtain a unique solution X i " of the multivalued SDE (16). Moreover, these processes Xz" are such that : for any 0 < t
0
< cc, P-a.s.,
I',
du
x ; " = x 0L -V I I ( X : " ) d s + l
0
< 00,
t"
(17)
(g"(s)Y:",O)ds+Wt.
We now want to let n go to cc in Definition 2.1 (notice that by (17), dK:" = VII(X:") d t ) . In the following, we choose T > 0 and we work on 2
the time interval [O,T].We know that for all n, supt>, - IXfnI 5 b. For any n 2 m, we have, by ItB's formula,
l2
d (X:" - X;"
= - (VII(X;")
VII(xfm)) . (X;" - X : " ) d t
-
+ (gn(t)q9" - gm(t)Yg".(t))
(X;" - X:")
dt.
Using the fact that, since II is convex, for any x and y E B , (VII(x) VII(y)).(z - y) 2 0, we obtain : V t E [O,T],
(X:"
- X:m
1'
1 t
5
( g n ( s ) Y g n ( s )- gm(s)Ygm(s))(X:"
- X.gm) d s
so that : V t E [O,T],
Ixin- x:"
1
I
2
t
5
lgn(s)Y;"
-
p(s)Y;"
0
5
J' (lgn(s)l jY;" 0
i1
t
5
- Y:*
I 1X.g"
1 + lY;m1
I2
lgl(s) IX:" - X;m d s
-
x:"
I ds
Ign(s) - g m ( s ) l ) 1X:"
+ 2b
I'
-
l g n ( s ) - g"(s)l d s .
Using Gronwall Lemma, we then obtain :
X:"
1 ds
214
From this inequality and the fact that g E Li ([0, TI),we deduce that there exists a continuous adapted process X : with values in B such that X:" --+ X : in L,"(L,oO([O,T I ) ) . One has the following estimate on the total variation of VII(XE")du on [O,T]:
By ItB's formula, we know that : W E [O,T],
Ix:"12
1 t
= IXOl2 -
Ix:"
IXpd s + 2
t
g"(s)
x;"Y,"" d s + 2 t + 2
1 t
x:".dWs,
which yields : V t E [0, TI,
Ji
It is obvious that s,' X;" . d W , + X , . d W , in L:(Lp([O,TI))-norm. Up to the extraction of a subsequence, we can suppose that this convergence holds for almost every w . Using this property together with (18) and ( 1 9 ) , we deduce that for a.e. w , the measure VII(X:")dt on [O,T]is such that
Jz
on
IVlr(Xig")ld t < C ( T , w ) where C(T,w) is a constant only depending w . 'One can thus extract a weakly converging subsequence of the other hand, taking the limit n --+ co in ( 1 7 ) ,
T and
1
du uniformly converges on [0, TI to K: satisfying :
1 t
K: =
(g(u)Y,, 0) du
+ Wt - (Xf- XO).
By identification of the limit, we have VII(X:") dt 2 dK: weakly. By Definition 2.1, the processes X:" are such that for any continuous (.Ft)-adapted process a t with values in R', for P-a.e. w , VO 5 s 5 t < co,
[
II(X5")d u 5
t
rI(a,)du +
1 t
(Xt"- a u ) . V r I ( X ~du. ")
(20)
215
One can pass to the limit n -+ co in (20), using the fact that Il(Xt")+ II(Xt) pointwise in u and that II(Xt")is uniformly integrable. Indeed, for any A
2 $, if we set
, we have (since x
Mu =
++
is decreasing on [e, +co)) :
1
T
so that
11n(xzn,12AII(Xtn) du + 0 uniformly in n when A
+ co. We
have thus obtained a continuous process Xf on [O,T]and a continuous process with finite variation Kf = Ji(g(u)Y,lv,0 ) d u Wt - (Xi - XO)on [0,TI such that for any continuous (Ft)-adapted process at with values in lR2, for IP-a.e. w , VO 5 s 5 t < TI
+
It
1 t
II(Xt)d u 5
II(a,) du
+
t
(Xt- a,).dKt.
This shows that we have built a solution to the rnultivalued SDE (9) on the time interval [0, TI. Since T is arbitrary, using Proposition 2.2, we have built a solution on lR+.Following again the arguments of the last section it is easy to show that : 0
for any 0
< t < 00,
IP-a.s.,
J" 1 - & d U < n , 0
dXf = -VII(Xi) dt
+ ( g ( t ) q g ,0 ) dt + d W t .
This shows that Xi is a solution to (6) and completes the proof of Proposition 2.1. 3. Does the solution reach the boundary ?
In this section, we want to determine whether or not the process Xi we have built in the previous section reaches the boundary of B . Should the occasion arise, we deduce that uniqueness does not hold for (6), a t least in the case g = 0. Throughout this section, we suppose that the initial condition is such that IP(IXo12< b) = 1. 3.1. Necessary and suflcient conditions
In this section, we want to analyze the event (3> 0, IXf12 = b}. We are going to prove :
216
Proposition 3.1. A s s u m e 9E
Jm+),
(21)
and that P(IX0J2< b) = 1. Let u s consider the process X : solution t o (6) built above. We have : 0
if b _> 2, t h e n P ( 3> 0 , IX:12 = b) = 0 , if b < 2 , then P ( 3> 0 , IX:12 = b) = 1.
In view of Proposition 2.2, we deduce immediatly
:
Corollary 3.1. If b 2 2 and P(IXOl2< b) = 1, t h e n trajectorial uniqueness holds for (6). Proof. First, by Girsanov Lemma, one can suppose g = 0. Indeed, let us consider the process X i we have built in last section. Under the probability Pgdefined by
+
the process (Gg,WE) = (& s," g(s)Y,S d s , Wt) is a Brownian motion and therefore ( X f ,y,",Kg, WE,lPg)tER+ is a weak solution of the SDE ( 5 ) with g = 0. Since this solution is with values in B,it is also a weak solution of the multivalued SDE (9), with g = 0, for which uniqueness in law holds. Since IPg and P are equivalent on F,we can then deduce the properties of Proposition 3.1 in case g E L2(R+) from the properties of Proposition 3.1 in case g = 0. In the following, we focus on the solution to (9) with g = .O, which we denote by X t = (Xt,Y,). We fix x E B and the superscript x means that we consider the solution to (9) with g = 0 such that Xo = x. Let us first suppose that 1x1 > 0. Let us consider the process R," = b - lXT12. We know that : b2 dR," = -dt - ( 2 + b) d t - 2X,".dWt. R," Let us introduce the stopping time
Let fix t > 0. By Girsanov Lemma, one shows that P - a s . , Indeed, by definition of r:,
P(lX~A,l= 0) = P(JXTI= 0 and t < r;).
217
Let:'FI
and
be defined by :
IE,"denote the corresponding expectation. By Girsanov Theorem,
+ W , - s;
VIl(X;,,,) du) s
IP(lXr1 = 0 and t
< T,") 5 IP(IBY1 = 0) = 0.
One can therefore show that IXrl
T" = lim
T,"
n+o3
(23)
> 0 on [O,T"),where
= inf { t 2 0, 1XF12 = b } = inf {t 2 0, R," = O}.
Thus, one can write, for t E [0, T " ) :
b2 dR: = -d t - (2 + b) d t + 2 d q d , & ,
R,"
where Pt is a Ft-adapted 1-dimensional Brownian motion. Let us now introduce the stopping time
S" = inf {t 2 0, R,"
4 (0, b ) )
We have, IP-a.s., S" 5 T". We refer here to I. Karatzas and S.E. Shreveg (see Section 5.5 p. 342-351). We introduce a scale function p such that :
(;
- ( 2 + b ) ) p ' ( r ) + 2(b - r)p"(r) = 0,
which leads to : p' ( r ) = C(b - r ) - l ~ - - ~ / ~ ,
where C > 0. We have thereforep(b-) = +m and ( b < 2 p(O+) > -m). Using this property of the scale function and the results of I. Karatzas and S.E. Shreve, one can conclude that : 0
if b
2 2, then IP (So= +m) = IP (T" = +m) = 1, if b < 2, then P ( lim IX:lz = b = 1. t+S"
1
(25)
218
In case b < 2, we can deduce from the second item that 5'" = T". We now want to know whether S" = +m or not in this case. Let us introduce the speed measure m on (0, b) defined by m(dr)=
I- b / 2 dr 2 dr 4(b-r)pt(r) 2C '
and the function v such that, for any r E (O,b),
We have p(b-) = +m and therefore v(b-) = +m. In case b < 2, it is easy to check that v(O+) < 00. Using again the results of I. Karatzas and S.E. Shreve, we can deduce from this that in case b < 2, we have
P(S" < m) = P(T" < 0 )= 1.
(26)
In case 1x1 = 0, the former results (25) and (26) still hold. Indeed, let us suppose that x = 0 and let us introduce the stopping time T = inf { t 2 0, IX:12 2 Obvisouly, one has :
i}.
IP (3> 0, IX:12 = b)
= IP (3> 0, IX:12 = b and T
< m) .
In case b 2 2, using the strong Markov property of X a (see E. C6pa4 p. 86), one has :
IP (3> 0, IX,"12= b) = IP ( 3> 0, lX:12 = b and = ( j t > O, IxyIz (lT
T
< 00) ,
= b,
lX=X,)
>
= 0. In case b < 2, we use the fact that, due to the proof of (23), IP(IX~,,,I = 0) = 0. By the strong Markov property and since IP-a.s., S U ~ ~IXt~0 I2 [ < ~ b, , we~ have ~ ~IP ( ~ 3 > 0, lX:12 = b ) = E (P (3> 0, Ix:lz = b ) lx=X1/J = 1. In case of a non-deterministic initial condition Xo with law po, we can deduce the properties of Proposition 3.1 from the fact that (by uniqueness of the solution) :
IP (3> 0, lXt12 = b) =
J
IP ( 3> 0, IX:12 = b) dpo(x).
0
Remark 3.1. In case g E L~o,(R+),what we can conclude is the following : 0
0
if b 2 2 , then if b < 2, then
IP (3> 0, IX:lz = b) = 0, IP (3> 0, IX:12 = b) > 0.
219
3.2. Non-uniqueness i n case b
<2
In this section, we suppose b < 2 and P(IX0l2< b) = 1. We restrict our attention to the case g = 0. We are going to construct another process Xt weak solution to (6) and such that lP(3t > 0, X t 4 B)= 1. In other words, we will build a solution to (6) which, unlike Xt, goes out of the ball B. This will show that (6) admits at least two different solutions. Let us consider the solution Xt to (6) we have built in Section 2. We know that IP-as., the process Xt reaches the boundary of B in finite time (see Proposition 3.1). Let us introduce the stopping time T = inf{t 2 0, IXtI2 2 b}. In polar coordinate, we write XT = (&, 6 0 ) : ( X T ,Y T )= (&os(60), &sin(eo)), where 60 E [0,27r) denotes the polar angle. We now want to construct a solution to (. 6.) , which takes ( X T ,Y T )as initial value, and lives outside of the ball B. Let us introduce a two-dimensional standard Brownian motion (Pi, rt) independent of Wt. We use a polar representation (fi,et) of the process we want to build. We consider the solution rt to the following multivalued SDE :
{ where f : R
-+I
drt TO
+ d f (rt)dt 3 (2 + b) dt + 2&dPt,
= b,
- co,+MIis the convex function defined by
:
-b21n(r - b) if T > b, otherwise.
so that af is a simple-valued maximal monotone operator with domain I = (b,co) (for all T > b, d f ( r ) = { V f ( r ) }= {&}). By E. C6pa4, there exists a unique process rt solution to (27). Following exactly the arguments of Lemma 2.1, one can show that this process rt is such that : for any 0 1
-
-
drt =
< t < co,P-a.s.,
11 5 1 t
1
du
< M, with
+W,
-&dt + (2 + b) dt + 2fidPt.
Let us now consider the process 6t defined by :
and the random process
Xt
x t =
in R2 defined by :
(fi cos(dt), fisin(&))
convention
220
By ItB's formula, we have
1 x t d X t = -21-1x,12 d t
:
+ (-sin(&), cos(8t))dyt+ (cos(&),sin(Ot))dpt.
b
Using Paul L6vy characterisation, one can show that
(- sin(&),cos(Ot))dyt
+ (cos(&),sin(Ot))d,Bt = d B t
where Bt is a two-dimensional Brownian motion, independent of W t. Let us now consider Xt defined by 2,= l o < t < T X t + l t > T x t - T and the process w t defined by wt = W t A T l t > F B t - T . It is obvious (for example by Paul L6vy characterisation) that Wt is a Brownian motion. In addition, the process X t is a solution to (6) with g = 0, such that IP(3t > 0 , X t @ B)= 1. This shows that the problem (6) with g = 0 does not admit a unique solution.
+
Remark 3.2. In case g E L E c ( R + ) using , the solution (rt,0,) of the multivalued SDE : ( T O , 00) = (b,190) and d ( r t , e t )+ ah(rt,ot)dt 3 ((2 + b) + rt sin(&)g(t),- sin2(Qg(t)) dt
+ (2&,
&)d(Pt,yt),
where h : R2 +]- co,+co] is the convex function defined by h(r,0) = f ( r ) (see formula (28)), one can by the same arguments prove that there is non-uniqueness in law for the solutions to (6). We have summarized in Table 1 some of the results we have obtained in the last two sections.
IP(IXo12 = b) = 0.
I IP(IXo12= b) > 0.
b < 2. Existence. IP (32 0, lXt12 = b) = 1. Non-uniaueness. Existence. Non-uniqueness.
I
I
b 2 2. Existence. IP ( 3 2 0, lXt12 = b) = 0. Uniaueness Existence. Non-uniqueness
Table 1.: Properties of solutions to (6) when g = 0. We suppose IP(I_XO~~5 b) = 1. In any case, uniqueness holds for solutions with values in B according to Proposition 2.2. The terminology uniqueness and non uniqueness relates to a solution that is not enforced to take values in B.
221
4. Invariant probability measure in case g = 0 and b
2
2
In this section we are interested in invariant probability measures for the SDE ( 6 ) with g = 0 in case b 2 2. The motivation for this study is twofold. First, since we consider a fluid which is initially a t rest, it is natural from a physical point of view to choose an invariant probability for the SDE ( 6 ) with g = 0 as law for Xo. Second, in the analysis of the coupled system (1-3), we are interested in the regularity of the stress 7 ( t , y ) = IE
(
xTT )
1- ( X , Y P + ( Y t ? l P
which, by Girsanov,
can also be written in the following form :
where X i = ( X i , & ) denotes (as in last section) the solution with values in to (6) with g = 0 (see Ref. This expression of the stress yields the following estimate (using Holder inequality) : for almost all y and t ,
w h e r e p = L. 9-1
It is thus important to estimate the quantities IE which is simple if we identify and start under an invariant probability measure (see formula (31)). The density po defined by : exp(-2II(x))
b
+2 (
= Jexp(-2II(x)) dx - 27rb
1--
b’2 lI=IZ
(30)
obviously solves div (-(V,II)po + $ ( V , p o ) ) = 0 and is therefore a natural candidate to be invariant. This is indeed the case as shown by :
Proposition 4.1. For b 2 2, po(x) dx is the unique invariant probability measure on B f o r the SDE (6) with g = 0 . This proposition is a consequence of the following lemma :
Lemma 4.1. Let b 2 2. For any x E B , t > 0 , the solution XT of the SDE (6) with g = 0 and XO= x has a density p ( t , x, y ) with respect t o the Lebesgue measure o n B . In addition, V t 2 0 , (2) d x dY-a.e., exP(-2WX))P(t, 2 , Y) = eXP(-an(Y))P(t, Y , x),
222
(ai) Vx E B , dy-a.e., p(t, x, y )
> 0.
Indeed, by (i), one easily checks that po(x)d z is invariant. By (ii), any invariant probability measure is equivalent to the Lebesgue measure on B which implies uniqueness (see Proposition 6.1.9 p. 188 of M. DuAo 7). With Proposition 4.1, it is then straightforward to prove that, if X Ohas the density po(x), then we have :
Let us now prove Lemma 4.1. Proof. In order to prove (i), we regularize the potential II so that the results of L.C.G. Rogers l 2 (see p. 161) apply. Let II, be defined by :
Un(x) = nn(Ix12),
(32)
and T, is increasing and C2(R+,R+), so that VII, is bounded with continuous derivatives of first order. Let t > 0 and x E R2. According to L.C.G. Rogers, the solution Xn7"of the SDE : rt
has a density p,(t, x , y) with respect to the Lebesgue measure on R2 which satisfies dx dy-a.e., exp(-211n(x))p,(t, x , y) = exp(-2IIn(y))p,(t, y , x). For x E B , let 7," = inf{t 2 0,1XT12 2 b ( l Since IP (X;'" # XF) 5 P(T," < t ) ,according to Proposition 3.1,
k)}.
n+cc Iim
P (X:'z # X r ) = 0.
(35)
We deduce that for a fixed x E B , p n ( t , x , y ) converges in L$(lR2) to p(t, x , y), which is the density of XF. As the non-negative potential II, converges pointwise to II in B , we deduce that exp(-211n(x))p,(t, x, y) converges to exp(-2II(x))p(t, x , y) in Lk,,(B x B ) and conclude that (i) holds. We are now going to check (ii) for a fixed x E B and t > 0. Let A be a Bore1 subset of B such that 1~dx > 0. We choose n E N' such that
223
1zI2< b ( l -
i) and S 1 A n dx > 0 where A , = A n B
Girsanov Theorem, under
where
7,”
IP:
defined by :
is as above, (X:AT;)s5t is a Brownian motion starting
from z a n d stopped at t h e boundary of B
IP: (X:AT; E A,) > 0.
Therefore, IP(X: E A )
2 IP (X&,,
&)> 0, which concludes the proof.
EE ( I A , , (xFAT;)
E A,)
= CI
Acknowledgments This work has partly been motivated by some remarks of Claude Le Bris.
Bibliography 1. R.B. Bird, R.C. Armstrong, and 0. Hassager. Dynamics of polymeric liquids, volume 1. Wiley Interscience, 1987. 2. R.B. Bird, C.F. Curtiss, R.C. Armstrong, and 0. Hassager. Dynamics of polymeric liquids, volume 2. Wiley Interscience, 1987. 3. M. BOSSY,B. Jourdain, T. Leli$vre, C. Le Bris, and D. Talay. Existence of solution for a micro-macro model of polymeric fluid : the FENE model. In preparation. 4. E. CCpa. Equations diffkrentielles stochastiques multivoques. ThGse, Universit6 d’orlkans, 1994. 5. E. CCpa and D. Lepingle. Diffusing particles with electrostatic repulsion. Probab. Theory Relat. Fields, 107:429-449, 1997. 6. M. Doi and S.F. Edwards. The Theory of Polymer Dynamics. International Series of Monographs on Physics. Clarendon Press, 1988. 7. M. Duflo. Random iterative models. Springer, 1997. 8. B. Jourdain, T. LeliBvre, and C. Le Bris. Numerical analysis of micro-macro simulations of polymeric fluid flows : a simple case. to appear in Math. Models and Methods in Applied Sciences. 9. I. Karatzas and S.E. Shreve. Brownian motion and stochastic calculus. Springer-Verlag, 1988. 10. H.C. Ottinger. Stochastic Processes in Polymeric Fluids. Springer, 1995. 11. D. Revuz and M. Yor. Continuous martingales and Brownian motion. Springer-Verlag, 1994. 12. L.C.G. Rogers. Smooth transition densities for one-dimensional diffusions. Bull. London Math. SOC.,17:157-161, 1985.
ON THE DISPERSION OF SETS UNDER THE ACTION OF AN ISOTROPIC BROWNIAN FLOW*
H. LISEI Faculty of Mathematics a n d Computer Science, Babeg-Bolyai University, Str. Koga"1niceanu Nr. 1, RO - 3400 Cluj-Napoca, Romania E-mail: [email protected]
M. SCHEUTZOW Institut fur Mathematik, MA 7-5, Technische Universitat Berlin, Straj3e des 17. J u n i 136, 10623 Berlin, Germany E-mail: [email protected]. de
We give a survey on results about the growth of the diameter of the image of a bounded subset X of Rd under the action of a stochastic flow. We provide a new proof of the fact that, under reasonable assumptions, the diameter of this image set will almost surely grow a t most linearly in time, and we establish an explicit upper bound for the linear growth rate which is both simpler and better than previous bounds. Our main tool is the Garsia-Rodemich-Rumsey Lemma.
1. Introduction
Imagine that at time t = 0 an oil slick on the surface of the ocean covers the set X and that each oil particle moves randomly according to a random differential equation or a stochastic differential equation. Let &(z) be the location of the particle at time t 2 0 which started at z E X at time 0. It is of considerable practical importance to predict some characteristics of the random set &(X) := { &(z), z E X}. We regard the particles as passive tracers, which means that we assume they are being carried by the fluid without interacting with the fluid or with other particles. This assumption is rather unrealistic for oil particles but is in good agreement with reality *This work is supported by the DFG-Schwerpunktprogramm Interugierende stochastische Systeme won hoher Komplexitat. 224
225
for light pollutants like dust. It has been conjectured by R. Carmona and Y. Sinai3 that under reasonable assumptions, the diameter of the set &(X) will grow linearly in t . Proving the conjecture consists in showing that the set will grow at most linearly, i. e. in giving an upper bound for the linear growth rate, and that it grows at least linearly, i. e. that it has a non trivial linear lower bound. A linear upper bound was proved for a certain class of stochastic flows by Cranston, Scheutzow and Steinsaltz' and by the authors" using somewhat different methods. In section 3 we will use yet another method - namely the Garsia-Rodemich-Rumsey Lemma (in short: GRR) - t o prove an upper linear bound which in fact happens t o be better than the previous ones. In addition, our proof seems to be shorter and more transparent. We state the GRR-Lemma in the appendix. Lower linear bounds have been proved under various assumptions by Cranston, Scheutzow and Steinsaltz5, Scheutzow and Steinsaltz12 and Cranston and Scheutzow4. We state a corresponding result for isotropic flows in section 4 but only provide an idea of the proof. The reader is referred to the references for more general results and detailed proofs. Finally we state some open problems.
time 0
time T
Figure 1. dispersion of an oil spot
226
2. Isotropic Brownian Flows We will first define the concept of an isotropic covariance tensor (or matrix) b, then we will introduce isotropic Brownian fields and finally isotropic Brownian flows (driven by an isotropic Brownian field).
Definition 2.1. Let b = ( b i j ( z ) ) i , j = l , , , , , d be a positive semidefinite real matrix for each x E Rd. We say that b is an isotropic covariance tensor or matrix if (i) (ii) (iii) (iv)
z H b(z)is four times continuously differentiable. b(0) = Ed (the identity matrix) H b(z)is not constant. b(z)= G*b(Gz)Gfor all z E Rd, G E O ( d ) .
z
(i) is a convenient and not too restrictive smoothness assumption, (ii) a normalization condition, (iii) is assumed to avoid rigid motions later and (iv) ensures that b is invariant under orthogonal transformations -justifying the term isotropic. Following Baxendale and Harris2, we define the longitudinal and transverse correlation functions Br, and BN by
BL(r) = bii(rei), BN(r) = bii(rej),
20 r 2 0, j # i,
r
where e k , Ic = 1 , .. . , d denotes the standard basis of Rd. Due to isotropy the functions BL and BN do not depend on the choice of i and j . For later reference, we introduce the strictly positive parameters
PL
:= -Bg(O),
PN := -BZ(O). If U ( z ) ,z E Rd is a zero mean, Rd-valued Gaussian vector field with covariance cov(V(y x ) , U(y)) = b(z),then it is easy to check that U has a continuously differentiable modification and we have
+
for any i
#j.
227
Definition 2.2. Let b =
( b i j ( ~ ) ) ~ , ~ = ~ , ,be , , , an d isotropic covariance tensor. An Rd-valued random field M ( t ,x), t 4 0 , x € Rd defined on some probability space (R, F ,P) is called an isotropic Brownian field, if
( t ,x) H M ( t ,x) is a zero-mean Gaussian process. COV(M(S, x),M ( t ,y)) = (S A t )b(x - y). ( t , x ) H M ( t , x ) is continuous for almost all w E R. From this definition it is easy to obtain the following properties of M .
Corollary 2.1. Let M be an Rd-valued isotropic Brownian field. Then the following holds:
t H M ( t ,x) is a d-dimensional standard Brownian motion for each x ERd. <'M(.,x),M ( . ,y) >t= b(x - y) t for each x,y E Rd. Next, we consider the Kunita-type stochastic differential equation (sde)
d X ( t ) = M ( d t ,X ( t ) ) ,
(1)
where M is an isotropic Brownian field. It wits shown by Kunitag, Theorem 4.5.1, that this equation does not only have a unique solution for every initial condition X ( 0 ) = x E Rd but that it even generates a stochastic flow of homeomorphisms, i. e. that there exists a family ( @ s l ) ~ j s , t < o oof random homeomorphisms of Rd such that = $tu. 0 $st for all 0 5 s, t ,u < 00 and all w E R. = IdlRd for all s 2 0 and all w E 0. For each s 2 0, z E Rd ($st(x))t>s - solves (1) for t 2 s with initial condition X ( s ) = x. The map ( s , t , x )H $,t(x) is continuous for all w E R. $szL
0 0
0
$ss
We will call any such stochastic flow of homeomorphisms (based on a Kunita-type sde driven by an isotropic Brownian field M ) an isotropic Brownian flow. It is easy to see that for each x E Rd, $ ~ t ( x )t , 2 0 is a standard d-dimensional Brownian motion starting in x. We point out however, that for x # y the RZd-valued process ($ot(z),$ot(y))t20 is not Gaussian. In the following we will write $t instead of $ot. We will need the following facts concerning isotropic Brownian flows (see Baxendale and Harris2): 0
For each z
# y, t
H
ll$t(x) - $t(y)ll =: pt is a diffusion on (0,m)
228
with generator A g ( z ) = (1 where g E C.;
-
BL(z))g”(z)
+ (d - 1) (1 - B N ( z )) S’(z),
Therefore pt satisfies the sde
w
0
where is a suitable standard Brownian motion For each x E Rd, v E Rd\{O} 1 1 X := lim -logII(D&)(x)vII = - ( ( d - l),Ojv - P L ) a. s. t’cc t 2 X is called top Lyapunov exponent of the flow.
(3)
3. The Upper Bound We will formulate and prove an upper bound under the following condition. Condition (C): ( C l ) ( t ,x) H &(x) is a continuous random field on [ O , o o ) x Rd such that there exist A 2 0, u > 0 and b > 0 such that for each x,y E Rd there exists a one dimensional standard Brownian motion W such that ll$t(x)
-
dt(Y)II L:
11%
+aV),
- YII
0I tI inf{s where W: := S U (C2) There exist we have
~
Ws. ~
<
~
~
2 0 : llds(x) - @S(y)ll = b } ,
~
A > 0 , B 2 0 such that for each x E Rd and each Ic 2 0
where r+ = r V 0 denotes the positive part of r E R. We recall the concept of upper entropy dimension (see e.g. HoffmannJ@rgensen8). Let X be a bounded subset of Rd and let N ( X , r ) be the minimal number of subsets of diameter a t most r which cover X. Then the upper entropy dimension A of X is defined as
A := lim sup log N ( X ,r ) 7-10
log f
.
Remark 3.1. In Cranston, Scheutzow and Steinsaltz6 and Lisei and Scheutzow” an upper linear bound was established under the assumption that the so called local characteristics of the flow are bounded and Lipschitz,
229
which implies Condition (C), see Cranston, Scheutzow and Steinsaltz', Lemma 5.1. for ( Cl) and Lisei and Scheutzow", equation (9) for (C2). Isotropic Brownian flows possess bounded and Lipschitz characteristics and therefore satisfy (C). In fact we can infer from (a), using It6's formula applied to logpt, that for an isotropic Brownian flow and E > 0 there exists some b > 0 such that condition ( C l ) holds with A = (A+€)+ and 0 = Since the one-point motion of an isotropic Brownian flow is a standard d-dimensional Brownian motion it follows that (C2) holds with B = 0 and A=l.
a.
Theorem 3.1. Assume that q5 satisfies condition (C) and that X c Rd i s a compact subset with upper entropy dimension A > 0. T h e n we have
where
where A0 = 2c2d d- A ) . For an isotropic Brownian flow with top Lyapunov exponent X 2 0 we get the result above with
+
5
Proof. Choose E > 0 and ro > 0 such that logN(X, r ) 5 (A E ) log for all 0 < r _< ro. Further, let y,T > 0 satisfy e-yT _< rg. Then N ( X , e - T T ) 5 exp{yT(A+e)}. Let Xi,i = 1 , . . . , N ( X ,e-TT) be compact sets of diameter at most e-YT which cover X and choose arbitrary points xi E Xi.Define
-
x := {Xi, i = 1,.. . , N ( X ,C ' T ) } . For
> 0 we have
K
+
P{ sup ll$t(x) - zll 2 KT b for some x E X } 5 S1 O
+S2,
where
S1 := exp{yT(A + E ) } maxP{ sup ll&(x) - 511 2 KT - eCYT} sEX
Wt
230
and S 2 := exp{yT(A
+
E)}
maxP{ sup diam($t(Xi)) 2 b } . O
Using (C2) we get
Our aim is to identify the infimum
k over all
r;
for which there exists some
y > 0 and E > 0 such that the upper bounds of both S1 and S, above decay to zero exponentially fast as T --+ 00. A simple Borel-Cantelli argument
k
will then show that is indeed an upper bound for the linear growth = B A m , where YO is the infimum rate. Observing (6) we get of all y > 0 for which there exists some E > 0 such that 5’2 decays to 0 exponentially fast as T .+ 00. Rather than identifying 70, we will instead provide some yo 2 70.Then
k
+
K : = B + A ~ will turn out to be an upper bound for the linear growth rate. We will estimate SZusing the Lemma of Garsia-Rodemich-Rumsey (see Lemma 5.1). Define
0
We will use the abbreviation c := -T(1 U2 2
We have
and we will use the following estimates
+ 6).
23 1
and
Therefore
We fix T
> 0 , y > 0 and i E
We choose ,B = ecT with
( 1 , . . ., N ( X , e - Y T ) } and define
< 2 -A.
Using (7) and ( C l ) we get
By the GRR Lemma 5.1 applied with the metric
d ( f ,9 ) = SUP Ilf(t) - g(t)II A b OStlT
and by (8) it follows that
where
I := "Texp 0
{/
s
}
d
t
.
232
We have
Define
U = U ( y , 6 ) :=
y-2fiJ72’(CGJ -a%( 1 6)
+
+
if y 2 a2ci(1 6) otherwise.
Then
1 5 2 exp{-UT) (1 Assuming betT
+ 6 ~ 2 0 2 4 1 6+) T )
2 121 we have
P(ZT 2 b) 5 P ( e x p {
{-}
2
7T Cd
g)
(v > -
Using Chebyshev’s inequality, we obtain
P ( ~ 2Tb) < E V Cd 8pexp{ Using
-
‘(i.g(E))2}. 4C 121
we have
1 lim sup - log S2 5 y(A T-oo
for some E (i) (ii)
+
E) -
2dy
+ N2 + (t2026
-
> 0 provided that
t + U > 0; 5 + A 2 0;
(iii) Ay
-
2dy
+ A>2 - (5 + w2 + (E2026 2+(1 + 6) < O.
If
A 2
2a2d(d - A) A ’
then it is easy to check that
70:= A + a 2 A + Jo4A2 60 := (70- A)2d - 1’ a27oA2 Eo := -U(yo, 60)
(5 + w2
2a2(1
+ 2A02A,
+ 6)
<
233
“,”
satisfy (i)-(iii) above, provided “>” and ‘(<” are replaced by and “5” in (i) and (iii) respectively, and that 60 > 0. Further, it is easy to see that yo is greater or equal than the infimum of all y’sfor which there exist 6 > 0 and ‘$ such that (i)-(iii) are satisfied. If on the other hand ~
2a2d(d - A) A ’ then it is again easy to check that
A<
A 1, a2d ‘$0 := -U(Yo, 60) so := -
+
satisfy (i)-(iii) above, provided “>” and “<” are replaced by “2”and “5” in (i) and (iii) respectively. Further, it is easy to see that yo is greater or equal than the infimum of all y’s for which there exist 6 > 0 and E such that (i)-(iii) are satisfied. Therefore, for each €0 > 0 we have
Using the Borel-Cantelli Lemma and letting €0 go to 0 we obtain 1 limsup - sup sup Ilqbt(x)II 5 K := B A m a. s. T-ca T x E X O
+
which proves (4) of Theorem 3.1. Formula (5) for the isotropic case follows from (4) by inserting A and A = A, and using formula ( 3 ) and Remark 3.1. B = 0, a =
=
1, [7
Remark 3.2. If & is a homeomorphism of Rd for every t 2 0 and if the upper entropy dimension A of the set X is greater than d - 1, then (4) and (5) remain true when replacing A in the definition of K by the smaller number d - 1. To see this, one can take a closed ball B which contains X and apply Theorem 3.1 with X replaced by aB, which has (upper entropy) dimension d - 1. Due to the homeomorphic property of 4t, the upper linear growth rate of & ( X ) is bounded by that of &(as). 4. The Lower Bound
In the following we will call a subset of Rd nontrivial if it contains a t least two points.
234
Theorem 4.1. Let (4t)t>O be an isotropic Brownian jlow on Rd, d 2 2. There exists a number c* > 0 such that for any nontrivial, connected,
compact subset X
c Rd we have
1 P liminf -diam(q5t (X)) 2 c* t+m t
{
diam(q5t (X)) = 0} = 1 (10)
and the first of the two probabilities is strictly positive. Since the two events in (10) are disjoint Theorem 4.1 says that for any subset X as above one of the following two cases will occur almost surely: either the diameter will grow to infinity with a t least linear speed c* or the diameter will shrink to zero. Even if the top Lyapunov exponent X is negative, linear growth will occur with strictly positive probability. Remark 4.1. It is easy to see that Theorem 4.1 will no longer hold if we either allow the set X to be finite or if d = 1. In the first case the diameter of 4 t ( X ) equals the maximum of the distance of a finite number of (correlated) Brownian motions in Rdwhich grows at most like a constant times (t log log t)'/2.In the second case the compact set X c R is contained in a compact interval [a,b ] , and hence diam (& ( 2 ) 5 ) &(b) - & ( a ) , which again grows at most like a constant times (t loglogt)1/2.
*
* St
St
1
t=O
Figure 2.
coordinate
t=l
1
linear expansion in t h e first coordinate direction
coordinate
235
Idea of the proof of Theorem 4.1. We sketch the competition and selection procedure to show that as long as the diameter of the set $ t ( X ) does not become too small, supzEx 4t(x) will grow a t least linearly in t (the upper index 1 stands for the first coordinate). Using isotropy of the flow, this implies that &(X) will grow a t least linearly in every direction, which is actually more than what we claim in the theorem. A complete proof (even for more general stochastic flows) can be found in Scheutzow and Steinsaltz12. Consider two points x and y in X such that 112 - yII = 1 and z12 y1 (assume that X has diameter at least 1). Since t H $;(x) is a martingale, we have
E (+:(~)lFo) = x1 = x1 V yl. Further it is plausible (and true) that there exists some p > 0 (not depending on the particular choice of z and y) such that
p (&Y)
2 4;(4
+ 1lFo) 1p
a. s.
Observe that a t this point we need the assumption d 2 2: for d = 1 it is impossible for a trajectory to pass another trajectory. Therefore
E(
4 m v 4ZY)IFo)
=E
( 4 x 4 + (4XY) - 4 : ( 4 ) + I F o )
2 x1v y'
+p
a. s.
Now we iterate the procedure by selecting x or y depending on whether $;(x) or $:(y) is larger. Assume that x is the winner. Then we pick a new competitor z E X for which Il$:(z) - $i(x)II = 1 and so on. Therefore in each unit time step the right frontier of the set 4 t ( X ) moves to the right by an average a t least p. Now a suitable version of the law of large numbers 0 for martingales (essentially) finishes the proof.
Remark 4.2. Under weaker conditions than in Theorem 4.1, Scheutzow and Steinsaltz12 proved much stronger results than 4.1, namely so-called bull chasing properties. We formulate one result for isotropic Brownian flows in dimension 2 or greater with a nonnegative Lyapunov exponent: there exist numbers c1 > 0 and c2 2 0 such that for any process II, : [ O , o o ) -+ Rd which is adapted to the filtration of 4 and which is Lipschitz continuous with constant c1, and for any nontrivial connected subset X ,there exists almost surely some x E X for which
lim sup t-cc
IlM.)
- +tll
log t
5 c2.
236
5. Open Problems
In this section we assume that (q&)t?o is an isotropic Brownian flow which has a nonnegative top Lyapunov exponent A. We list some open problems. Is it true that for any (reasonable) nontrivial compact subset Rd the limit 1 lim - diam((bT(X)) T-cc
XC
T
exists almost surely? If so, is it deterministic? If the answer to both questions is yes, does this limit depend on the set X (e.g. on its dimension A)? Since our upper bound depends on A we conjecture that the linear growth rate will depend on A in general. Let X be a curve in Rd of finite length LO > 0, and let LT be the length of the curve ~ T ( X )How . does LT grow as T + oo? I t seems reasonable to conjecture that l i m $ l o g L ~ = X almost surely but we conjecture that LT will grow faster, namely that 1 lirn - log LT = X
T-oo
T
PL +almost surely. 2
It has been shown by G. Dimitrofl, using martingale arguments, that 1 log LT 5 lim sup 1 log LT 5 X PL X 5 lim inf a. s. T-too T T-m T 2 Let X be a compact subset of positive d-dimensional Lebesgue measure and let V, be the d-dimensional Lebesgue measure of 4 t ( X ) . It is not hard to see that (V,)t?o is a (nonnegative) martingale (see Baxendale and Harris2). By the martingale convergence theorem V, converges almost surely to a (finite) random variable V,. We conjecture that V, > 0 almost surely. Is it true that q5t(X) becomes dense in Rd as t -+ 03 in case X is a nontrivial, connected and compact subset of Rd? More precisely we can ask if
+
lim P {w : &(u, X)n B
t-,
# 0}
=
1
holds for any nonempty open subset B C Rd. Let X be a compact subset of Rd with nonempty interior and denote by Ad the d-dimensional Lebesgue measure. Does there exist a function I : [0, oo)--$ [0, m) such that 1 lim -logXd {x E X : I I ~ T ( w , z ) I I L y T } = -I(y) a. s.? T-+m T
237
If such a function I exists, then it will take the value +00 for sufficiently large values of y by Theorem 3.1. The following simple observation shows that if such a function I exists, then I ( y ) 2 y2/2 for all y 2 0: using Chebychev's inequality and Fubini's theorem, we get for any E > 0
Since
$.
we get I(?) 2 If the flow is volume-preserving, equation (11) provides an upper bound for the probability that the amount of oil (say) which is found outside a ball of radius yT at time T exceeds
11
the value exp - - E T (since it is easy t o find an explicit H 7 2 * upper bound for P { \ I ~ T ( x ) ~ ( 2 yT}). This bound does not use any information about the correlation of several tracers and it is likely that it can be improved considerably by using such information.
Appendix We state the following lemma which is originally due to Garsia, Rodemich and Rumsey and which we briefly refer to as the GRR-Lemma. A proof (of a more general version) can be found in Arnold and Imkeller'.
Lemma 5.1. Let B be a compact subset of Rd, ( E , d ) a metric space, Q : [0,00) -+ [0,00)a right-continuous and strictly increasing function satisfying Q(0) = 0 and assume that f : B -+ E is continuous. If
then we have
where Cd denotes the square of the volume of a ball of radius 1 in Rd.
238
References 1. L. Arnold and P. Imkeller, Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory, Stoch. Proc. Appl. 62, 19-54 (1996). 2. P. Baxendale and T. Harris, Isotropic stochastic flows, Ann. Probab. 14,11551179 (1986). 3. R. Carmona and F. Cerou, Transport b y incompressible random velocityfields: simulations and mathematical conjectures, in: Stochastic partial differential equations: six perspectives, eds. R. Carmona and B. Rozovskii, AMS, 1999. 4. M. Cranston and M. Scheutzow, Dispersion rates under finite mode Kolmogorov flows, Ann. Appl. Probab., 12,511-532 (2002). 5. M. Cranston, M. Scheutzow and D. Steinsaltz, Linear expansion of isotropic Brownian flows, Electron. Commun. Probab. 4, 91-101 (1999). 6. M. Cranston, M. Scheutzow and D. Steinsaltz, Linear bounds for stochastic dispersion, Ann. Probab. 28, 1852-1869 (2000). 7. G. Dimitroff, forthcoming Ph. D. thesis, Technische Universitiit Berlin. 8. J. Hoffmann-J~rgensen, Probability with a view toward statistics, Vol. II, Chapman & Hall, 1994. 9. H. Kunita, Stochastic flows and stochastic differential equations, Cambridge University Press, 1990. 10. M. Ledoux and M. Talagrand, Probability in Banach spaces, Springer, 1991. 11. H. Lisei and M. Scheutzow, Linear bounds and Gaussian tails in a stochastic dispersion model, Stochastics and Dynamics 1,389-403 (2001). 12. M. Scheutzow and D. Steinsaltz, Chasing balls through martingale fields, Ann. Probab. 30, 2046-2080 (2002).
STOCHASTIC BURGERS EQUATION IN D-DIMENSIONS A ONE-DIMENSIONAL ANALYSIS: HOT AND COOL CAUSTICS AND INTERMITTENCE OF STOCHASTIC TURBULENCE
A. TRUMAN*, C. N . REYNOLDS AND D. WILLIAMS Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, Wales, UK
We give a one dimensional analysis of the solution vp of the stochastic Burgers equation in d dimensions, with viscosity p2 0, as obtained by Davies, Truman and Zhao. Our analysis shows how the graph of a simple action functional in one space variable can be used to decompose the caustics into hot and cool parts. The inviscid limiting Burgers velocity field has a jump discontinuity across a cool part but is continuous as you cross the hot part. Our analysis also enables us to get a hold on the intermittence of stochastic turbulence in terms of the recurrence of a one dimensional stochastic process C simply related to the reduced action. Some detailed examples are discussed. N
1. I n t r o d u c t i o n
Burgers equation has been used to model the large scale structure of spacetime (Shandarin and Zeldovich and in a noisy environment in studies of turbulence ( E, Khanin, Maze1 and Sinai Here we develop some related results. We begin by giving a brief account of the results of Davies, Truman and Z h a ~ ~Let > ~Wt . be a B M ( R ) process on the probability space 3,P ) with
(a,
lE[WtW,] = (s A t ) . Consider the stochastic viscous Burgers equation for up = u p ( z , t ) , IC E Rd,
t > 0.
* [email protected] 239
240 VP(Z,0) =
VSO(Z)
,
Wt being white noise, pz coefficient of viscosity. We are interested in
Burgulence, that is the advent of discontinuities in v’(z, t) = lim v ~ ( zt), . P
V
The corresponding Stratonovich heat equation is
U Y X ,0) = exp
(-so(x)/p2)To(Z),
the convergence factor TObeing related to the initial Burgers fluid density. Here the connection is the Hopf-Cole transformation V P ( Z , t) =
- p 2 v lnuP(z, t)
.
Following Donsker, Freidlin et a1 l1 we expect as p -pz lnuP(z, t)
4
\0
+
inf [So(X(O)) A ( X ( O ) ,z, t ) ]= S ( x ,t ) , X(0)
where
A ( X ( O ) ,Z, t ) = inf
A [ X ],
X(S) X(t)=x
1 t
t
X 2 ( s ) ds
A [ X ]= 2-1
-
1 t
c ( X ( s ) )ds
-E
k , ( X ( s ) )dW,
This gives the minimal entropy solution of the Burgers equation
12.
Set
+
d [ X ]:= A [ X ] S o ( X ( 0 ) ). Then necessary conditions for X to be an extremiser of A are :
0
=
dX(s)
+ V c ( X ( s ) )ds + &VlCs(X(s))dW,
,
X ( 0 ) = VSo(X(0)) . Minimising A [ X ] over X ( 0 ) gives S(z,t) which satisfies the HamiltonJacobi equation
dSt
+ (2-1
+
(VStj2 c(z)) dt
+ &(x)
dWt
=0
,
St=o(.)= So(.) . Definition 1.1. We define the stochastic wavefront Wt in x by {x : S ( Z , t) = O}.
24 1
For small p , the heat equation solution u p switches continuously from being exponentially large to small as we cross the wavefront. up can also switch discontinuously. Define the classical flow m a p Q S : Rd 4 Rd by d&s with
+ VC(@,)ds + E V ~ ~ (dWs @ , )= 0
= id, 60= 0 5 ' 0 . So, since by definition
X(s)=@,q1X
I
X ( t ) = x,
]
where we accept that xo(x,t ) = @,lz is not necessarily unique. Given some regularity, the global inverse function theorem gives the is a random diffeomorphism. caustic time T ( d )such that for s < T(w), Moreover, for t < T ( w ) , UO(5,
t ) = d&@;lz
)
is a classical C1 solution of Burgers equation. From the method of characteristics, we expect non-uniqueness of xo(z,t ) to be associated with discontinuities in vo(x,t). The simplest way €or this to arise is if a positive, infinitesimal volume of points is focused into zero volume by @t.
Definition 1.2. Det- ax(t)= 0 Detax(t) 8x0 When @,'{z} critical point,
1
Pre-caustic in zo, QT'C~,
8x0
Caustic in x, Ct.
=0 so=zo(z,t)
= { ~ ~ (t )z,z,z ( x ,t ) ,. . . , xE(x,t ) } , for a non-degenerate n
u p ( z ,t)N
C
~i
exp (-~:(z, t ) / p 2 ,
i=l
where for i = 1 , 2 , . . . , n
+
SA("Cl t ) = So(x9(z1 t ) ) A(sA(z, t ) ,z, t ) I and Oi is an asymptotic series in p2. When zo(x,t)is unique, t < T ( w ) , the asymptotic series for up = vp(x, t ) is given for each integer m 2 0 by
242
c
(
m
w p (x,t ) =
p2jvj (x,t ) - p2V lnE{ exp
t
V . um(y;, t - s) ds
-
j =O
Here Uj(X,t) =
V S j ( 2 , t ),
Sj satisfying
for j = 0 , 1 , 2 , . . .with the convention 2-lAS-l explicitly5. Moreover, the Nelson diffusion process
=
-c-&lctWt, can be found
j =O
yo” = x .
Here, we see as p
-
0, the leading term V P ( Z ,t )
-
VSO(Z, t )
+O ( 2 ),
where SOis the solution of the Hamilton-Jacobi equation, which minimises the action A. When @T’{x} = {xA(x,t ) ,xg(x,t ) ,. . . , $(x, t ) } , there is a similar asymptotic series 64 for the ith term in the series. Since
S(x,t) = min Sh(x,t) , z=1,2, ...,n we define the zero level surface by
H,“ = {x : ~ h ( r ct, ) = o , for some i} , where H: includes the wavefront. The dominant term for wo(x,t)comes from the minimising 20(x,t)(assumed unique) and we obtain the corresponding Burgers velocity field VO(X, t ) = & q ’ z =
&tS,(x,t)
Two x;(x,t)’scan coalesce a.nd disappear as we cross the caustic. When this corresponds to the minimiser jumping, up=o has a jump discontinuity and we say that this part of the caustic is cool.
243
Example 1.1. (Cusp and Tricorn). In two dimensions c = 0, kt z 0, &(z, y) = The multiplicity of 20’s changes as we cross the caustic.
9.
cusp
3t 2 1 -zo - 2 t
.
z ( z 0 , t )= s(ii
Ht” :
Y(X0,t)= Figure 1. Multiplicity of
Evidently, n, the multiplicity of zo(z,t), depends on z and t . This multiplicity changes by multiples of 2 as we cross the caustic surface. This is associated with level surfaces of Hamilton’s principal function having cusps on the caustic caused by 2 different zo(z,t)’s coalescing. This is illustrated in 1-dimension by considering
I(x,t) =
.I
G(zo)eiF(xO, a ? t ) / P zdzo
,G
E
C,-(R)
7
R
where i = G. Consider the graph of the phase function F,,t(so) = F ( z o , z , t )as z crosses the caustic. (D,F” > 0 in neighbourhood of (l).)
244
Cusped Side of Caustic
On cool part of Caustic
Beyond Caustic
Begin moving x in direction n.
2 coalesce at the point of inflection. Zo(x,t) is the global minimiser of Fz,t(.) .
New ZO(Z,t ) here.
i). Zo(x,t ) jumps from position (1) to position (2). This causes u p and up to change discontinuously as we cross the caustic. ii). This only happens when a point of inflexion is the global minimiser of &(.I. iii). Some parts of the caustic (the cool parts) will be jump discontinuities in wo and uo.Coalescing 2x0’sis associated with level surfaces of Hamilton’s function having cusps on the caustic. I t is therefore important to know when Ht has a cusp on the caustic. We investigate this in the next section.
1.1. W h e n does Ht have cusps?
I
\
I4
\
,, ,
,, I I
Figure 2: Pre-Caustic and Figure 3: Cusp, Tricorn and Pre-Level Surface Line Pair An important insight about where Ht has cusps is that the cusped part
245
where @FICt is the pre-caustic and @ T I H tthe pre-level surface, determined algebraically. If you want to find cusps on the level surfaces of Hamilton's principal function you look for images of intersections of the corresponding pre-level surfaces with the pre-caustic.
Figure 4: Pre-Level Surface and Pre-Caustic
Figure 5: Level Surface and Caustic
As we shall see, if you want nc(t),the number of cusped curves in (Ct n H,) to change, the simplest way is for the pre-surfaces @FICt and @FIHtto touch, or for the Burgers velocity field to be zero on the caustic, or orthogonal to the caustic. The turbulent times t are when nc(t) changes. For the stochastic Burgers equation, such times t are the zeros of a stochastic process C, i.e. times t satisfying c(t) = 0. Typically these zeros form a perfect set - an infinite set containing no isolated points. At such times the geometry of the surface of discontinuity of vo can change infinitely rapidly - reflecting turbulent behaviour of the fluid. If C is recurrent to 0, the scale of random fluctuations varies,in a random periodic way. This will be seen as intermittence of stochastic turbulence, when the cusp is on the minimising part of the level surface of the Hamilton Jacobi function i.e. on the cool part of the caustic. We shall see this can be investigated by the one-dimensional graph above. Our analysis also shows which part of the caustic corresponds to discontinuities in vo. in both deterministic and stochastic cases.
246
2. Some Geometrical Results
= 1)
(E
We investigate the geometrical relationship between curves on level surfaces of the Hamilton Jacobi function and caustics for Burgers equation. In 2 dimensions the curves are the level surfaces themselves. In 3 dimensions we think of them as arising by taking planar cross sections.
Definition 2.1. A curve x = x(y), y E N(yo,6) is said to have a generalised cusp at y = yo, y being an intrinsic variable such as arc-length, if
Consider first the deterministic case
E
= 0. Here
where
The corresponding Euler-Lagrange equations read
X ( s ) = -Vc(X(s)) , and X ( t ) = 2, X(0) = ZO. The free case corresponds to c = 0,
Consider the level surface H t obtained by eliminating
A(Q, x,t)= 0
and
i3A
-(xo, ax,.
x,t ) = 0 ,
20
between
cli =
1 , 2 , . . .d .
Eliminating 5 alternatively gives the pre-level surface @T1H:. Similarly the pre-caustic (and caustic) are obtained by eliminating 2 (or XO) between Det(e(xo,x,t)) = 0 ax0
and
-i3A (xo,~,t)=O, ax;
a = 1 , 2, . . .d .
247
We denote these by @T'Ct and Ct. (In passing, we point out that the processes of determining @FICtand @L'Ht are algebraic. So @;'Ct and @T'Ht are algebraic inverse images not the topological inverse images @F (Ct) and @F1(Ht) .)
'
In the free case the equation for the zero pre-level surface is the eikonal
equation
t
+ So(x0) = 0 ,
- /VS0(zo)l2
2
and the derivative map DDt(xo) : T,,
D@t(zo)= ( I
+ T,
is given by
+ tV2So(zo)) .
The following elementary identity is the key to the free case
Vx,
{5
IPS0(zo)l2
1
+ So(x0)
=
(I
+ tV2So(xo))VSo(z0)
The next lemma and proposition illustrate the scope of our results in 2 dimensions.
Lemma 2.1. A s s u m e the pre-level surface meets the pre-caustic at xo where [ ( I tV2So(zo))VSO(xo)(# 0 and dim (Ker ( I + t V 2 S o ( z o ) ) )= 1. T h e n the tangent plane t o the pre-level surface T,, i s spanned by K e r ( ( I + tV2So(xo))).
+
Proof. At the point of intersection, the normal to the pre-level surface is a linear combination of the eigenvectors of ( I tV2So(zo))corresponding t o non zero eigenvalues. Let eo be the eigenvector corresponding to the eigenvalue zero. This normal is orthogonal to eo, so T,, =< eo >.
+
P r o p o s i t i o n 2.1. A s s u m e that [(I+tV2So(xo))VSo(xo)I # 0 , so that xo i s not a singular point of @;'Ht. T h e n @t(xo)can only be a generalised cusp, if @t(xo) E Ct, the caustic. Moreover, if z = @txoE @t(@;lCt n @;'Ht), x will indeed be a generalised cusp of the level surface. Proof. We have normal
# 0 and
n(z0)
%(7)I
For this to be zero it is necessary that Det ( I BTlCt. Trivially from Lemma (2.1)
l Y =Yo
# 0 and from above Y=YO
+ tV2So(xo))= 0 , so zo E = 0 , since &( dy
) 11 eo.
248
It is very easy t o generalise the above to d dimensions and t o include noise. Let the stochastic action be defined by
where X , = X ( s ) = X ( s ,xo,po) E Rd and d X ( s ) = - V C ( X ( S ) )ds
-
E V ~ ( X ( SS)) dW, ,
,
s E [O, tl
1
with X ( 0 ) = xo, X ( 0 ) = P O ; z o 1 p o E Rd.We assume X, is F,-measurable and unique. If du, dX, = 0, we have from ItB’s formula
In particular this is true when us = for any a! = 1 , 2 , . . . ,d . Using Kunita6, mild regularity gives with above Equation (1). = 1 , 2 , . . ., d ,
Q
almost surely. This gives:
Lemma 2.2. Assume SO,c E C2 and k E C2io,V c , V k Lipschitz, with Hessians V 2 c ,V 2 k and all second derivatives with respect to space variables of c and k are bounded. Then, for po possibly xo dependent, we have -(xo,po,t) dA ax,.
= X(t).%
ax,.
-
X,(O)
a!
7
= 1 , 2 , . . ., d
Now let
4x015 1 t )=
A ( X o 1 Po, t)lpo=po(zo,z,t) 1
where PO = p o ( x o , z , t ) is the (random) minimiser (assumed unique) of A(zo,po,t) with X ( t , x o , p o ) = z. (Here we need the map po H X ( t ,x0,po) E Rd to be onto for all 20. Methods of Kolokoltsov et a18>’ guarantee this for small t.)
Theorem 2.1. The classical stochastic flow m a p d ax,.
-[SO(XO)+ A(zo, 2,t)l = 0 so that x
= Qtxo.
7
@t
a! =
is defined by 1 , 2 , . . .d
,
249
Assume now that A(x0,x , t ) is C4 in space variables and Det Then we can show that:
(a) # 0.
Lemma 2.3. The random classical flow map has Frechet derivative a s .
Proposition 2.2. The random pre-level surface at a point xo is obtained by eliminating x between A(xo,x , t ) = c and ~d d( x ox , t),O, , (Y = 1 , 2 , . . . ,d . dx, Then the normal to the pre-level surface at the point xo is
We content ourselves here by quoting a result in 3 dimensions. Theorem 2.2. Let x E cusp(^^) = { x E (@;lCt n @ ; ' H ~ ) , x = a t x O n , ( x o )# o } . Then in 3 dimensions in the stochastic case, T,, the tangent space t o the level surface at x , is at most one-dimensional.
at
Proof. On the caustic at Qt(xo),Det
(s) =
0, so there exists eo E
Ker ( @ ( x o , 2,t ) ) , eo # 0. From the above eo . n = 0, so eo E T,,, the tangent plane to the pre-level surface. Similarly ( n A eo) E Txo. From the explicit form of D@t(xo)we see that D@t(xo)eo= 0. Therefore, T, is spanned by DQt ( n A eo). The above explains the geometry of level surfaces of the Hamilton Jacobi function. We know that u p changes dramatically as we cross Cusp(Ct n H t ) in the cool region. What about discontinuities in up as p O? Let us now see how a simple one-dimensional analysis reveals all. N
Definition 2.2. The classical flow map
is globally reducible if
Yo = (Yo , Vo, . . . > Yo 1 y& = y k ( y , y o1 , y o2 , . . ., yyo'-',t) , r = d , d - l , d - 2 , . . . , 2 .
Y = %Yo , Y
=
(Y' , Y 2 , . . . , Y
d
1
1
2
d
1
Given some differentiability and non-vanishing of derivatives this will be true locally. We want a global result. We want C 2 functions yo",yod-', . . . , such that Yy," =
Y~(Y,Y~,Y~,...,Yod-l,t)
250
where y,”( ) = y$(y, yo, 1 yo, 2 . . . yt-’, t ) . No root is repeated so second derivatives of A do not vanish. (Evidently we are assuming a favoured ordering of coordinates and a corresponding decomposition of @ t , so that non-uniqueness is reduced to the level of the y: coordinate.) Proposition 2.3. Assume the reduced action
f
(YA, 9, t ) = 4
@t
map is globally reducible. Define the
Y k Y 3 Y >YIL t ) ,V03(Yj YIL YE(Y, Y h , t ) ) ,
1
Y, t )
Then ,
af
a). T(yA, 8Yo
y, t ) = 0 and Equations (2)
af
* y = atyo
a2f 2(YILYlt) = 0 ii). Equations (2) and 7 ( y A 1y1t ) = dY0 (aY; 1 y = a t y o is such that the number of solutions yo of this equation changes.
*
Lemma 2.4.
where the last term is f”(yh, y, t ) and the first ( d - 1) terms are non-zero as above.
25 1
The above results follow by applying the principle of stationary phase t o
% (yh, y, t ) = 0 and y ayo
is
such that q ( y h , y, t ) # 0, then the first equation will have n roots yh
=
For instance by stationary phase, if we assume (ago)
a : ( y , t ) ,ai(y, t ) ,. . . , aA(y, t ) . If we vary y now so that %(yi,
(ad)
9, t ) = 0,
typically two of the above critical points will coalesce - a local maximum and a local minimum forming a point of inflection. Then, if D, a z f , (yh , y, t ) # (ad) 0, D, directional derivative, we have the picture shown below.
Here the picture deforms as we move in direction n.
Here 2 a:'s coalesce, say Here the point of inflection at (1) has disapand uA. a n1 P l ( y , t ) = uA(g,t) = peared. a : ( y , t ) , a repeated root.
UA-~
Because the value f (a:(y, t ) ,y, t ) < mini=1,2,...,,-2 f (ai(y, t ) ,y, t ) the = a; is the minimising one. So the minimiser disappearing root jumps from (1) to (2). Hence wo is discontinuous and uo is exponentially discontinuous. Hence the function fy,t(yi) = f (yi , y, t ) gives a complete analysis of the discontinuities. A similar analysis may be given if you only have local reducibility. This explains how to analyse hot and cool parts of the caustic. 3. Intermittence of Stochastic Turbulence
Here we illustrate how turbulent times and turbulent processes C can be determined when at is globally reducible. For simplicity we work in two dimensions.
252
Proposition 3.1. Assume @t is globally reducible. Let f(x,t,(xA), the reduced action, be defined as above, so that f(x,t,
where x =
(4)= 4:, xo2(x,x:, t ) ,2, t ) ,
(z:), (xi) xo =
and
50
x
=
@txo s f{x,t)(xA) = o and xg
=x i(.,
xo, 1 t) .
W h e n x E Ct, the random caustic, let f[x,t,(xh)= 0 have the repeated root xi = xg(x,t), Let X H xt(X) be a parameterisation of Ct, X E R, such that X = A0 corresponds t o a cusp o n the caustic, or a point o n the caustic where the Burgers velocity field is zero or orthogonal t o the caustic. T h e n the processes f o r stochastic turbulence at x t ( X 0 ) are given by
<
= f(z,(x,,,t)(.;;(.t(XO),t))
- c,
for c E R.
Remark 3.1. The ( 0 processes are just the stochastic action evaluated at the relevant points on the caustic and their inverse images. Similar results hold in &dimensions and for more general noise. Proof. Firstly, X H x2;(xt(A),t ) (the equal root vector (xh,xi(x,xA,t ) ,. . .) evaluated at x; = zE(x,t ) , x = xt(X)) is a parameterisation of the precaustic @FICt. Hence, the number of cusps on the level surface 5' = c is given by #{A E
R : f(xc,(x,,t,(.2;(xt(X),t)) = c } .
Differentiating our last equation with respect to X gives X < c ( t ) = 0.
= Xo
and
The above suggests the nomenclature for the three kinds of turbulence - cusp, zero and orthogonal turbulence. We expect orthogonal turbulence to be the most important. Similar results hold in higher dimensions. 4. Some Analytical Results (Small E )
Here we summarise some of the (small E ) analytical results of Davies, Truman and Zhao3i4. Consider dv
+ (u.V) dt = -VC(Z) dt
-
~ V k ( 2dWt ) ,
253
and the corresponding stochastic classical mechanics dXi"(zo, S) = -VC(X'(ZO,s ) )ds - E V ~ ( X ' ( Z s)) O , dW9
,
with X'(z0,O) = zo and Xi"(xo,O)= VSo(zo),0 < s < t . Let X o ( z ,s) = @:xo satisfy the deterministic ( E = 0) version of the above equation and let 4 be given by Bi, = { X , " ( u ) , X ! ( s ) }8(s - u),the product of the Poisson bracket { } and the Heaviside function 8.
Lemma 4.1.
G satisfies the matrix Jacobi equation
with boundary condition
Let X ' ( ~ Os ), = @;ZO
-
EL'
SO, s , u ) V k ( @ ; z o dW, ) ,
for s E [O, t]. This is the first term in, the perturbation expansion for X ' . Theorem 4.1. Given some mild conditions on continuity and boundedness of c and k and their derivatives, there exists a constant A4 > 0 such that for a n y 6 > 0 and suficiently small E > 0
and
In particular,
and
V X " ( x 0 ,s ) - VX'(z0, s ) = O(&+) , as
E
\ 0 in probability.
254
It is not difficult to prove from the above that the pre-caustic surface of the stochastic mechanics converges to the pre-caustic surface of classical mechanics as E \ 0 in probability. Caustic surfaces are stable in probability. What about the stability of level surfaces of the Hamilton-Jacobi function? We can prove: 2
Theorem 4.2. Let @, be the minimiser of 2-1
Jot c(@,xo)ds
satisfying Qtxo
So(x, t ) and let
@:
Ji l&sx~l ds+So(@txo)
-
x, with corresponding minimum
=
be the minimiser of 2-1
s," I&:xo/
2
ds
+ So(@zxo)
-
s," c(@:xo)d s - E J i
k(@zxo)dW,, satisfying @zxo = x for almost all w with corresponding minimum S E ( x , t ) .Then we have for almost all w
EL
E R, ER
t
So(x, t)-
k(@;xo)dW,
In particular, as
E
I SE(x, t ) 5 So(x, t )- E
\ 0, S E ( xt, ) + So(x, t) as.
Finally, if we assume there exists a unique z o for fixed t and x such that Qtxo = x, then the first approximation is rt
S'(Z,t ) = S 0 ( z t, ) - E
lo
k ( @ . , ~ odW, )
+
O(E)
,
where So(x, t ) is Hamilton's principal function for the path X o ( x o ,s). Similar results hold for xh(z,t ) and corresponding S'.
5. Some Applications We give two elementary results illustrating the kind of applications now accessible.
5.1. Hot and Cool Parts of the Caustic Recall that when the level surface of Hamilton's principal function (with a cusp at the point of intersection with the caustic) is the minimiser of the action at the point ( x , t ) of intersection we say the caustic is cool. The corresponding solution of the heat equation will have a jump discontinuity here because fio(z,t ) will jump.
Theorem 5.1. (Polynomial Swallowtail in 2 dimensions). Let c = 0 , kt(x,y) = x, So(xo,yo) = x; xgyo. We have global reducibility and
+
255
y o ( y ,XO) = ?/
-
txg. T h e n the graph in question is
(
t
x; xzo ~ ( z o= ) 2t - -+ t
as,
-dyo (xo,Yo(Y,xo)))
2
7 1Wsds t
2 -
where
-
As expected
(;)
= Qt
(;:)
f’(X0) = 0
and yo
2
= y - txo
Moreover, additionally ( 5 ,y ) E
ct * f ” ( Z 0 )
=0 .
Analysis of the graph o f f ( . ) yields the hot and cool parts of swallowtail, as shown in Figure 6, where =
(
-EL (-&-EL
+ 8v@) 18000
-t5(3
and K =
t
1
Wsds,--+ 2t
t
t3 ( 9 - 4) 450
W s d r , - - ~ + ~. ) 2t
50
Proof. I t may be shown7 that, for k t ( z , y ) 3 x, the effect of noise is to bodily translate the whole picture in the direction ( - 1 , O ) . Hence we need only consider the deterministic setting, in which
f(xo)= Z; We consider the roots of
-
t
4
2
2t
x2
-xi + -(1+ 2 t y ) - -+ -
f’(x0)for
502
t 2t the following two cases.
+
3 Case 1 : y < -2t1 or y > --%l &. Since (x,y ) E Ct we know f’(x0) = 0 has only one solution, namely the repeated solution x2;(( z, y ) , t ) . Thus f (zo) has only one stationary point which is a point of inflection and so one side of this part of CL is cool. Case 2 : -%1 5 y 5 -&
+ &.
We adopt the labelling scheme for the caustic shown in Figure 6. On branch (A), f’(zo) = 0 will have one solution which is repeated and as in Case 1 one side of this part of Ct is cool.
256
Figure 6. Hot and Cool Parts of the Polynomial Swallowtail
is a point on branch (D) then f‘(zo) = 0 will have three solutions xA((x,y), t), z6((x, y), t ) and zi((z,y), t ) where the middle one is repeated. This implies f ( x 0 ) will have three stationary points occurring from left to right as maximum, inflection and minimum. Hence f(z6((x,y),t)) > f ( x g ( ( zy), , t ) ) meaning that the coalescing cusped level surfaces do not correspond to the minimiser and so one side of branch (D) is hot. Full details of the analysis for branches (B) and (C) are omitted for the sake of brevity. It may be shown that one side of branch (B) is hot, whilst on branch (C) there exists a point X at which Ct will switch from cool to hot. This is found by solving the four equations:If
(2, y)
y), t ) . Solving these yields in four unknowns z, y, zZ;((x,y), t) and zi((x, =
(
+
-t5(3 +8&) 1 , -18000 2t
450
257
A similar numerical study works in three dimensions. The effect of the noise here is to bodily translate the whole picture in the direction (-l1010) by E W, ds. (See Reynolds
s,”
5.2. Intermittence of Stochastic Turbulence
-
a simple
example in two dimensions
+
Theorem 5.2. Let c = 0 , kt(x,y ) = x and So(x0,yo) = f(x0) g(xo)yo, where f l g , f’ and g‘ are zero at xo = a , g”(a) # 0 . The turbulent times t at which nc(t),the number of cusps on the zero pre-level surface of the Hamilton- Jacobi function changes are the zeros of the stochastic turbulence process (0
{t :
Then with probability one there exists a sequence of times (a,) with a, / such that Y,,, = 0 for every n.
00
Proof. We begin by finding a sequence of times tending to infinity at which yt 2 0. Define f ( r ) := T for 0 5 r 5 1 so that clearly f is absolutely continuous] f(0)= 0 and f’(u)’du < 1. Thus f ( r ) is a Strassen function] f E K. Hence by Strassen’s Law of the Iterated Logarithm we know that after throwing away a null set of paths, we can path-wise find a sequence t, such that if
Jt
h ( t ) := ( 2 t l n l n t ) i
258
then
h(tn)-lWrt,
+
f(r)
7
uniformly over r in [ O , 1 ] . We show that for each w with t , = t n ( w ) we have h(t,)-2t;1yt, + $. Let us consider each of the terms that comprise the stochastic process y t ( w ) . i) .
a&h(tn)-2t,lWt,
+
0.
ii) .
iii).
Combining the above we see that for each w with t, = t n ( w ) we have
To conclude we must find a sequence of times tending to infinity at which yt 5 0. If c > 0 then we simply choose times when Wt = 0. For cI 0 we must choose a Strassen function such that
Taking
it may be easily shown that f E K and f(1)J : f(u)du = 5.3.
-&
< 0.
C process f o r small noise in 2 dimensions
We perturb an underlying deterministic classical mechanical system by adding a small noise potential term ~IC~(z)r't., to see its effect on stochastic turbulence at the displaced cusp z t ( X 0 ) of the deterministic caustic. We use
259
map, wit,* the above notation for the globally reducible deterministic X: = @;zo, and with X: = z ; ( z , t ) ,z = X: = zt(Xo), z; the repeated root vector. When Xf = 0, there is a very simple result for the small noise stochastic turbulence process. (There are numerous examples with X: = 0 in the free case.) Let the corresponding processes be for stochastic turbulence a t the cusp on the deterministic caustic, z t ( X 0 ) . These are simple deterministic functions coming from the reduced classical action,
<:
<
ft’,t),)A.(
x;
= zE(z, t ) ,z = z t ( X 0 ) .
Proposition 5.1. If X: = 0 , formally correct t o first order in stochastic turbulence processes are given by
<
E,
the
t
M t ) = e ( t )- E
l
t ) ) )d W s ,
k s (Q;(.;;(.t(Xo),
c E R,
0
where z.~(zt(Xo),t ) is the repeated root vector evaluated at zt(X0) the cusp o n the deterministic caustic, (2 the reduced classical action at zt(Xo) and .;;(zt(Xo), t ) . Remark 5.1. Observe that Cc(t) i s possibly not Markov if Ic(s,t) = k, (@;(zg(~(X), t ) ) )depends upon t.
Proof. A simple consequence of Theorem 4.2 and a calculation. Question: What properties of the underlying system give rise to recurrence of and the intermittence of stochastic turbulence? We include an example here, very similar to the above, to show that the explanation of intermittence of stochastic turbulence is sometimes very simple.
<
Example 5.1. (Harmonic Oscillator Potential). Let kt(z,y ) = z and c= y)Q2(z,Y ) ~ where , R2 is a real symmetric 2 x 2 positive definite matrix with
i(z,
wi
0
if i = j , otherwise .
+
If we take So(z0,yo) = f(z0) g(z0)yO where f , g , f’,g’, f”’ and g”’ are zero at 20 = ai and g”(ai) # 0 for i = 1 , 2 , . . . ,n, then the zeros t ( w ) of the stochastic process [ t ( w ) := - - a1 i w2l
4
1 sin(2wzt) sin(2wlt) - -wz 4 g/’(ai)
+
(f”(Qi)
sin(wl(r - t ) )0 aW,
-
w1 Cot(Wlt))2
260
sin(wl(r - t)) o aW,
will be turbulent times. We show that there exists an increasing sequence {t,} with t, 700 such that
sin(2w2t) cosec2(wlt){sin(cjit).Y(cr,)
+ w1 cos(w1t)I2
1 2 +ecosec(wlt)Rt(w) - -a,wl sin(2wlt) - c , 4 where &(w) is a stochastic process well defined for all t. As t 4 we have cosec2(wlt) + m. Let { t k } denote an increasing
sequence a t which cosec2(wltk) = 00,then limt,t,
[t = -m
if 4&cSd >0 d’(4
but limt+tk [t = +m if s ’ n ( 2 w z t k ) < 0. However, we can find an infinite g” (at) increasing subsequence {tk,} such that It is continuous on (tk, , tk,+l) and sgn ( s i n ( 2 ~ Z t k ~ =) - sgn (~in(2w2tk,+~) , so that limt->t,
st,
sgn (sin
(F) ) ,
is the same for all k E Z+. This will only be the case if for some n E Z.
% = 2n7r, namely
w2 = nwl,
We conclude with an elementary result in the direction of Proposition 5.1. Assume that in Proposition 5.1, k , (@y(zL(zt(Xa), t))) = k x 0 ( s ) ,is independent of t. (See Reynolds for examples like this.) Then, for small noise, for a BM(R) process B, Sdt) = <,OM- &B(V(t)), where v(t) =
s,” k:,(s)
ds. This gives:
Proposition 5.2. Assume that v(t) is bounded and that v(t) /” co as t /” 03. Then a suficient condition for Cc to be recurrent is that <;(t)/(2v(t) loglogv(t))+ + o as t /” 00.
261
Remark 5.2. This means that t h e stochastic turbulence at cusp zt(Xo) will be intermittent as long as xL(xt(Xo),t ) is t h e minimising critical repeated root.
Proof. A simple consequence of t h e Law of t h e Iterated Logarithm.
[7
Needless to say most of t h e above results can be extended to d-dimensions a n d t o more general kinds of noise. However, we should add t h a t t h e physical interpretation of t h e small noise process is fraught with difficulty.
C
Acknowledgement
It is a pleasure for one of us (AT) to acknowledge helpful conversations with Professor Costas Dafermos (Brown), Professor Mark Freidlin (Maryland) a n d Professor Oleg Smolyanov (Moscow). References 1. S. F. Shandarin and Ya. B Zeldovich, The large-scale structure of the universe: turbulence, intermittency, structures in a self gravitating medium, Rev. Mod. Phys. 6,185-220 (1989). 2. W.E, K. Khanin, A. Maze1 and Ya Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. Math. 151, 877-960 (2000). 3. I.M. Davies, A. Truman and Huaizhong Zhao. Stochastic heat and Burgers equations and their singularities I - geometrical properties, J. Math. Phys. 43,3293-3328 (2002). 4. I.M. Davies, A. Truman and Huaizhong Zhao. Stochastic heat and Burgers equations and their singularities - geometrical and analytical p r o p erties (the fish and the butterfly, and why.), UWS MRRS preprint, http://www.ma.utexas.edu/mp~arc-bin/mpa?yn=Ol-45, 2001. 5 . A. Truman and H.Z. Zhao. Stochastic Burgers’ equations and their semi classical expansions, Comm. Math Phys. 194,231-248 (1998). 6. H. Kunita. “Stochastic Differential Equations and Stochastic Flows of Homeomorphisms” in Stochastic Analysis and Applications, edited by M. A. Pinsky, Advances in Probability and Related Topics (Marcel Dekker, New York, 1984), Vol. 7,pp. 269 - 291. 7. C. Reynolds. On the polynomial swallowtail and cusp singularities of stochastic Burgers equation, PhD thesis, University of Wales, Swansea, 2002. 8. V. N. Kolokoltsov, R. L. Schilling, A. E. Tyukov. Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations, to appear in Revista Matematica Iberoamericana. 9. V. N . Kolokoltsov, A. E. Tyukov. Small time and semiclassical asymptotics for stochastic heat equation driven by LBvy noise, Stoch. Stoch. Rep. 75, 1-38 (2003). 10. K.D. Elworthy, A. Truman and H.Z. Zhao. Stochastic elementary formulae on caustics I: One dimensional linear heat equations, UWS MRRS preprint.
262
11. M. I. Freidlin and A. D. Wentzell, R a n d o m Perturbations of Dynamical systems, (Springer-Verlag, New York, 1998). 12. C. Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften 325, (Springer-Verlag, Berlin, 2000).
A VERSION OF THE LAW OF LARGE NUMBERS AND APPLICATIONS
ARMEN SHIRIKYAN Laboratoire de Mathe'matiques Universite' de Paris-Sud X I , Bhtiment 425 91405 Orsay Cedex, France E-mail: [email protected] We establish a version of the strong law of large numbers (SLLN) for mixing-type Markov chains and apply it to a class of random dynamical systems with additive noise. The result obtained implies the SLLN for solutions of the 2D Navier-Stokes system and the complex Ginzburg-Landau equation perturbed by a non-degenerate random force.
1. Introduction
We study the 2D Navier-Stokes (NS) system perturbed by an external random force:
li - Au + (u, 0 ) u + Vp = q(t,z), div u = 0, u = 0,
x
E
z E D,
dD.
(1) (2)
Here D c R2 is a bounded domain with smooth boundary d D and 7 is a random process of the form
k=l
where v k are i.i.d. random variables in L 2 ( D , R 2 )and S ( t ) is the Dirac measure concentrated at t = 0. It was established in that, if the distribution of T k is sufficiently non-degenerate, then the family of Markov chains associated with the problem (l),(2) has a unique stationary measure p and possesses an exponential mixing property. Namely, for a large class of functionals f and any solution u(t) of (1) - (3), the average of f ( u ( k ) )converges exponentially, as k t 00, to the mean value o f f with respect to p: 511110>1196913114
I ~ f ( u ( k )) (f,p)l
I conSte+, 263
k
2 1,
(4)
264
where /? > 0 is a constant not depending on u(t). Moreover, as was shown in 7 , the strong law of large numbers (SLLN) for stationary processes combined with the coupling of solutions constructed in lo implies an SLLN for solutions of the problem (1)-(3): for any solution u ( t ) ,with probability 1 we have k-I
We note that similar properties were established for perturbations of the NS system by a random force smooth in x and white in t (see >. We refer the reader to 12,’ for a detailed discussion of the results obtained in this direction. The aim of this article is to derive the SLLN (5) from the mixing property (4) without using the coupling of solutions and to estimate the rate of convergence. To this end, we establish a simple version of SLLN for a class of Markov chains (Section 2) and show that it applies to the problem in question (Section 3). We note that the result of this paper remains valid for the 2D NS system perturbed by a random force white in time and smooth in the space variables. 5,3,4,2112,7
Notation Let H be a real Hilbert space with norm 1) . 1). We shall use the following notation: B H ( R )is the ball in H of radius R > 0 centred a t zero; B ( H ) is the Bore1 a-algebra in H ; P ( H ) is the family of probability measures on ( H ,B ( H ) ) ; C ( H ) is the space of continuous functions f: H -+ R; Cb(H)is the space of bounded functions f C ( H )endowed with the norm
llflloo := SUP lf(.>I. uEH
C ( H ) is the space of Lipschitz-continuous functions f
E
Cb(H) with norm
If f : H 4 R is a B(H)-measurable function and p E P ( H ) ,then we denote by (f,p ) the integral of f over H with respect t o p .
265
2. Strong law of large numbers for mixing-type Markov chains 2.1. Formulation of the result
Let (R, F,P)be a probability space and let H be a real Hilbert space with norm 11 . 11. We consider a family of Markov chains ( U k , P U ) in H with transition function pk(U,r)= P,{?lk E I?}, u E H , r E B ( H ) . Recall that the corresponding Markov semi-groups are defined by the formulas yk :
Cb(H)
Cb(H),
q k f (U)
=
1
pk(%
d v ) f(v),
A measure p E P ( H ) is said to be stationary for the family
(Uk,Pu)
if Q l p = p.
Definition 2.1. We shall say that the family ('ilk, P,) is uniformly mixing if it has a unique stationary measure p E P ( H ) and there is a continuous function p : R+ + R+ and a sequence { Y k } of positive numbers such that, for any f E C ( H ) and u E H,we have l?kf(U)
- ( f , P ) I 5 ’-Ykp(llull)llfllL, 2 0 .
(6)
The following theorem shows that “sufficiently fast” mixing combined with a dissipation property implies an SLLN. Theorem 2.1. Let in H such that
(Uk,
P,) be a uniformly mixing family of Markov chains
k=O
Suppose there is a continuous function h : R+ -+ R+ such that pkp(u) := &p(IIukll) 5 h(llU11) for all k 2 0 ,
(8)
where lE, is the expectation with respect to P,. Then there exists a constant D > 0 such that for any f € C ( H ) , u E H,and S > 0 the following statements hold:
(i) There is a P,-a.s. finite random integer K ( w ) 2 1 depending on f , u , and S such that
266
(ii) For 0 < r < 36, we have E d T 5 1+
& IIsll~~(llull)~
(10)
We note that Theorem 2.1 remains valid (with trivial modifications) for Markov processes with continuous time. Moreover, under some additional assumptions, one can take in (9) functionals f with polynomial growth at infinity. We also note that inequality (9) immediately implies the following estimate:
where M ( w ) = D
+ 2 K ( w )g-'.
2.2. Proof of Theorem 2.1 Let us fix an arbitrary function f E L ( H ) and set k-1
There is no loss of generality in assuming that
Ilfllm I 1 and (f,p ) = 0.
Step 1. We first show that
k2,I c ~ ~ ~ f l l L ~ ( l lk~2l 1. ~)~-ll
(11)
Here and henceforth, we denote by Ci positive constants that do not depend on f, u,lc and 6. Let us note that
By the Markoa-property,
Hence using (8) and the inequality
267
Substitution of this inequality into (12) results in (11) with C1 where C is the constant in (7).
=
2C,
Step 2. We now prove (9). To this end, we fix 6 E (0, f ) and set
where [a] is the integer part of a 2 0. Let us consider the events
G, = { W E R : ISk,l > K ' } ,
2
?I1.
Using (11) and the Chebyshev inequality, we derive
P(Gn) 5 n 2 & I s k , 1 2
5 CZllf11Lh(11.11) n-l-'.
(13)
Hence, by the Borel-Cantelli lemma, there is a Pu-a.s. finite random integer m ( w ) 2 1 such that ISk,(W)l
I
n-l
for n
2 rn(w).
(14)
We shall assume that m ( w ) 2 1 is the smallest integer satisfying (14). In particular, if m ( w ) 2 2, then
Isk,(w)I > n-l To estimate lsk
(Ski
-
Sk,
for kn-l < k
for n = m ( w ) - 1.
< k,, we note that
1 5 (i - k)lsk, I +
s k -Sk,,l
_-
Since k nkn-1 - k n - 1 < C3n-' and n-l 5 kn and (16) that ~
lSkl
5
l s k - sk,l
+ ISk,l
(15)
5 2 k n - krn - 1 .
(16)
1
'''
= k i B f 6 , it follows from (14)
+
5 2 "ic,, n-' 5 (2c3 + 1)n-l k -kn-l
5 (2c3 + l ) k i 5 + 6 5 (2c3 + 1)k-i+6, where n 2 m ( w ) and kn-l K ( w ) = [m(w)3+P].
< k < k,.
Thus, inequality (9) holds with
Step 3. It remains t o establish (10). To this end, we first note that, for 0 < q < 0, M
M
1=1
1=2
268
where we used inequalities (13), (15) and the definition of m ( w ) and G,. Since K = [m3+O], we see that, for 0 < T < 36,
E,KT 5 E,mT(3+O) 5 1
+ a - r (c4 3+p)
w-11)IlfllL.
5 1 + 3(&
w4)IlfllL
The proof of Theorem 2.1 is complete. 3 . Applications
3.1. Dissipative P D E ’ s perturbed by a bounded kick force
Let H be a real Hilbert space with norm 11 . 11 and orthonormal base {ej}. We consider the random dynamical system (RDS) uk
= s(uk-1)
+ qk,
(17)
where S : H + H is a continuous operator such that S(0) = 0 and { q k } is a sequence of i.i.d. random variables. As was explained in RDS of the form (17) naturally arise in the study of dissipative PDE’s perturbed by the random force (3), and in this case S is the time-one shift along trajectories of the unperturbed equation. We assume that S satisfies the following three conditions introduced in 8,10: 8,9110,
(A) For any R > T > 0 there are positive constants a = a ( R , r ) < 1 and C = C(R)and an integer no = no(R,r ) 2 1 such that IIS(u1) - S(m)115 C ( R ) \ l U l - U Z I I IISn(u)ll 5 max{aIIu.II,r}
for all ul, uz E B H ( R ) , for u E BH(R),n 2 no
(B) For any compact set K c H and any bounded set B c H there is R > 0 such that the sets A k ( K , B ) defined recursively by the formulas do(K,B) = B and d k ( K , B ) = S ( d k - l ( K , B ) ) K are contained in the ball B H ( R )for all k 2 0. (C) For any R > 0 there is an integer N 2 1 such that
+
l l Q N ( s ( ~ 1)
S(uz))llI illui - uzll
for all
~ 1 , 2 1 2E B H ( R ) ,
where Q N is the orthogonal projection onto the subspace spanned by { e j , j 2 N 1).
+
We note that the above conditions are satisfied for the resolving operators of the 2D Navier-Stokes system and the complex Ginzburg-Landau equation. As for the i.i.d. random variables q k , we assume that they have the form 00
qk
=
bjtjkej, j=1
(18)
269
where
bj
2 0 are some constants such that 00
j=1
and [jk are independent scalar random variables whose distributions rj satisfy the following condition:
(D) For any j 2 1 there is a function of bounded variation p j ( r ) such that r J ( d r )= pj(r)dr, where dr is the Lebesgue measure on R. Moreover, s u p p r j and E > 0.
c
[-1,1] and J , , < E p . ( r ) d r> 0 for all j I 3
2
1
Let (uk,pu)be the family of Markov chains that is associated with the RDS (17) and is parametrized by the initial condition u E H . We denote by Pk(U, r) the corresponding transition function and by q k and pz that, if the Markov operators generated by Pk. It was proved in conditions (A)-(D) are fulfilled and 10,11i6
(20) where N 2 1 is sufficiently large, then the RDS (17) has a unique stationary measure p , and for any f E C ( H ) we have bj#0
for j = l , . . . , N ,
JPkf(4- (f&))L P(ll~ll)llfllLe-pk, k 2 11
(21)
where p : R+ 4 R+ is a continuous increasing function and ,D > 0 is a constant not depending on f and u.Thus, the family (uk,Pu)is uniformly mixing, and condition (7) is satisfied. We claim that (8) also holds. Indeed, let us define the compact set
where bj 2 0 are the constants in (18). It follows from condition (D) that the support of the distribution of rlk is contained in K. Therefore, by assumption (B), there is a continuous increasing function R = R(d),d 2 0 , such that pu{IIukll
5 R ( d ) } = 1 for llull 5 d, k 2 0.
Hence, since p is increasing, for ))u)) _< d we obtain ';PkP(.)
iP(W)),
= ~uP(llUkll)
which means that (8) holds with h ( d ) = p ( R ( d ) ) . Thus, Theorem 2.1 applies, and therefore inequalities (9) and (10) hold for the RDS (17).
270
3 . 2 . T h e Navier-Stokes s y s t e m perturbed b y an unbounded
kick f o r c e We now consider the problem (1)-(3). It is assumed that V k are i.i.d. random variables of the form (18), where b j 2 0 are some constants for which (19) holds, and [jjk are independent scalar random variables satisfying the following condition (cf. (D)):
(D') For any j 2 1 the distribution of cjjk possesses a density p j ( r ) (with respect t o the Lebesgue measure) that is a function of bounded variation such that A e p 2 p j ( r ) d r5 Q , where Q
p 3 ( r )> o for all r E
IR,
> 0 is a constant not depending on j .
The problem (1)-(3) reduces to an RDS of the form (17). Namely, let us introduce the Hilbert space (endowed with the L2-norm)
H
=
{u E L 2 ( D , R 2 :) divu = 0, (u,")IaD
= 0},
where v is the unit normal to 6'D (see l5 for further details on the space H). Let S : H 4 H be the time-one shift along trajectories of the NS system (l), (2) with 7 z 0. Setting Uk = u ( k ,z), we obtain (17) (see ',lo for details). Let (uk,pu) be the family of Markov chains associated with the RDS (17). As is shown in '114, if the non-degeneracy condition (20) is satisfied for N >> 1, then the family (uk,pu)has a unique stationary measure p , and (21) holds with p(d) = C , ( l + d ) , where C1 and p are positive constants not depending on f, u,and k . Moreover, by Theorem 1.3 in ', we have Kdllukll)
I c2(1+ llull)
for all
k
2 0.
Thus, the conditions of Theorem 2.1 are fulfilled, and we obtain the SLLN for solutions of the NS system (1)-(3). References 1. J. Bricmont, A. Kupiainen, and R. Lefevere, Ergodicity of the 2D NavierStokes equations with random forcing, Comm. Math. Phys. 224 (2001), 65-
81. 2. J. Bricmont, A. Kupiainen, and R. Lefevere, Exponential mixing for the 2D stochastic Navier-Stokes dynamics, Comm. Muth. Phys. 230 (2002), no. 1, 87-132.
271
3. W. E, J. C. Mattingly, and Ya. G. Sinai, Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation, Comm. Math. Phys. 224 (2001), 83-106. 4. J.-P. Eckmann and M. Hairer, Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise, Comm. Math. Phys. 219 (2001), 523-565. 5. F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Comm. Math. Phys. 172 (1995), 119-141. 6. S. Kuksin, On exponential convergence t o a stationary measure for nonlinear PDE’s, perturbed by random kick-forces, and the turbulence-limit, The M. I. Vishik Moscow PDE seminar, AMS Translations, 2002. 7. S. Kuksin, Ergodic theorems for 2D statistical hydrodynamics, Rev. Math. Physics 14 (2002), no. 6,585-600. 8. S. Kuksin and A. Shirikyan, Stochastic dissipative PDE’s and Gibbs measures, Comm. Math. Phys. 213 (2000), 291-330. 9. S. Kuksin and A. Shirikyan, Ergodicity for the randomly forced 2D NavierStokes equations, Math. Phys. Anal. Geom. 4 (2001), no. 2, 147-195. 10. S. Kuksin and A. Shirikyan, A coupling approach to randomly forced nonlinear PDE’s. I, Comm. Math. Phys. 221 (2001), no. 2, 351-366. 11. S. Kuksin, A. Piatnitski and A. Shirikyan, A coupling approach t o randomly forced nonlinear PDE’s. 11, Comm. Math. Phys. 230 (2002), no. 1, 81-85. 12. S. Kuksin and A. Shirikyan, Coupling approach t o white-forced nonlinear PDE’s, J. Math. Pures Appl. 81 (2002), 567-602. 13. N. Masmoudi and L.-S. Young, Ergodic theory of infinite dimensional systems with applications t o dissipative parabolic PDEs, Comm. Math. Phys. 227 (2002), 461-481. 14. A. Shirikyan, Exponential mixing for 2D Navier-Stokes equations perturbed by an unbounded noise, J . Math. Fluid Mechanics, to appear. 15. R. Temam, Nauier-Stokes Equations. Theory and Numerical Analysis, NorthHolland, Amsterdam-New York-Oxford, 1977.
COMPREHENSIVE MODELS FOR WELLS
MARIAN SLODICKA Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Ghent, Belgium E-mail: [email protected]. be, web page: http://cage.rug.ac. be/-ms
The aim of this paper is to present various mathematical models for wells. Special attention is paid to a non-standard description using nonlocal boundary conditions (BCs). We also develop numerical algorithms t o handle nonlocal BCs. The choice of the appropriate model depends, of course, on concrete situation.
1. Introduction
Many ground-water hydrologists are interested in the determination of water-table elevations resulting from inputs and outputs such as natural replenishment, artificial recharges and pumping. Some of them are interested in the general flow pattern in the whole aquifer, other study the details in a vicinity of a well. Here, wells represent inputs or outputs, which affect flow in a soil matrix. These sinks/sources are concentrated, i.e., their diameters are relatively small compared with the whole aquifer. This feature makes the modeling more complicated. Of course, wells are not only used in the ground-water hydrology, but also by oil extraction or soil venting, which is used for soil remediation (for cleaning of unsaturated zone from chlorinated hydrocarbons or other volatile organic compounds). The main difference among all these applications are (a) the substance (water, oil, gas) for which wells are used, (b) different geological conditions.
2. Point sources
Let us consider the steady-state case with a single extraction well with an infinitely small diameter located at the origin. We suppose that our domain is infinite in all directions and we consider a homogeneous unconfined aquifer with the conductivity KO.Then a fundamental solution of
-v
. (KOVUO) = ss 272
273
(classical outside the origin) for a single point sink is given by
uo(x)=
rL
lnlxl 25KO 4.rrKoI 2 I
in 2D in 3D.
(1)
This solution so far has not included any realistic BCs and it generates drawdownsa everywhere. Further, the seepage face at the well is omitted because of a negligible well radius. This is not realistic for a small vicinity of the wellbore. Method of images is a simple technique to create some basic BCs. Adding imaginary wells to the real point sink at strategic locations allows to generate infinitely long straight equipotentials or no-flow boundaries (cf. For the analytical description of a single-phase flow caused by a single extraction well for a perfectly layered subsurface we refer the reader to '. Bounded domains. We consider a bounded domain R E Co?l in IRN ( N = 2,3) with boundary r = I'D U r N , where J?D has a positive measure. We study 'i2).
-V . ( K V u ) = SS in R u=O o n r o K V u . v = 0 on r N . Problem (2) is linear, but the right-hand side does not belong t o the H-l(R) (dual space to H1(R)), thus we cannot directly apply the theory of linear elliptic equations. When the conductivity K is Holder continuous (with the coefficient a , Q > 0 in 2D, a > in 3D) near the well, then one can use the method of subtraction of singularities. Then ( 2 ) is rewritten in terms of an new unknown function ii = u - U O , uo being defined by (1). The reformulated problem will contain the right-hand side from Lz(R), due to the Holder continuity of K near the origin. The case when the conductivity is not Holder continuous is more difficult. Such a situation can appear, e.g., when a well is located at an interface of two different layers, or there is a rock at the well tube. In such situation, we cannot suppose the regularity of K , thus the right-haad side of the modified problem (after subtraction of singularities) will not belong to the Lebesgue space. Nevertheless, one can overcome this using the so-called very weak solution as it has been proposed in '. Here, the solution is defined aPumping from a phreatic aquifer removes water from the void space leaving there a certain quantity of water which is held against gravity. As a result, the watertable at each point is lowered with respect to its initial position by a vertical distance called drawdown.
274
in terms of an adjoint problem. The author also describes the numerical schemes based on finite elements.
3. Wells with a non-negligible r a d i u s Method of images helps in some cases to model BCs. For more complicated but also more realistic situations we have to use variational calculus, where the differential equation can be equipped by various types of BCs. It always depends on the concrete case which BC has to be chosenb. We briefly discuss typical cases and later we focus our attention to nonlocal conditions.
Pressure Condition. Pressure is prescribed on the well, i.e., we speak about a Dirichlet type condition. This is frequently used for passive wells by soil venting. Here, clean air enters the contaminated domain. One can suppose that a constant atmospherical pressure is given on passive wells. Flux Condition. Flux through the well boundary is prescribed pointwise, i.e., we consider a Neumann type condition. This case is doubtful in many real cases, because the flux distribution is completely unknown. This cannot be used for inhomogeneous vicinity of the well or in the case when the well is located near a boundary (e.g., lake, river, . . . ). Signorini Condition. When a well diameter cannot be neglected, then a storage capacity of the well tube has to be taken into account (see 6,7,8). Then one part of the probe discharge comes from the soil matrix and the other one from the well tube. By this situation the waterhead inside and outside the extraction tube can be different, i.e., the seepage face can appear (cf. Figure 1). The length of the seepage face depends on the well diameter. For a large well radius one can observe a very small seepage face. This can be explained by a large storativity of the well tube. This model can be mathematically described as (cf. 9 ) : Find p such that
bModels describing air-, water- or oil-pumping wells differ from each other.
275
R
rN
(impervious layer)
Figure 1. A vertical cross section through a well
with the initial and BCs
u(0) = d o in 52 q ( t ). u = 0 on r N u ( t ) = do on r D p(t) 5 0, q ( t ). v 2 0, p(t)q(t). v = 0 for z 2 w ( t ) for z < w(t) p(t) = w(t)- z
(4)
Continuity equation for water inside the well tube is
nR2&W(t>= 2nR
ID
q .u -Q
,
(5)
where 0 denotes the saturation, K conductivity, p pressure, q the mass flow, R the well radius, Q the discharge of the well. D is the thickness of the aquifer. The Neumann, Dirichlet and Signorini boundaries (see lo) are denoted by r N , r D , rs,respectively. (D) Discharge Condition. It is assumed that a constant but unknown pressure builds up on the well boundary such that the prescribed discharge is obtained. Such a type of BC can be used for active wells by soil venting. Here, the total discharge of the well is given and one can assume that the pressure along the well tube is constant. We demonstrate this on the following study case:
v . (-KVU) = f
in R On r D -KVU, on r N u = unknown constant on r, U = gD Y = gN
G(u)=
L,
(-KVu) . v = s
ER
(6)
276
Here, apart from a standard Dirichlet BC on r D and Neumann BC on r N , there is a nonlocal BC on F,, where the total flux through (a well) is given along with a condition that the solution (pressure) is constant but unknown on r,. One can define a suitable variational framework for (6) by introducing the subspace V of H1(R)
r,
v = {'p E H'(R);
p = o on r D ,
'p = const
on
rn}.
(7)
Adopting standard assumptions on the data-functions appearing in (6), one can show the well-posedness of a weak solution. The choice (7) of the test space V is not standard. Therefore, there could be problems by a space discretization, due to the nonlocal BC on I?,. One can show (cf. 11) that the solution can be obtained via a linear combination of solutions to the following two BVPs with standard BCs:
V . (-KVV)= f
r D
V =gN
v=o
on
r,
21
-KVV.
R
in on On
=gD
(8)
r N
and
V . (-KVz) = 0 z=o -KVz .V = 0 z=1
in on on on
R r D
(9)
r N
rn.
The problems (8) and (9) are well-posed. Now, applying the principle of linear superposition, we see that u, = v+az for any Q E R solves
V . (-KVU,)
=f
in on On on
'& = g D -KVU,'U=gN
u,
=Q
R r D
rn.
The total flux through F n is a linear operator, thus G(u,) aG(z). Setting G(u,) = s we get
a= Thus, taking this value of
Q
(10)
r N
= G(v)+
s - G(v)
G(z)
'
for u,,we see that u, solves (6).
277
(R) Robin Condition. This type of BC is developed for a well in a confined aquifer. It also represents a kind of nonlocal BC, where the pressure along the well-boundary is assumed to be constant (as by the discharge condition), but the total discharge is also unknown. Its dependence on the pressure inside the well is known. Next section is devoted to the study of this case. 4. Robin type boundary condition
An aquifer that is sandwiched between two impermeable layers is called a confined aquifer if it is totally saturated from top t o bottom. If a recharge area for the aquifer is located a t a higher elevation that the top of the aquifer, and a well is drilled into the aquifer, the water will rise above the top of the well without additional forces. Such an aquifer is known as artesian. Similar situation can appear by oil pumping. The oil is usually stored in a large deepness under the soil surface. In fact, it is a mixture of oil and gas. By standard pumping one creates an under-pressure a t a well and in this way oil or water come out from the soil. This situation can be described in various ways, e.g., by a Dirichlet or by a discharge condition. The question is how to describe a flowing well, see Figure 2. Here, the liquid is flowing
t
Figure 2.
Cross-section through a well
out without pumping. The pressure at the bottom of the well is unknown.
It varies in time depending on situation in the aquifer. One can measure the total flux through the well tube, but we cannot expect that it will remain constant. The total flux through the well clearly depends on the pressure at the bottom of the well. This can be taken as a space constant (along the well boundary at the bottom). In fact, this value can change in time and it is a priori unknown. But the dependence of the total flux on
278
the well-pressure can be known. It is a nonnegative function, monotonically increasing, zero up to a given point PO. Roughly speaking, po is the minimal value of pressure which has to be achieved in order to push the fluid up to the soil surface. Therefore, the situation at the bottom of the well (suction area) can be described as follows
rn
p
= unknown
space constant
r
where q is the flux vector and u is for the outer normal vector at rn. The derivation of a flow equation for water in a confined aquifer can be found in 1 3 . We assume that the flow is governed by Darcy’s law. Thus, the flow equation for a saturated flow reads as
where S is the storativity, K is the conductivity tensor, p stands for the water density, f describes possible spatially distributed sources, and g denotes the gravitation vector. If the confined aquifer is located horizontally, then (12) will be independent of the gravitation vector. To avoid un-necessary technical details we study the following problem
+f
in R P=PD on r D vp.v=o on rN p = unknown space constant on r,
8tP = AP
(13)
in R, where R c RN for N 2 2 is a bounded domain with a Lipschitz continuous boundary I?. This is split into three mutually disjoint parts r D , rN and I?,, which describe the Dirichlet, Neumann and nonlocal boundary part. We assume that all three parts have a positive measure. For the function g describing the total flux through the well we adopt the following assumptions ( L is the Lipschitz constant of 9 )
279
4.1. Variational formulation, well-posedness
sM
We denote (w, z),,,, = wz,and the corresponding norm I I w Let the Hilbert space H l ( 0 ) be equipped with the norm
d m .
~ ~ =~ , ~
be defined by(7), Now, we give the variational formulation of the
Problem 1. Find a esuple (u,a) such that
lrnl
rn.
denotes the measure of the boundary part Taking into The symbol account (14) and (7), one can easily deduce the well-posedness of a weak solution p t o the IBVP 1. Throughout the paper we tacitly assume that the data-functions appearing in the problem setting are sufficiently smooth. In that follows C, E and C, denote generic positive constants depending only on the data, where E is a small one and C, is a large one. 4.2. Numerical scheme We divide the time interval [O,T]into n E N equidistant subintervals ( t z - l r t zfor ) t, = ir, where T = Applying the discretization in time (Backward Euler method) we get
E.
1
(Szz, 4 0
+ P z z , VF), + (dzz), P)r, Irn I
= (fz, d n
(17)
for i = 1 , .. ., n, Sz, = -and the starting datum zo = p a . The well-posedness of (17) is guaranteed by the theory of monotone operators. One can use an iteration scheme by computations at each time step t o avoid the nonlinearity. There are more possibilities. Newton like iteration
280
schemes need to start close to the exact solution. This implicitly means that the time step T is small. One can use the following linearization scheme, which is robust and converges for any initial datum pi,^ ( k E N,p E V )
We define p , , ~= pi-]. This choice can diminish the number of relaxation iterations. Similar relaxation schemes have also been used in The problem (18) is linear and well-posed. This follows from the V ellipticity of the left-hand side and from the Lax-Milgramm lemma. For a given time-index i we perform relaxation iterations for k = 1 , .. . , ki,maa: until the stopping criterion 12114.
IlPi,k - P i , k - l \ I o , r ,
5 7'
(19)
is achieved for some 77 > 0. Then we set p i = p i , k t , m a z and we switch to the next time step. Now, we introduce a sequence of auxiliary nonlinear elliptic BVPs, which are defined in terms of pi, for i = 1, . . . , n in the following way
The existence and uniqueness of a weak solution ui E V for i = 1,.. . , n follows from the theory of monotone operators. For convenience we define uo = Po. We show that relaxation iterations pi& converge towards ui as k 4 00 in appropriate function spaces. Let us note that ui differs from z i because we stop the iteration process after a finite number of steps.
Lemma 4.1. There exist positive constants COand TO such that for any r 1. TO and for all k E N the following estimates hold:
Proof.
(2)
We subtract (20) from (18) and set p = p i & - u i . We get
28 1
for P ( s ) := g ( s ) - Ls. Using the Cauchy-Schwarz and Young's inequalities to the right-hand side, and the Lipschitz continuity of the function p, we deduce
Thus 2 lrnl
2
2
IIpi,k - uiI10,n +2 IrnI I I V ( P ~ , ~ - ui)llo,n+ L I I P ~- ,uiIIi,r, ~ T 2 5 L l l p i , k - l - uillo,r,, *
(21)
The generalized Friedrichs inequality implies that the following relation is valid for any w E H1(R) (due to the fact that > 0) ll'Ulll0,n
5 c ll~'UlIl0,n~
(22)
The combination of the trace inequality and (22) yields for some Co > 0
COI I
P~,~
2
-
utIIo,r,
2
5 coI I P ~ ,-~ uiIIo,r i 2 IrnI I I V ( P ~-, ~ui)11;,n.
(23)
Now, we deduce from (21) and (23)
( L + CO)i h , k
2
- uiIlo,r,
5 L lIpi,k-l
2
-
ui))o,r,.
(24)
This iterative relation gives rise to the following estimate
(ii) The desired result is a consequence of the part (i) and (21). We point out that the choice of the time step r is free. Next, the relaxation iterations can start from any starting datum from H1(R) and they converge in the H1(R)-norm to a function u, E V , which is defined by (20). Please note that u, = p ( t , ) . We stop the relaxation iterations if l l p z , k t , m a z - P z , k t , m a z - i /lo,r, I 7'. Moreover, we know from (24) that Ih,k -
for some 0
uzllo,r,
I 4 l l ~ z , k - l - uzllo,r,
< q < 1 and for any k. Thus
\ \ p z , k t , m a z - 1 - uzl\o,r,,
5 I l p z , k z , m a z - l - ~ z , k ~ , m \\o,r, az + 5 7' + 4 Ilp,,kz,maz-l - utllo,rn.
The last inequality yields
I I P ~ , ~ ~ uzllo,rn ,~~~ -
282
This estimate together with (21) for k = ki,maz imply llPd%maz- uillo,R 5 [lPi,kt,maz-l-
%l l o , r ,
I CTV,
(25)
and
c
IIv(Pi,k%,maz - ui)llo,n I ~ ~ P i , k z , m a s- lUiII o,r,
I CTV1
(26)
which are valid for any i = 1,.. . , n. Now, we derive suitable a priori estimates for ui. Lemma 4.2. L e t q
2 1. We a s s u m e (19)
i=l
for all i = 1 , .. . , n. T h e n
i=l
take place f o r all m = 1 , .. . , n
Now, we set
'p = u i r
and sum the relation up for i = 1 , .. . , m. W-e get
The lower bound for the left-hand side is (the last term is nonnegative)
c i= 1
c m
m
1 2 ( ~ ~ ~ m+l ~ ~ , n ui-ll/i,n + IIU~ -
i=l
)
.
I I ~ ~ ~ I I : , ~ T
Applying the Cauchy and Young inequalities and (25) to the right-hand side we easily get the upper bound
283
The rest is a consequence of Gronwall's lemma. (ii) We start from the relation (27). Setting cp = 6uir and summing up for i = 1,.. . , m we get
Now, we introduce the convex function Q g ( z ) =
g(s) ds. According to
the properties of g we have
which holds for any z1, zz E
R. Moreover, one can prove
Using these properties we can write
Thus, the lower bound for the left-hand side of (28) is
Applying the Cauchy and Young inequalities and (25) to the right-hand side of (28) we easily get the upper bound m
m
C&(1
+
3q-1))
+ €Cll~uzll:,n
7
5 C&+
i=l
which is valid for any
E
ll~~ill02,n 7, i=l
> 0. Therefore, we have
m.
m.
m
i=l
i=l
i=l
Fixing a sufficiently small positive E , we conclude the proof.
CI
284
4.3. Convergence of the scheme Now, let us introduce the following piecewise linear in time function
and the step function En -
un(0)= uo,
z,(t) = uil
for t E (ti-l,ti].
Exactly in the same way we also define the step functions ji, and7, as well as the piecewise linear function p,. Using this notation we rewrite (27) into
Lemma 4.3. Let the assumptions of Lemma 4.2 be fulfilled. Moreover we assume that 7 2 Then there exists a positive number C such that
g.
Proof. We subtract (16) from (29), set over the time interval (0, t ) and get
'p
= Ti, - p , integrate the equality
The last term on the left-hand side is nonnegative due to the monotonicity of g. Further, Lemma 4.2(ii) implies
285
Using the Cauchy-Schwarz inequality and (25) we deduce
i”
The Cauchy-Schwarz inequality and &u,, & p E L2 ( ( 0 ,T ) ,L2(R)) imply
The last term of (30) containing the function f can be estimated analogously taking into account the properties of f . Therefore, we can write llUn(t) - P(t)ll&2
+ J’0
t
t
llV(% - P)ll:,sl I c (T
+
IIun - PIli$)
Applying the Gronwall argument we arrive at
from which we easily conclude the proof. Now, we are in a position to derive the error estimates for p , Theorem 4.1. L e t t h e a s s u m p t i o n s of L e m m a 4.3 be satisfied. T h e n there exists a positive n u m b e r C s u c h t h a t
Proof. The desired result is a consequence of Lemma 4.3, (25) and (26)u The error estimates from Theorem 4.1 fully correspond to the known results for semilinear problems starting from po E H1(R). When starting from more regular initial datum po E H2(R), assuming that po is compatible with BCs and taking 7 2 2 , one can get the convergence rate 0 (r2).
286
References 1. H.M. Haitjema. Analytic Element Modeling of Groundwater Flow. Academic Press, Inc, New York, 1995. 2. D.J. Wilson. Modeling of Insitu Techniques for Treatment of Contaminated Soils: Soil Vapor Extraction, Sparging, and Bzoventing. Technomic Publishing Company, Inc., Lancester-Basel, 1995. 3. R. Nieuwenhuizen, W. Zijl, and Van Veldhuizen. Flow pattern analysis for a well defined by point sinks. Bansport Porous Med., 21:209-223, 1995. 4. M. SlodiEka. Finite elements in modeling of flow in porous media: How to describe wells. Acta Mathematica Universitatis Comenianae, LXVII( 1):197214, 1998. 5 . M. SlodiEka. Mathematical treatment of point sources in a flow through porous media governed by Darcy's law. Tramport Porous Med., 28(1):5167, 1997. 6. I.S. Papadopulos and H.H. Cooper Jr. Drawdown in a well of large diameter. Water Resources Research, 3(1):241-244, 1967. 7. S.P. Neumann and P.A. Witherspoon. Analysis of nonsteady flow with a free surface using the finite element method. Water Resources Research, 7(3):611623, 1971. 8. E.A. Sudicky, A.J.A. Unger, and S. Lacombe. A noniterative technique for the direct implementation of well bore boundary conditions in three dimensional heterogeneous formations. Water Resources Research, 31 (2):411-415, 1995. 9. S. Schumacher, M. Sloditka, and U. Jaekel. Well modeling and estimation of hydraulic parameters. Computat. Geosci., 1(3,4):317-331, 1997. 10. C. Baiocchi and A. Capelo. Variational and Quasivariational Inequalities. John Wiley and Sons, Chichester . New York, 1984. 11. M. SlodiEka and R. Van Keer. A nonlinear elliptic equation with a nonlocal boundary condition solved by linearization. International Journal of Applied Mathematics, 6(1):1-22, 2001. 12. M. SlodiEka. Error estimates of a n efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition. M 2 A N , Math. Model. Numer. Anal., 35(4):691-711, 2001. 13. J.W. Delleur (Editor-in Chief). The handbook of groundwater engineering. Springer, Heidelberg, 1998. 14. M. SlodiEka. A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. S I A M J . Sci. Comput., 23(5):1593-1614, 2002.
STOCHASTIC CASCADES APPLIED TO THE NAVIER-STOKES EQUATIONS
ENRIQUE THOMANN AND MINA OSSIANDER Department of Mathematics Oregon State University Cornallis, Or 97331 E-mail: [email protected]. edu In this paper a representation of the Fourier transform of solutions to the NavierStokes Equations are obtained in terms of a stochastic recursion generated by a branching random walk. The notion of majorizing kernel is introduced and used to study regularity and existence of solutions of the Navier-Stokes equations. Similar representation of solutions to other equations are also discussed and its corresponding multiplicative recursion in the physical space are presented. This is joint work with R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, C. Orurn and E . Waymire.
1. Introduction The study of properties of solutions of the Navier Stokes equations remains one of the most notable problems in mathematics. While a substantial body of literature is available on this subject, see e.g. Ternamlo and Galdi4, the recent work of LeJan and Sznitman' has opened new opportunities for analysis and a novel application of branching processes. Indeed, in their work, LeJan and Sznitman obtained a representation of the Fourier transform of the solution of the Navier-Stokes equations as an expected value of a multiplicative functional defined on a branching random walk. This representation uses exponential random variables with means depending on wave number in a way naturally related to the equation. The distribution of the offsprings at each branching is on the other hand determined by a kernel conveniently introduced to use the quadratic nonlinearity of the equation. Three basic extensions of this approach are presented in this paper. First, the notion of majorizing kernels is introduced in order to analyze and control the regularity of solutions of the Navier-Stokes equations. While the solutions determined in the work of LeJan and Sznitman have to be understood in a weak sense, we show that it is possible to use an appropriate majorizing kernel to maintain or improve the regularity of the solutions. 287
288
Second, we remove the restriction to three space dimensions present in the work of LeJan and Sznitman. Using the example of the Kolmogorov Petrovskii and Piskunov (KPP) equation holding in one space dimension, a representation for the Fourier transform of its solution is obtained also as an expected value of a multiplicative functional. From the work of McKeang it is known that such a representation is available in physical space. In this paper we establish a direct relation between both approaches. Third, solutions to a linearly damped Burgers equations as an expected value of a multiplicative functional defined on a branching process in physical space are obtained. An outstanding problem is to obtain representations in physical space for the Navier-Stokes equations. While no solution for this problem is suggested in this paper, recent work of E Frolova contains some related ideas. While not developed in this paper, it should be noted that a similar approach applies to other evolution equations including linear parabolic equations such as Schrodinger equation with a potential that is the Fourier transform of a complex measure and to evolutions equations that involve a fractional power of the Laplace operator. Further details and examples of the methods presented in this paper can be found in Bhattacharya et a1.l and Chen et aL3. The organization of the paper follows the three extensions described above. In the next section, the stochastic branching process and multiplicative functional corresponding to the Navier-Stokes equations are introduced. Also in this section, the notion and basic properties of majorizing kernels are developed. Finally, a correspondence with more standard Picard iteration schemes is made. In section 3, the example of the KPP equations is considered both in physical and Fourier space. The relations between the corresponding representations is also described. Section 4 includes a treatment of the multiplicative functional and corresponding branching process for the damped Burgers equation as well as concluding remarks.
2. Applications to the Navier-Stokes Equations
Recall that the 3d incompressible Navier-Stokes equation can be expressed in the Fourier domain as follows:
289
where for complex vectors w, z
w CQ z = - i ( q . z ) I I c ~ w ,
ec
=
E MI
(1)
v > 0 is the viscosity parameter, and IIELW is the projection of w orthogonal to and ij is the Fourier transform of known exterior body forces. For # 0, LeJan and Sznitman' rescale the equation (FNS) to normalize the integrating factor e-'lc12s to the exponential probability density v)<)2e--vlf12s. The resulting equation is precisely the form for a branching random walk recursion for G(E, t)/vlE12 for a transition probability kernel naturally constrained by normalization requirements to dimensions d 2 3. To extend this approach introduce non-negative measurable functions h such that
<
<
h * h(E) I Bl
E # 0, B > 0.
(2)
Refer to such a function h as a majorizing kern.el with, constant B or in the case B = 1 as a standard majorizing kernel. Note that if h is a majorizing kernel with constant B then is a standard majorizing kernel. Also, if h ( [ ) is a majorizing kernel then so is ce".ch(<) for arbitrary fixed vector a and positive scalar c. To avoid unnecessary technicalities regarding their supports, attention is restricted in this paper t o positive majorizing kernels h ( ( )defined for E # 0. Such kernels are said to be fully supported. Examples of majorizing kernels are given by the following proposition.
5
Proposition 2.1. For 0 #
E
E R3,
0
5 p 5 1,CY > 0,
defines a majorizing kernel. Given a majorizing kernel we consider the Fourier transformed equation (FNS) rescaled by factors of the form for # 0. Namely, we consider
&, <
where
290
<
Notice that for each fixed with h * h(<)# 0, the convolution h * h(<) simply normalizes the product h(ql)h(772) to be a probability kernel on the set 71 772 = (. In particular, while a majorizing kernel need not be integrable, it is required that the convolution h * h(<)be finite for each E R3\{O}. It is then possible to show the existence of globally defined solutions of the (FNS) equations, the regularity of which depends on the particular majorizing kernel applied as follows. Introduce the Banach space Fh,T as the completion of the set
+
<
in the given norm. In the case h = ho this is the Besov type space introduced by Cannone and Planchon2. Note that considering h = hz with p < 1, from Proposition 2.1, the Banach space corresponding to such a majorizing kernel contains initial data 210 which are infinitely differentiable functions of compact support. One of the main results obtained using majorizing kernels is the following theorem. Theorem 2.1. Let h(J) be a standard majorizing kernel. Fix T > 0 and suppose that ICo(<)I 5 ( ~ ‘ % ) ~ ; h ( ( ) ,and I@([,t)l 5 (~%i)~(3~1<1~h(<), ( # 0,O 5 t 5 T . Then (FNS) has a unique solution in the ball of radius R = (&)’v/2 centered at 0 in the space Fh,T. Theorem 2.1 illustrates how majorizing kernels can be used to maintain regularity of the solutions. For example, if the majorizing kernel h = h g ) for p > 0 is being used, the solution remains infinitely differentiable. A further example is that it is possible t o obtain spatial analyticity of the solutions, for t > 0, provided the initial data satisfies for some majorizing kernel h and appropriate constants A, C independent of v ICo(<)l
5 Ch(<)ve-A’v.
In the case that h ( ( )= 1/1[12 G ho((),this result was obtained by LemariQRieusset’. However, the following proposition shows that there are majorizing kernels that exhibit a stronger singularity at the origin and decay slower at infinity than ho(<). Proposition 2.2. For
< E R3 such that C,”=,S t j , o < 2 } , let
291
T h e n H3 i s a majorizing kernel, and
H3 = G(
lim G(w) = 00
w-iv
where v is an element of the standard basis of R3 Using the majorizing kernel obtained in Proposition 2.2 with Theorem 2.2 exhibits solutions of the Navier-Stokes equations with initial data whose Fourier transform blows up at the origin a t a faster rate than l/l<12. See Bhattacharya et a1 for details. The solution obtained in Theorem 2.1 can be obtained as an expected value of a multiplicative functional defined on a branching process. The following subsection provides the main points of this idea.
2.1. Stochastic Recursion Denote by V the vertex set of a complete binary tree rooted a t 8 coded as
v = u,o~,~{i, 2}j = {el< 1 >, < 2 >, < 11 >, . . .},
n~o{l,
(6)
where {1,2}O = {O}. Also let aV = 2) = (1, 2}N. A stochastic model consistent with (3) is obtained by consideration of a multitype branching random walk of nonzero Fourier wavenumbers <, thought of as particle types, as follows: A particle of type # 0 initially at the root O holds for an exponentially distributed length of time So with holding time parameter A(<) = V I J ~ ~ ; i.e. ESo = When this exponential clock rings, a coin KO is tossed and either with probability the event [ K O = 01 occurs and the particle is terminated, or with probability f one has [ K O = 11 and the particle is replaced by two offspring particles of types 71, r/2 selected from the set 71 r/z = according t o the probability kernel
<
&.
+
<
This process is repeated independently for the particle types 71,772 rooted at the vertices < 1 >, < 2 >, respectively. Now, recalling (4), for given initial data and forcings xo(<)and p(<,t ) , ( # 0, t 2 0, define a functional X(O, t ) by the following stochastic recursion: X(O,t) =
where 71
+
{
xo(Ee), if so 2 t cp(t - S O , < ) ! if SO< t , K O = 0, m((o)X(<1 >,t - So) @,cs X(< 2 >, t - So) else
are distributed according to K c s ( d q l , d 7 2 ) and r<1>, 7<2> are the trees defined by re-rooting a t the vertices < 1 >, < 2 > of 72
=
292
new types 71, 7 2 , respectively. Standard results on critical branching show that this recursion will terminate in finite time with probability one. In particular there can be no explosion of the branching random walk in finite time. Thus X(0, t ) is a finite random variable for each time t and wavenumber 5. Indeed, for the evaluation of the stochastic functional X(e,t), for a given (0 = E , it is useful to identify a particular tree structure intrinsic to the stochastic branching model. Let
where Ivl-1
B~ = 0,
B,
=
C s,,~, e # v E v.
(9)
j=O
Then the stochastic functional X(0, t ) on a particular tree is obtained as a product of m’s, X O ’ S , and cp’s appropriately evaluated a t the nodes of this tree. Moreover, decomposing the functional X in terms of the events [SO 2 t ] ,[So < t , no = 01 and [SO< t , no = 11, one may check the following consequence of the strong Markov property.
Theorem 2.2. If EIX(Q,t)I < co,for each solves (3).
E # 0, then x(<,t ) = EX(0,t )
Theorem 2.1 is obtained from this by simply noting that if m(() 5 1, 5 1 and Ixo(J)I 5 1, then the finite number of factors appearing in the product functional t)l are bounded by 1, and consequently Ix(<,t)l <_ 1 for all and t . In this sense the notion of majorizing kernel as described above simply exploits sufficient bounds on the stochastic times functional X(e, t ) . However, the essential property of the majorizing kernel is the finiteness of the convolution h*h(<)for normalization t o a probability. In particular, this suggests that significantly sharper results are possible by so relaxing (2) and more detailed analysis of the stochastic structure of the branching random walk. Icp(E,t)l
<
[I(@,
2.2. Successive Iterations of a Contraction Map It is possible to relate the stochastic theory presented so far with an iterative method based on Picard iterations such as the one considered by Kato For this, write (FNS) as
293
where
Now, consider the iteration Gn+l(t,t) =
Q[an;.clo,Gl(E,t), (12) t ) , for u(')([,t ) = e-'IE12tii~(<).Note the
where iil(t,t ) = Q[u(O); GO,GI(, particular initialization of the iteration, which is the one utilized by Kato', and it is the appropriate one to relate the iteration to the stochastic process introduced in subsection 2.1 To establish this relation, define the replacement time of a vertex v as IVI
k=O
and let
A,(@,t ) = [lvl 5 n Vv E ~ ( t n ) ][R, > t Vv E {u E 7 e ( t ) : 1
~ =1 n}],
with l [ n ;19,t]being the indicator of the event A,(O, t ) . Prop 2.1. Let uk(t,t) = h(E)Xk(t,t)
=
h ( < ) E d l [ k E, tIX(I9,4 )
and denote by & ( t ,t ) the Fourier transform of the kth iterate of the iteration scheme defined in (12). Then W k ( < , t ) = file([, t ) .
A consequence of the proposition is that the convergence of the iteration scheme (12) and the existence of the expected value in Theorem 2.2 are essentially equivalent. 3. Application to KPP Equations
Recall that from the work of McKeang the solution t o the initial value problem
is given by
r Nt
1
294
where Bv(t)is the location of a branching Brownian motion defined recursively as follows. Let Xe(t),denote the location at time t of a standard Brownian motion, and let Te be an exponential random variable with parameter 1 independent of this Brownian motion. If To 2 t , set Bs(t) = z X,(t).Else, start two independent Brownian paths X<1>and X each with its own independent exponential time T<1>and T respectively and iterate on this process. Let
+
Ivl-1
ye(z, t ) = {V E
V : Rv =
IVI
C Tvlj< t I C Tvlj}
j=O
j=O
Then for v E ye(z,t ) ,
Bv(t)= z +
c
Ivl-1
XVlj(TVlj)+ Xv(t - RV).
j =O
Finally, let M ( y e ( 2 ,t ) )= max{ IvI : v E ye} and let 1[k;z, t] the indicator of the event [ M ( y o ( < , t )5) k]. Let ~ ( zt ),= Ex
[n
[uo(Bv(t))l l[k : z, ti] .
(15)
It follows that
u ( z , t )= lim u k ( z , t ) k+cc
On the other hand, consideration of the Fourier transform of the KPP equation leads after a simple integration to the integral equation
where 1 A(<) = 1 + ~ 1 < 1 2 . Proceeding as done with the Navier Stokes equations, scale (16) by l/h(<) to obtain
where
295
Define the recursive functional
where < 1 >, < 2 > are re-rooted trees a t vertices of types <,Ec2> respectively and the distribution of types is given on v~ q 2 = & by
+
Note that the only difference with the recursive functional corresponding t o the Navier-Stokes equations is the node operation which for the K P P equations is standard multiplication. Using the strong Markov property it follows that the solution of (16) is given by
fi(<, t )
h(t)EIX(TB(t, t ) ) ].
provided the expected value is finite. The analogue of a majorizing kernel for the KPP equation is given by
1 ( h * h)(E) I B(1+ 21
It is simple t o check that Cauchy densities,
are majorizing kernels. Finally, the relation with the McKean representation of the solutions is furnished by the following proposition which is identical to proposition 2.1 Recall that A,(<, t ) defined above Proposition 2.1 denotes the event that all vertex on a tree rooted a t E are of length less than or equal t o n and those vertex of exactly length n are replaced after time t. As in that proposition, let l [ k ;E, t] denote the indicator of A k ( [ , t ) .
denote the Fouriuer transform of the function
296
4. Damped Burgers Equation, Some Open Problems
An application of the Duhamel principle shows that the solution of the damped Burger equation
au -- -1 _ at 2ax2
1au2 2
(17)
+----‘11,
ax
with initial data u(x,0) = U O ( X ) , satisfies the integral equation
where
It then follows, using integration by parts on the second integral, and
that J
+I”/
e-5
(E) g(t
1
-
s)dyyds. s, 2 - y)-u2(y, 2
Using the same branching Brownian process used by McKean for the K P P equation as done in section 3, it is possible to define a recursive multiplicative functional such that the solution of (17) is obtained as an expected value. Indeed, let
Then, using the strong Markov property it follows that U ( X , t ) = EX(x,t ) .
It should be remarked that the introduction of the damping term in the equation (17) is only done for simplicity of presentation. The main point of this example is to illustrate that recursive multiplicative functionals can be used to obtain solutions of nonlinear partial differential equations. As indicated in the introduction, a similar representation for the Navier-Stokes equations is not presently available. The major difficulty to be overcome appears to be the projection on divergence free vector fields used to eliminate the pressure term. In the Fourier space, this projection
297
becomes part of the node operation as it is a local operator and is given by l$= 1 @ given in (1). By contrast, the same projection in the
& &
physical space involves t h e Riesz transforms t h a t are nonlocal operators. Despite t h e results of Gundy and Silverstein5, that provides a probabilistic interpretation of the Riesz transform in terms of Brownian motions, the representation of solutions of the Navier-Stokes equation as a n expected value of an appropriate stochastic functional remains an open problem.
Acknowledgments The work presented here is joint work with R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, C. Orum and E. Waymire and it is partially funded by US NSF Grant 0073958.
References 1. R. N. Bhattacharya, L. Chen, S. Dobson, R. B. Guenther, C. Orum, M. Os-
siander, E. Thomann, and E. C. Waymire, Majorizing Kernels & Stochastic Cascades With Applications To Incompressible Navier-Stokes Equations. To appear in Transactions of the AMS. 2. Cannone, M. and F. P1anchon:On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations Revista Matema’tica Iberoamericana 16 1-16, (2000). 3. Larry Chen, Scott Dobson, Ronald Guenther, Chris Orum, Mina Ossiander, Enrique Thomann, Edward Waymire. On ItB’s Complex Measure Condition For a Feynman-Kac Formula. To appear in IMS Lecture-Notes Monographs Series, Papers in Honor of Rabi Bhattacharya, eds. K. Athreya, M. Majumdar, M. Puri, E. Waymire. 4. G. Galdi, “An Introduction to the Mathematical Theory of the Navier-Stokes Equations” Vol 1 and 2. Springer Tracts in Natural Philosophy, Vol 38 and 39. Springer 1994. 5. Gundy, R and M.L. Silverstein: “On a probabilistic interpretation for the Riesz Transforms” in Functional analysis i n Markov processes, Lecture Notes in Mathematics, 923, 199-203. Springer 1982. 6. LeJan, Y. and A.S. Sznitman: Stochastic cascades and 3-dimensional NavierStokes equations, Prob. Theory and Rel. Fields 109 343-366, (1997). 7. LemariB-Rieusset, P.G. Une remarque sur l’analyticitB des soutions milds des Bquations de Navier-Stokes dans R3,C.R. Acad. Sci. Paris, t.330, SBrie 1, 183-186, (2000). 8. Kato, T.: Strong L p solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Z., 187 471-480, (1984). 9. H. McKean, Applications of Brownian motion to the equation of Kolmogorov, Petrovskii and Piskunov. Comm. Pure and Applied Math. Vol 28, 323-331, (1975). 10. R. Temam, “Navier-Stokes equations and nonlinear functional analysis”. SIAM 1995.
STOCHASTIC BURGERS EQUATION WITH LEVY SPACE-TIME WHITE NOISE
AUBREY TRUMAN AND JIANG-LUN W U D e p a r t m e n t of Mathematics, Unversity of Wales Swansea Singleton Park, Swansea SA2 8PP, UK E-mail: A . D u m a n @ s w a n s e a . a c . uk, J . L. Wu@swansea. ac. uk T h e purpose of this paper is t o investigate t h e Cauchy problem for t h e following stochastic Burgers equation
with suitable initial condition (for all ( t ,x) E [0,03) xW), where Ft,, is a L6vy spacetime white noise. The problem is interpreted as a stochastic integral equation of jump type involving the heat kernel. We obtain existence of a unique local solution in the L2 sense and show t h a t it gives rise t o a (local) stochastic flow (in time). Mathematics Subject Classification (1991): 60H15, 35R60. Key Words and Phrases: Stochastic Burgers equation, LBvy space-time white noise, stochastic integral equations of jump type, local existence and uniqueness, flow property.
1. Introduction
This paper is mainly concerned with the Cauchy problem for the following stochastic Burgers equation
on the given domain [0, m ) x R with L2 initial condition, where Ft,z is the socalled Lkvy space-time white noise consisting of Gaussian space-time white noise (i.e. a Brownian sheet on [0, m) x R) and Poisson space-time white noise (see 52 for the definition). There has recently been increasing interest in solving stochastic partial differential equations with non-Gaussian white noise (see, e.g. Bertoin5i6, Giraud18, Winke143, Mueller31, M ~ t n i kand ~~ Shlesinger et a136 and references therein). In particular, Gaussian white noise driven parabolic SPDEs have been intensively studied (see e.g. W a l ~ and h ~ references ~ therein). SPDEs driven by Poisson white noise are less well known and were first investigated in 298
299
Albeverio et all. Let us also mention that Saint Loubert BiB 35 formulated a parabolic SPDE driven by a Poisson random measure in a different way from Albeverio et all and he obtained very.interesting results on the existence of the unique solution. Moreover, parabolic SPDEs driven by L6vy space-time white noise are studied in Applebaum and Wu' and besides the existence of the unique solution, the flow property of the system obtained is also discussed. While very interesting studies of heat equations driven by Qstable LBvy noise have been carried out by Mueller31 and by Mytnik3'. On the other hand, as is well-known (see e.g. Burgers7), the Burgers equation
au +--=la(u2) at 2 ax
d'u
8x2
has been used extensively, under the name of Burgers turbulence, to model a variety of physical phenomena where shock creation is an important ingredient. The solution to Burgers equation is then called Burgers turbulent fluid flow. In recent years there appears to be a great interest to investigate Burgers turbulence in the presence of random forces, that is, to study stochastic Burgers equations with (Gaussian) white noise as random forces and/or with random inital data, see e.g. Bertini et a13, Bertini and Giacomin4, Bertoin5y6, Da Prato et als, Da Prato and Gatarekg, Da Prato and Zabczykl', Davies et all', E et all', Giraud17>18,Gyongy and Nualart2', Holden et a123,24,Kifer27, Le6n et a12', Sinai37>38,Tribe and Z a b o r o n ~ k iTruman ~~, and Zhao40,41,Winke143. Burgers equation has,also been used to study efficient stock markets, see Hodges and Carverhill" and references cited there. One of the main investigations of Burgers equation is based on the intriguing connection between the (nonlinear) Burgers equation and the somehow simpler linear heat equaiton, via the celebrated Hopf-Cole transformation. This technique can be still adapted to stochastic Burgers equation with additive Gaussian white noise (see e.g. Bertini et a13, Holden et a123124),but it is no longer available in the case of stochastic Burgers equations driven by more general Gaussian white noise (for instance, multiplicative Gaussian space-time white noise). Another method is used successfully, e.g. in Da Prato et a18, Da Prato and Gatarekg, Da Prato and Zabczyk" and Gyongy and Nualart" (here we just mention a few references), to study the mild solutions to stochastic Burgers equations driven by Gaussian space-time white noise. In this paper we introduce a stochastic Burgers equation driven by L6vy space-time white noise which generalizes all stochastic Burgers equations with white noises considered in the literature mentioned above. We will
300
prove existence of a unique, local, mild solution to the stochastic Burgers equation we posed above. The paper consists of three sections. In the next section, we set up what we call Poisson white noise and the corresponding stochastic integrals. In Section 3, in order to make the problem we are considering precise, we first elucidate briefly what L6vy space-time white noise is and then interpret the (heuristic) stochastic Burgers equation driven by Lkvy space-time white noise (weakly) as a rigorous jump type stochastic integral equation which involves evolution heat kernels. We present existence of a unique local L2solution. Namely, for any initial function from L2(R), we obtain a local solution with c&dl&g(i.e., right continuous with left hand limits in the time variable t E [0, cm)) trajectories in L2(R)). Finally, we discuss the flow property of the local solution. Our approach is based on combining the methods for solving stochastic Burgers equations driven by Gaussian space-time white noise in Da Prato et a18, Da Prato and Gatarekg, Da Prato and ZabczyklO, GyOngylg, Gyongy and Nualart20, Gyongy and Rovira2' with the techniques for solving parabolic SPDE's driven by L4vy white noise in Albeverio et all, Applebaum and Wu2, Mueller31, M ~ t n i k Saint ~ ~ , Loubert Bi635.
2. P o i s s o n White Noise and S t o c h a s t i c I n t e g r a t i o n
In this section, we set up some notations and recall some facts for our later presentation. We start with a general account of Poisson white noise in an abstract setting. Let ( R , 3 , P) be a given complete probability space and (U,B ( U ) ,v ) be an arbitrary a-finite measure space. Definition 2.1. Let ( E , & , p )be a a-finite measure space. By a Poisson white noise on ( E ,E , p ) we mean an integer-valued random measure
w>, ). x (0,F ,P )
N : ( E ,&, p ) x (U,
N u (01 u I..)
with the following properties: (i) for A E & and B E B ( U ) , N(A, B , .) : (R, 3,P ) 4 N U (0) U {cm} is a Poisson distributed random variable with e-fi(A)u(B) [p(A)v(B)]" n! for each n E N~{O}u{co}. Here we use the convention that N ( A ,B,.) = 03, P - a s . whenever p ( A ) = 00 or v(B)= a; (ii) for any fixed B E B ( U ) and any n 2 2, if A l , . . . , A , E & are all disjoint of one another, then N(A1, B , .), . . . , N(An,B,.) are mutually
P{w E R : N ( A , B , w )= n } =
301
independent random variables such that
Clearly, the mean measure of N is E [ N ( A ,B , .)] = p ( A ) u ( B ) , A E I ,B E B ( U ) . Moreover, N is nothing but a Poisson random measure on the Cartesian product measure space ( E x U , & x B ( U ) ,p @ v ) as formulated e.g. in Ikeda and Watanabe25. Hence, by a similar argument to that of Theorem 1.8.1of Ikeda and Watanabe25, we have the following existence result for Poisson white noise, namely, given any a-finite measure p on the measurable space (E,&), there exists a Poisson white noise N on ( E ,&, p ) with mean measure E[N(A,B , .)] = p ( A ) v ( B )A, E E , B E B ( U ) . In fact, N can be constructed as follows 7 n( W )
N ( A ,B , W ) :=
CC ,EN
l ( A n E , , ) x ( B n l r n ) ( [ ~ ) ( ~ ) ) l ( w E R : a n ( W ) ~ i ~ ( ~(2) )
j=1
w E R for A E & and B E B ( U ) , where (a) { E n } n Ec~& is a partition of E (i.e., En, n E N, are disjoint of one another and U n E ~ E , = E ) with 0 < p(E,) < m , n E N, and {U,},,N c B ( U ) is a partition of U with 0 < v(Un)< a, n E N; (b) V n ,j E N, ),:[ : s1 E, x U, is .F/& x B(Un)-measurable with ---f
where &, := & n E, and B(U,) := B ( U ) n U,; (c) V n E N,7, : R --+ N U {0} U {m} is a Poisson distributed random variable with
P{w E R : vn(w)= k } = (d)
e-p(En)V(Un)
u,)]
[p(~~)v(
k!
, k E N U {0}u {a};
and qn are mutually independent for all n,j E N.
Thus, given any a-finite measure p on ( E ,&) and any a-finite measure Y on (U,B( U ) ) ,there is always a Poisson random measure N on the product measure space ( E x U,& x B ( U ) ,p @ v )which can be constructed in the above manner. We call such a N canonical Poisson random measure associated with p and v.
302
Now let us give an example of Poisson white noise. Take ( E ,&, p) = ( [ O , c o ) x Rd,B([O,co)) x B ( R d ) , d t18d x ) , d E N. Then the Poisson white noise N can be well defined. Denote
Nt,,(B, w ) := N([O,tl x (-00,x],B , w ) , ( B ,w) E
WJ)x d
for t E [O,m) and x = (xj)15j5d E Bd, where (-co,x] := n j = l ( - m , x j ] . We can define (formally) the Radon-Nikodym derivative
for (t,x) E [O, m) x Rd. We call Nt,z Poisson space-time white noise. In the sequel in this section, we will take the measurable space (I?,&) in Definition 2.1 to be a product space ( [ O , m) x E , B([0,m)) x &) where ( E , & ) in the latter is a Lusin space. Let p be a o-finite measure on ( E , & ) (note that from the next section onwards, ( E , & , p )will be taken to be (EX,B(B), d x ) ) , then there exists a Poisson white noise N on ([O, m ) x E , B([0,m)) x E , d t 8 p ) with mean measure
E{N([O,t] x A, B , .)} = t p ( A ) v ( B )(,t ,A, B ) E [0,03) x & x B(U). Let { 3 t } t E [ 0 , 0 0 ) be a right continuous increasing family of sub a-algebras of 3,each containing all P-null sets of 3,such that the Poisson white noise N has the property that (i) N([O,t ] x A, B , . ) : R + N U (0) U {co} is Ft/P(N U (0) U {co})-measurable V ( t ,A, B ) E [0,m) x & x B ( U ) and (ii) {N([O,t + s] x A, .) - N([O,t ]x A , . ) } ~ > o , ( A , B ) E E ~ Bis( uindependent ) of 3 t for any t 2 0, where P(NU (0) U {m}) is the power set of N U (0) U {co}. (For instance, we may directly take 3 t := m ( { N ( [ O ,t] x
A, B , .) : ( A ,B ) E E x B ( U ) } )V N , t E [0, a)
where N denotes the totality of P-null sets of F.) In what follows, let us set up related stochastic integrals by following the procedure of Section 11.3 of Ikeda and Watanabe25 (see also Applebaum and Wu')). First of all, for those integrands f : [0,00)x E x U x R + R which are {Ft}-predictable and satisfy
for some ( A ,B ) E & x B ( U ) , the stochastic integral
is well defined as the usual Lebesgue-Stieltjes integral.
303
Now we define the (compensating) martingale measure
M ( t , A, B , U ) := N([O,t], A, B , w ) - t p ( A ) v ( B )
(3)
for any (t, A, B ) E [0,co)x E x B(U) with p(A)v(B)< co. Obviously,
E[M(t,A, B , ,)I = 0 and
E ( [ M ( tA, , B , .)I2) = tP(A)V(B).
(4)
For any {Ft}-predictable integrand f : [0, co) x E x U x R satisfies
E
I" s,
If(s,X I Y,.)ldsp(dz)v(dy) < 00,
a.s.
--+
R which
vt > 0
for some ( A ,B ) E & x B ( U ) , we can define the stochastic integral
Moreover, stochastic integrals with respect t o M are also well defined for {&}-predictable integrands f satisfying
for some ( A , B ) E E x B(U) by a limit procedure (see the argument in Section 11.3 of Ikeda and Watanabe25) and t E [ O , o a ) H
304
Ji'
S, S, f(s,x,y , . ) M ( d s ,dx, d y , .) E R is a square integrable {.Ft}martingale with the quadratic variation process
On the other hand, it is clear that A4 defined by (3) is a worthy, orthogonal, {&}-martingale measure in the context of Walsh4'. Thus stochastic integrals of {Ft}-predictable integrands with respect to M can be also well defined (alternatively) by the method in Section 11.3 of Ikeda and WatanabeZ5. Furthermore, the following stochastic Fubini's theorem for changing the order of integration in iterated stochastic integrals with respect t o M was presented in Applebaum and Wu2:
3. Burgers Equation Driven by LQvySpace-Time White
Noise Let ( R , 3 , P ) be a given complete probability space with a usual filtration {3t}t~[0,~). In this section we will consider the Cauchy problem for the following stochastic Burgers equation
( g &)u(t,x,w ) + + g [ u 2 ( tx,1 w ) l = -
+qt, 2,u(t,5 , w))Ft,,(w)
I
u(O,x,w ) = uo(x,w ) , (x,w ) E R x R
a(t,x,u(t,x,w ) ) + ( t ,5 , w ) E (0, 03) x R x R
(9)
305
where a , b : [0, 00) x R x R ---f R are measurable and the initial condition uo is .&measurable, and F is an Lkvy space-time white noise, which includes terms not only controlled by a Gaussian space-time white noise but also by a Poisson space-time white noise, so that in fact the noise we shall consider has a formal structure similar to that of a Lkvy process:
where c1, c2 : [O, 00) x R x U 4 R are measurable, Wt,, is a Gaussian space-time white noise on [O, 00) x R used initially by W a l ~ h(formally, ~ ~ Wt,, := *, where W ( t ,x) is a Brownian sheet on [0, co) x R), Mt,, and Nt,z are defined formally (in the previous section) as Radon-Nikodym derivatives as follows
for ( t ,x) E [ O , o o ) x R,while as given in Section 2, N is the Poisson white noise on ([0,co)x R,B([0, 00) x R)) with respect to a given o-finite measure space (U,B ( U ) ,v), and UOE B ( U ) with v(U\Uo) < 00, M is the associated (compensating) martingale measure. Following Walsh4’, let us introduce a notion of (weak) solution t o Equation (9). An L2(R)-valued and {&}tEio,w)-adapted cAdlAg (in the variable t E [ O , c o ) ) process u : [O,m) x R x R + R is a solution t o (9) if for any cp E S(R),the Schwartz space of rapidly decreasing Cw-functions on R,
306
holds for all t E [0, oo). Based on this notion, we can present a weak (but rigorous) formulation of Equation (9). Let Gt be the Green’s function for the operator in the domain [0,oo) x R, which is given by the following formula:
&
G t ( z ,2)
z=
1 (x - z ) 2 -ezp{ - ___ }
rn
4t
6
(13)
for t > 0; and Go(x,z ) = S(z - 2). We will need the following facts: (i) J , G t ( z ,z ) d z = 1, Jw[Gt(zl z ) I 2 d t = (27rt)-? , Vt E [0, oo),Vz E R ; (ii) Gt(z,t)= Gt(z,z) , t E [0, oo), 2,t E R; (iii) J, G t ( x ,z’)G,(z‘, z)dz’ = Gt+s(x,z ) , s, t E [0, co), z, z E R ; (iv) Vm,n E N U {0}, there exist some constants K , C > 0 such that
For the property (iv), cf. e.g., Friedman14, or Ladyzhenskaya et a128. Based on (iv), we have the following particular estimates which will be used later on
and
for all t E (0, oo),z, z E R. The following heuristic discussion paves the way for us t o give a rigorous (weak) formulation of Equation (9). (Of course, our derivation can be made rigorous in the sense of Schwartz distributions.) First of all, we notice that the solution of (9) can be (formally) written in terms of the Green’s function as follows
1a u 2 u(t, 5, W ) = [G * (210 - - - CL bF)](t, Z, W ) . 2 In other words, if u solves the following convolution equation
at + +
307
~ ( Z, t ,U ) = (G * U O ( . , w ) ) ( t ,Z)
1 a u y , ., w ) + {G * [-z az
+a(*,., U ( . , * , w ) ) -k b(., ., u(., .,w))F.,.(U)]>(t,
7
(14)
then u satisfies (9). Furthermore, (14) is formally equivalent to the following stochastic integral equation
since from (10) we have the following (heuristic) derivation for the second term in the right hand side of (14)
308
Pt
P
x M ( d s ,dz, d y , w )
x N ( d s ,dz, dy, w). where we have used “integration by parts” for the first term. Moreover, by observing that v(V \ Vo) < 00 and using formula (6), we see that equation (15) can be written in the equivalent form
+
Gt-s(x:rz)b(s,z , ~ ( s - ,z , w ) ) [ci(s, z ; y)lu0(Y) +CZ(S,
z ; V)1U\Uo(Y)]M ( d s ,dz, dY, w ) .
(16)
Based on the above discussion, without loss of generality, we shall consider the equations in the following form
309
where f,g : [ O , o o ) x R x R --t R and h : [ O , o o ) x R x R x U + R are measurable and the coefficient function q : [O,co)x R x R -+ R is measurable and satisfies the following growth condition Iq(t,z,
.)I I Kl(Z) + K2(z)I4 + const.l.12
(18)
for all ( t , z , z ) E [ O , o o ) x R x R, for some nonnegative functions K1 E L1(R) and KZ E L2(R)”. Clearly, the term containing the quadratic u2 in Equations (15) and (16) satisfies the above growth condition for q with q ( t , z , z ) = z2. Moreover, the case that replacing u2 by a more general form of \ulr for T E [1,2] also satisfies the conditions posed for q (with q(t, 2,z ) = 1 . ~ 1 ~ ) . Therefore, the condition for the coefficient q we posed above covers at least these two important and interesting cases. Also, it is obvious that q ( t , z, z ) = z is another special case under our growth condition, which corresponds to the second term containing the linear u instead of u2 on the right hand side of Equations (15) and (16). Clearly, Equation (17) is a weak (but rigorous) formulation of the following (formal) equation
= f(t,5, u(t,2 ,w ) )
+g(t,
5,
u(t,2 , w ) ) W t , z ( w )
+s,
q t , z,u(t,2 , w ) ; Y)Mt,z(dY,w ) .
Let us now give a precise formulation of solutions for Equation (17). By a (global) solution of (17) on the set-up (0,F,P ; {Ft}tE[~,cro)), we mean an {Ft}-adapted function u : [0, co)x R x R + R which is c&dl&gin the variable t E [ O , o o ) for all z E R and for almost all w E R such that (17) holds. Furthermore, we say that the solution is (pathwise) unique if whenever u ( l ) and u ( ~are ) any two solutions of (17), then u(l)(t, z, .) = u ( ~ ( t), z , . ) ,pa.e., V ( t ,x) E [0, m) x R. Moreover, one can formulate a (global) solution over a finite time interval [O,T]for any 0 < T < co in the same pattern. Furthermore, an {.Ft}-adapted function u : [O,T]x IR x R + R which is c&dl&gin t E [O.T]is called a local solution to Equation (17) if there exists an increasing sequence { T ~ } ~of~ stopping N times such that W E [0,TI and Vn E N,the stopped process u(t A T,, 2 , w ) satisfies Equation aHere and in the sequel, by “const.” we mean a generic positive constant whose value might vary from line to line.
310
(17) almost surely. Clearly, a local solution becomes a global solution if := s u p n E W ~= , T . Moreover, a local solution to Equation (17) is (pathwise) unique if for any other local solution fi : [O,T]x R x R R, u(t,x , w ) = fi(t,2 , w ) for all ( t ,x,w)E [0,T, A),? x R x := { ( t ,x , w ) E [0,T ]x R x R : 0 I t < T,,(w) A?,(w)}. We have the following main result: T , ,
Theorem 3.1. Let T > 0 be arbitrarily fixed. Assume that there exist (positive) functions L1, L2, L3 E L 1 ( R )such that the following growth conditions
Idt, 2, .)I2
+
If(t,z, z)J2I L ~ ( z ) const.)zI2,
(19)
+
(20)
/
U
Ih(t,2, z ; Y)l2Y(dY)I L2(x)+ const.1zI2
and Lipschitz conditions Iq(t, 5 , a )- 4 ( t ,2, z2)12 + If ( t ,x,z1) - f (t,5, .2)12 I [L3(x) const.(lzl12 Iz21')]1z1 - z2I2
+
+
Ig(t, x,z1) - g ( t , x,z2)I2 2
5 const.jz1
+
- z21
L
(21)
Ih(t,2, z i ; Y ) - h ( t , x ,z2; Y ) ~ ~ Y ( ~ Y ) (22)
hold f o r all ( t , x ) E [O,T]x R and z , z l , z 2 E R. Then f o r every Fomeasurable uo : R x R -+ R with EJw(luo(x,.)12)dz < co, there exists a unique local solution u to Equation (17) with the following property
for any t E [0,TI.
We need some preparation before the proof to Theorem 3.1. For any fixed n E N,let the mapping 7rn : L 2 ( R )4 B, := {u E L2(R) : 11ullL2 5 n } be defined via
Clearly, for any n E N,we have II7rn(u)IILz
5 TL.
Moreover, it is clear that the norm ll7rnllL2
:=
sup l l 4 l L Z l l
ll7rn+
51
311
that is, 7rn : L2(R)-iL2(R)is a contraction. Notice that if u is a solution to Equation (17), then u is L2(R)-valued, {&}-progressive process. Thus, by Theorem 2.1.6 in Ethier and Kurtz13 (page 55), for any n E N,
defines a stopping time. It is clear that {T,},~N is an increasing sequence of stopping times determined by u. Moreover, for any fixed n E N,the stopped process u(t A 7,) satisfies the following equation
On the other hand, any solution to Equation (23) is a local solution to Equation (17). Therefore, the existence of a unique local solution to Equation (17) is equivalent to the existence of a unique solution to Equation (23). Hence, we will focus our attention on showing the existence of a unique solution to Equation (23). The following proposition is a reformulation of some estimates obtained in Gyongy and NualartZ0 in a way convenient for our setting here. One can, alternatively, verify them by utilizing inqualities of Holder, Minkowski and Young.
Proposition 3.2. (i) For u : [O,T]x lR 3 R, the following estimates hold
and
312
in particular,
(ii) For 0 5 tl 5 t 2 5 T , there exist Q E (0, ( 0 , l - :), such that the following estimates hold
i ) ,E ~(4,co) and ,8 E
Proof of Theorem 3.1 We will carry out the proof by the following three steps. Step 1 Suppose that u : [O,T]x R x
f2 -+ R is an L2(R)-valued, {Ft}-
adapted, c&dl&gprocess. For any fixed n E
with
and
N,set
313
By (24) in Proposition 3.2, the condition (19) and Schwarz inequality, we have
5 const.ti 5 const.T? < 0 0 . Notice that here and after the constant “const.” depends also on n (of course on T as well). By (25) in Proposition 3.2 and the condition (18), we get
5 c0nst.t’ 5 const.T2 < 0 0 .
314
On the other hand, by our Proposition 2.2 (Fubini’s theorem) and It6’s isometry property for stochastic integrals with respect to (both continuous and c&dl&g)martingales, we have
and
Thus, by the condition (20), we get
Therefore, we obtain that
for any fixed t E [0,TI.
> 0 be arbitarily fixed. For any L’(IR)-valued, {&}adapted, c&dl&gprocess u : [O,T] x IR x R 4 IR with initial condition u(0, z, w ) = UO(Z, w ) , we define
Step 2 Now let X
315
Clearly, 11 . IIx is a norm. Let B denote the collection of all L2(W)-valued, {Ft}-adapted, c&dl&gprocess u : [0, TI x R x R -+ W with initial condition u(O,x,w ) = u ~ ( zw,) , such that
Then ( B ,(1. Ilx) is a Banach space. Now Vu E B , Ju is well defined and for any fixed t E [O, TI
E(L/(Ju)(t,z,.)l'dz Thus
00
<_ const.
( t i + t')e-xtdt
I const.(A-$
+A - ~ ) <
fi
= const.[-A
2
-3
+2~-31
00
that is, J u E B, which implies that J : B + B.
B, by (24), (25), (26), (27) and (28) in Proposition 3.2 and the Lipschitz condition (21), together with utilizing Fubini's theorem, the Young inequality and the Schwarz inquality, we get for any t E [0,TI Step 3 Now Vu,v E
5 c0nst.E
[l(t L - s)-i
(L3(z)
+ const.[(nnu)2(s,z,
a))
316
317
and by It6’s isometry for both stochastic integrals with respect to W ( d s ,d z ) and M ( d s , dz, dy, w),we have
318
Now let us take X large enough so that const.
(5+ 5)
<1
which implies that J : B 4 B is a contraction. Therefore there must be a unique fixed point in B for J and this fixed point is the unique solution for our Equation (23). To see that this gives us a local solution to Equation (17), let us denote by u, the unique solution of Equation (23) for each n E N. For this u,, let us set the stopping time T,(w) := inf{t E [0, T ]:
s,
u;(t, z, w)dz 2 n 2 } .
Clearly by the contraction property of J , we have for all j 2 n and for almost all w E R
Therefore we define
for any ( t ,z, w ) E [0, ~ , ( w ) ) x
IR x R and (w):= sup 7, (w).
Too
nEN
Then {4t7X,W)
: ( t , z , w )E
[0,7,(W))
x JR x 0)
is a local solution to Equation (17). Finally, for the uniqueness of the local solution to Equation (17), suppose that there are two local solutions u and v to Equation (17). Then u and v must satisfy Equation (23) for any fixed n E N. On the other hand, by the uniqueness of solution to Equation (23), we get U(t,X,W)
=W(t,X,W),
V(t,X,W)
E [O, 7,(w)) x
R x a.
319
Now let n -+ co,we deduce
u(t,2 , w)= v(t, 2 , w), V(t, 2 , w)E [O, 7,(w))
x
IR x R .
Hence we obtain the uniqueness. Q.E.D.
--
Remark 3.3. In the case M 0, Equation (17) becomes a Burgers equation with Gaussian (space-time) white noise. Unique global L2 solutions are obtained by Gyongy and Nualart2' in the whole space line and by Da Prato, Debussche and Teman in Da Prato et al' (see also Da Prato and Zabczyk'') in bounded space intervals. Their methods depended critically on some uniform estimates which employed Burkholder's inequalities for continuous martingales. We observe that we are unable to follow this route herein as the corresponding inequalities for cadlag martingales (see e.g Jacod and Shiryaev26) do not behave so nicely. Finally let us consider the flow property of the local solution to Equation (17) starting with an L2 function as the initial condition. We refer to the references Fujiwara and Kunita15 and Fujiwara" for investigations of LBvy flows associated with (ordinary) stochastic differential equations of jump type. For ( T , t , w)E { ( T , t , w)E [0,TI x [0,TI x R : T F t < T,(w) I TIw E Q}, and 'p E L2(R), we define
for z E R. Then it follows clearly by Theorem 3.1 that almost surely : L2(R) + L2(R)
for all 0 5 r _< t < r m ( w ) 5 T . Furthermore, we have
320
Proposition 3.4.
act = identity,
V t E [0,t m ( w ) ) , P - a s . for w E R ;
(30)
I r F r' I t < T,(w), P - a s . f o r w E R.
(31)
and 0
a:,.,
= a:',
V0
Therefore we conclude that { @ ~ t } o ~ r ~ t < T , ( wforms ) , w Eao(local) stochastic %ow.
since Go(z, z ) = 6(z - z ) and N ( { t } ,{z}, dy, w ) = 0, P - a s . for w E Thus (30) comes directly from (29) and (32). To verify (31), by (29), it suffices to show the following equality
R.
x M ( d s ,d z , d y , W ) , P - U.S. (33) is obtained by a straightforward derivation using the (usual) Fubini's theorem (see, e.g. Theorem 7.8 in R ~ d i n ~integration ~), by parts, Theorem 2.6 of W a l ~ h our ~ ~ Proposition , 2.2, property (iii) of the Green's function G, and our equality (32). Q.E.D.
321
Acknowledgements
We thank Ian M. Davies for the great help in setting of the manuscript. References 1. S. Albeverio, J.-L. Wu, T.-S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stoch. Proc. Appl. 74 (1998), 21-36. 2. D. Applebaum, J.-L. Wu, Stochastic partial differential equations driven by LBvy space-time white noise, Random Oper. Stochastic Equations 8 (2000), 245-259. 3. L. Bertini, N. Cancrini and G. Jona-Lasinio, The stochastic Burgers equations, Commun. Math. Phys., 165 (1994), 211-232. 4. L. Bertini, G. Giacomin, Stochastic Burgers equations and KPZ equations from particle systems, Commun. Math. Phys., 183 (1997), 571-607. 5. J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Commun. Math. Phys., 193 (1998), 397-406. 6. J. Bertoin, Structure of shocks in Burgers turbulence with stable noise initial data, Commun. Math. Phys., 203 (1999), 729-741. 7. J.M. Burgers, The Nonlinear Diffusion Equations, Reidel, Dordrecht, 1974. 8. G. Da Prato, A. Debussche, R. Teman, Stochastic Burgers equation, NoDEA, 1 (1994), 389-402. 9. G. Da Prato, D. Gatarek, Stochastic Burgers equation with correlated noise, Stochastics Stochastics Rep., 52 (1995), 29-41. 10. G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems. LMS Lect. Notes 229, Cambridge Univ. Press, 1996. 11. I.M. Davies, A. Truman, D. Williams and H.-Z. Zhao, Singularities of stochastic heat and Burgers equations and intermittence of stochastic turbulence, Swansea preprint, 2001. 12. W. E, K. Khanin, A. Maze1 and Ya Sinai, Invariant measures for Burgers equations with stoachastic forcing, A n n . Math., 151 (2000), 877-960. 13. S.N. Ethier and T.G. Kurtz, Marlcov Processes: Characterization and Convergence. John Wiley and Sons, New Yrok, 1986. 14. A. Friedman, Partial Differential Equations of Parabolic Type. Prentice-Hall Inc., Englewood Cliffs, NJ, 1964. 15. T. Fujiwara, H. Kunita, Stochastic differential equations of jump type on manifolds and LBvy processes in diffeomorphisms group, J . Math. Kyoto Univ. 25 (1985), 71-106. 16. T. Fujiwara, Stochastic differential equations of jump type on manifolds and Lkvy flows, J. Math. Kyoto Univ. 31 (1991), 99-119. 17. C. Giraud, Genealogy of shocks in Burgers turbulence with white noise initial velocity, Commun. Math. Phys. 223 (2001), 67-86. 18. C. Giraud, On regular points in Burgers turbulence with stalbe noise initial data, A n n . Inst. H. Poincare' Probab. Statist. 38 (2002), 229-251. 19. I. Gyongy, Existence and uniqueness results for semilinear stochastic partial differential equations, Sdoch. Proc. Appl. 73 (1998), 271-299. 20. I. Gyongy, D. Nualart, On the stochastic Burgers equation in the real line, A n n . Prob., 27 (1999), 782-802.
322
21. I. Gyongy, C. Rovira, On LP-solutions of semilinear stochastic partial differential equations, Stoch. Proc. Appl. 90 (2000), 83-108. 22. S. Hodges and A. Carverhi11,~~Quasi mean reversion in an efficient stock market: the characterisation of economic equilibria which support Black-Scholes option pricing, The Economic Journal 103 (1993), issue 417, 395-405. 23. H. Holden, T. Lindstr~m,B. 0ksendai, J . Ub@eand T.-S. Zhang, The Burgers equation with noisy force and the stochastic heat equation, Comm. PDEs, 19 (1994), 119-141. 24. H. Holden, B. Bksendal, J. Uboe and T.-S. Zhang, Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach. Birkhauser, Boston, 1996. 25. N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha, 1981. 26. J. Jacod, A.N. Shiryaev, Limit Theorems for Stochastic Processes. SpringerVerlag, Berlin, 1987. 27. Y . Kifer, The Burgers equation with a random force and a general model for directed polymers in random enviroments, Probab. Th. Rel. Fields 108 (1997), 29-65. 28. O.A. Ladyzhenskaya, N.A. Solonnikov, N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs 23, AMS, Providence, RI, 1968. 29. J.A. Le6n, D. Nualart and R. Pettersson, The stochastic Burgers equation: finite moments and smoothness of the density, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 3 (2000), No.3, 363-385. 30. E.H. Lieb, M. Loss, Analysis (2nd Ed.). Graduate Studies in Mathematics 14, AMS, Providence, Rhode Island, 2001. 31. C. MueIIer, The heat equation with LBvy noise, Stoch. Proc. Appl. 74 (1998), 67-82. 32. L. Mytnik, Stochastic partial differential equation driven by stable noise, Probab. Th. Rel. Fields 123 (2002), 157-201. 33. D. Nualart and B. Rozovskii, Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise, J. Funct. Anal. 149 (1997), 200-225. 34. W. Rudin, Real and Complex Analysis. (2nd edition) McGraw-Hill, New York, 1974. 35. E. Saint Loubert BiB, Etude d’une EDPS conduite par un bruit poissonnien, Probab. Th. Rel. Fields 111 (1998), 287-321. 36. M.F. Shkesinger, G.M. Zaslavsky and U. Frisch (Eds.), Le‘vy Flights and Related Topics in Physics. Lecture Notes in Physics, 450, Springer-Verlag, Berlin, 1995. 37. Ya.G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Commum. Math. Phys., 148 (1992), 601-621. 38. Ya.G. Sinai, Two results concerning asymptotic behavior of solutions of the Burger equation with force, J. Stat. Phys., 64 (1992), 1-12. 39. R. Tribe and 0. Zaboronski, On the large time asymptotics of decaying Burgers turbulence, Commun. Math. Phys., 212 (ZOOO), 415-436. 40. A. Truman and H.-Z. Zhao, On stochastic diffusion equations and stochastic Burgers’ equation, J. Math. Phys., 37 (1996), 283-307.
323
41. A. Truman and H.-Z. Zhm, Stochastic Burgers equation and their semiclassical expansions, Commun. Math. Phys., 194 (1998), 231-348. 42. J.B. Walsh, An introduction to stochastic partial differential equations. In: Ecole d 'e'te'de Probabilite's d e St. Flour X I V , pp. 266-439, Lect. Notes in Math. 1180,Springer-Verlag, Berlin, 1986. 43. M. Winkel, Burgers turbulence initialized by a regenerative impulse, Stoch. Proc. Appl., 93 (2001), 241-268.
A COMPARISON THEOREM FOR SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH TWO REFLECTING BARRIERS AND ITS APPLICATIONS
T.S.ZHANG Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England.
In this note, we prove a comparison result for the solutions of backward stochastic differential equations with two reflecting barrier processes. The result is then applied t o obtain an existence result for solutionsof a backward SDE with reflecting barrier processes under weak assumptions on the coefficients.
AMS Subject Classifications: Primary 60H20. Secondary 60H10,60H30.
1. Introduction
The notion of backward stochastic differential equation was introduced by Pardoux and Peng' in (1990), there they obtained existence and uniqueness of adapted solutions under suitable conditions on the coefficients and terminal random variables. Certain backward SDE were also independently used by Duffie and Epstein in (1992) to study stochastic differential utilities in economics models. This subject has attracted a lot of attention and has developed rapidly in recent years, which is partly due to the applications found in the theory of partial differential equations and mathematical finance, etc. See 2,5,8 and references therein. We particularly mention the paper by J.Cvitanid and I.Karatzasl, which motivates our work here. In this paper they studied backward stochastic differential equations with two reflecting barrier processes and obtained existence and uniqueness of solutions under various conditions. They also proved that the solution of a backward SDE with two reflecting barrier processes is the value function of certain Dynkin game. If the coefficients depend also on the state variable, the Lipschitz condition is required. The aim of this paper is to prove a comparison theorem for solutions of backward stochastic differential equations with two reflecting barrier pro324
325
cesses. The theorem is applied to obtain a new existence result for solutions of a backward SDE with reflecting barrier processes. 2. Backward SDE With Two Reflecting Barrier Processes
In this section we follow closely the notations in '. Let ( R , F , P ) be a complete probability space. Bt , t 2 0 denotes a standard &dimensional - the argumentation of the natBrownian motion. Denote by F = (.F~}o
<
L ( t ) 5 U ( t ) , YO 5 t 5 T and L ( T ) 5 5 5 U ( T ) a s (1) These two processes will serve as two reflecting barriers. Consider the backward SDE with two reflecting barrier processes.
+
+
d X ( t ) = - f ( t , w , X ( t ) ) d t - d K 1 ( t ) dK2(t) Y'(t)dB(t)
As in
we introduce the following
-
(2)
Definition 2.1. We say that ( X ,Y,K 1 ,K 2 ) : X : [0,TI x R 4 R , Y : [0, T ] x R 4 Rd, and K 1 ,K 2 : [0,T ] x R R is a solution of the backward SDE (2) with reflecting barriers U ( . ) ,L ( . ) and terminal condition if the following holds (i) X , Y,K 1 and K 2 are continuous and F-progressive, (ii) K1(t),K 2 ( t ) t, 2 0 are increasing with K1(0) = K 2 ( 0 )= 0, (iii)
X(t)=
<+
1
T f(S,
<
+ K 1 ( T )- K 1 ( t )
w,X(s))ds
- ( K 2 ( T )- K 2 ( t ) )-
/
T
Y'(s)dB(s), 0 I t IT,
0
(3)
(iv) LJt) 5 X ( t ) I U ( t ) , 0 5 t 5 T, (v) J, ( X ( t )- L ( t ) ) d K l ( t )= J?(U(t) - X ( t ) ) d K 2 ( t )= 0, almost surely. Let fl(s,w , x) and f 2 ( s , w , x) be two P@B(R)-measurablefunctions. Let ( X i , K:, K?) be a solution to equation (2) with f replaced by fi, terminal condition
Theorem 2.2. Assume one of fl and condition in x uniformly w.r.t. (s,w ) , i.e., lf2(s,w7x1) -f2(s,w,x2)1
f2,
say
f2,
satisfies a Lipschitz
I c151 - 5 2 1
326
for some constant c. If 51 L then
E2
and f ~ ( s , wx), L f i ( s , w , x ) almost surely,
Xl(t) 5 X 2 ( t ) , Vt E [O,T], a.s.
(4)
Proof. Choose a sequence { & , n 2 1) of functions that satisfy & E C 2 ( R ) ,&(x) = 0, for 5 5 0, 0 5 4L(x) I 1, &(x) 2 0 and &(x) ,/ x+. This is always possible, see ,for example, the proof of Theorem 3.2 in Observe that
Applying It6’s formula, we have
This gives
327
Taking expectation we see that
where we have used the fact that Letting n + 00,we obtain
5 &, and (iv), (v) in the definition 2.1.
Iterating the above inequality n times, we arrive at the following
1 E[(Xl(S) - XZ(S))+]I M--c"(T - t)" n!
(9)
where M = suptcT E[IXl(t) - Xz(t)(].We complete the proof of the theorem by sending t o +00.
n
328
3. Existence of Solutions of A Backward SDE with Two
Reflecting Barriers In this section, [ denotes a F+ measurable random variable with E[l[I2]< 00, which will be the terminal condition. L ( t ) ,U ( t ) , t 2 0 denote the reflecting barrier processes as in section 2. Let g ( t , w) : [0,T ] x R + R be a P-measurable process. Consider the backward SDE with reflecting barriers
L ( t ) ,U ( t ) ,t 2 0 : d X ( t ) = - g ( t , w)dt - d K l ( t )+ dK2(t)+ Y ' ( t ) d B ( t )
X ( T ) = 5.
(10)
Condition 3.1. The backward SDE with reflecting barriers L ( t ) ,U ( t ) , t 2 0 admits a unique solution ( X ,Y,K 1 ,K 2 ) for every ?-measurable process g with E[JTg ( t , ~ ) ~ d < t ]00. Remark 3.2. Condition 3.1 is fullfilled if L ( t ) < U ( t ) , 0 I t < T and LWX{t
See
+ [ X { t = T } I E[EIFtT,]I U ( t ) X { t < T }+ 5 X { t = T }
(11)
for details.
Let f : [0, T ] x R x R as in section 2.
+
R be a P 8 B(R)-measurable function described
Theorem 3.3. Suppose that f is bounded and f(t,w,.) : R + R is uniformly continuous on bounded intervals uniformly with respect t o ( t ,w) E [0, T ] x R} . In addition, condition 3.1 holds. Then there exists a solution to the backward SDE (2) with two reflecting barrers L , U , terminal condition [ and the coefficient f . Proof of the theorem. Choose a decreasing sequence fn : [0, T ]x R x R + R , n 2 1 of P 8 B(R)-measurable functions that satisfy
Ifn(t,w,.)
- fn(t,w,Y)I B CnI.-Yyl
for some constant c, and that for any fixed ( t ,w ) E [0, T ] x R, f n ( t ,w ,). converges to f ( t ,w , x) uniformly on bounded intervals in R. It is easy to see that such a sequence f,, n 2 1 exists under our assumptions on f . It was proved in that when f is repaced by fn the equation (2) has a unique solution. Let us denote it by ( X n ( t ) ,Y,(t), KA(t),K:(t)). By Theorem 2.1, we have
'
X , ( t ) 2 X2(t) 2 X 3 ( t ) 2 . . . 2 X n ( t ) 2 . . . 2 X M ( t ) ,
a.s
(12)
where X M stands for the solution to the backward SDE with reflecting barriers corresponding to the coefficient given by the lower bound of f (t,w ,.).
329
This shows that the sequence { X n ( t ) } n > lhas a limit, which we will denote by k(t).Observe that for fixed ( t , w ) , k n ( t )lies in a compact interval of R for all n 2 1. Hence, fn(t,w,X n ( t ) )- f ( t ,w,X ( t ) )
+ f ( t , u,X n ( t ) )- f(t,w , R t ) )
= f n ( 4 w,X n ( t ) )- f(t,w,X n ( t ) )
as
+O,
(13)
7 2 4 0 0
which yields, by the dominated convergence theorem , that n-+0 lim
Ell
T
m)2
( f n ( 4w , X n ( t ) )- f ( 4 w,
dtl
=0
(14)
Let ( X ,Y,K 1 ,K 2 ) be the unique solution to the backward SDE with reflecting barriers with g ( t , w)= f ( t ,w,X ( t ) )replacing f ( t , w , .). Then
X(t)=I
+
1
T
+
f ( s , w ,X ( s ) ) d s K 1 ( T )- K 1 ( t )
- ( K 2 ( T )- K 2 ( t ) )-
/
T
Y ' ( s ) d B ( s ) , b'0 5 t 5 T ,
(15)
t
L ( t ) 5 X ( t ) 5 U ( t ) , "0 5 t 5 T ,
l T ( X ( t )- L ( t ) ) d K l ( t= )
LT
(16)
( U ( t )- X ( t ) ) d K 2 ( t )= 0 ,
(17)
almost surely. Next we show that X n ( t ) converges to X ( t ) ,and hence, X ( t ) = X ( t ) . From the It6 rule,
+2
LT
( X n ( S )- X ( S ) ) ( Y ' ( S )
- YL(s))dB(s)
LT
lY'(s) - Y;(s)l2ds.
Keeping in mind that L ( s ) 5 X n ( s ) 5 U ( s ) and L ( s ) 5 X ( s ) 5 U ( s ) we have
330
- 2 L T ( X n ( S )- X ( s ) ) d K ' ( s )
1 1
T
=
-2
=
-2
( X n ( S )- L ( s ) ) d K l ( s )+ 2
T
4'
( X ( s )- L ( s ) ) d K l ( s )
( X n ( S )- L ( s ) ) d K l ( s )
10
Similarly, it can be seen that
Therefore] limn+mE[(Xn(t) - X ( t ) ) ' ] = 0. So, X ( t ) = X ( t ) . Consequently, (XIY,K ' , K 2 ) is a solution to the backward SDE with reflecting barriers with coefficient f ( t , w , z), and the terminal condition (. This ends the proof.
33 1
Acknowledgements: Financial support of the British EPSRC (grant no. GR/R91144/01) is gratefully acknowledged. References 1. J.Cvitanic and LKaratzas: Backward Stochastic Differential Equations With Reflection and Dynkin Games, The Annals of Probability 21:4 (1996) 20242056. 2. Duffie.D and Epstein.L, Stochastic differential utility, Econometrica 60 (1992) 353-394. 3. N.Ikeda and Swatanable, Stochastic Differential Equations and Diffusion Processes, North-Holland and Kodansha,Amsterdam and Tokyo, 1981. 4. LKaratzas and S.E.Shreve, Brownian Motion and Stochastic Calculus,2nd ed. Springer, New York, 1991. 5. N.E1 Karoui, C.Kapoudjian, E.Pardoux, S.Peng and M.C.Quenez, Reflected solutions of beckward SDEs and related obstacle problems for PEDs. Preprint,l995. 6. E. Pardoux and S. Peng, Adapted solutions of backward stochastic differential equations, Systems and Control Letters 14 (1990) 55-61. 7. T.S.Zhang, On the strong solution of one-dimensional stochastic differential equations with reflecting boundary, Stochastic Processes and Their Applications 50 (1994) 135-147. 8. T.S.Zhang, On the quasi-everywhere existence of the local time of the solution of a stochastic differential equation, Potential Analysis 5 (1996) 231-240.
BURGERS EQUATION AND THE WKB-LANGER ASYMPTOTIC L2 APPROXIMATION OF EIGENFUNCTIONS AND THEIR DERIVATIVES
A. TRUMAN Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Email: A . [email protected]
H.Z. ZHAO Department of Mathematical Sciences, Loughborough University, Loughborough, L E l l 3 T U , UK Email: [email protected] In this paper we study the WKB-Langer asymptotic expansion of the eigenfuncV(z). Applying these asymptotic tions of a Schrodinger operator H = -+ti2& formulae we prove that the exact L2 eigenfunction Q J I E ( N , (and ~) its derivative h Q & ( N , f iof ) ) the Schrodinger operator with a well-shaped analytic potential are approximated up to arbitrary order hm by the semi-classical WKB-Langer approximate eigenfunction QE, ( ~ , f i ) (and , ~ its derivative hQ& ( N , h ) , m ) respectively in n
+
L 2 , i.e. IIwE(N,fi)-Q'E,(N,fi),mttL2= o(hm+l),
l l h Q & ( ~ , h -)h w & n ( ~ , f i ) , m I I L 2
=
O(hm+l) uniformly for any N. Here E m ( N , h ) approximates E ( N , h ) up t o m-th order (in h) and satisfies the m-th order quantization condition. There are applications of this limit to Burgers equations, turbulence and the large scale structure of the universe.
1. Introduction
Consider the following second order Schrodinger operator on
1 a2 H = --h22 ax2
IR1
+V(x)
We are interested in studying approximate eigenfunctions and their derivatives by using our Hamilton Jacobi methods, extending the results in Truman and Zhao 21 to more general potentials than linear harmonic oscillator ones. It is well known that the WKB method leads to approximate eigenfunctions. However, the WKB solution is not finite at the turning points. Langer therefore introduced Bessel functions to study the wave functions 332
333
near the turning points (Langer The WKB-Langer semi-classical solution associated with the N-th Bohr-Sommerfeld quantization rule gives the approximation for the exact eigenfunction associated with the N-th exact eigenvalue. As we will explain soon, however, the exact eigenfunction is not dominated by the WKB or WKB-Langer solution pointwise. The approximation of the WKB or WKB-Langer semi-classical solution t o the eigenfunction needs t o be justified mathematically. One way to do this is to compare the error of the approximation with the modulus function M ( z ) of Airy functions (Olver 15). However, M ( z ) is related to an Airy function which is exponentially large for large 1x1 and is excluded in the approximation of the eigenfunction. In this paper by using the Hamilton-Jacobi continuity equations we will prove that the exact eigenfunction Q E ( N , R ) is approximated by the WKB-Langer semi-classical approximate solution Q E , ( N , ~ ) , ~in L2(R) up to m-th order in h. Moreover, we will prove its derivative hQk(N,h)is also approximated by hQ’,m(N,h),m in L2(R) up to m-th order in h. Here E,(N, h) approximates E ( N ,h) up to rn-th order (in h) and satisfies the rn-th order quantization condition. Note that the modulus function used in Olver l5 is not in L 2 ( R ) ,so Olver’s result does not lead to the approximation in L 2 ( R )studied in this paper. Our results are valid for large quantum number N as long as N and fi satisfy a rigid relation such that E ( N ,h) is bounded and V(z) satisfies some minor condition at the turning points. Our proof is simple. We do not need to use the pseudo-differential operator theory of Helffer (Helffer, Martinez and Robert 7 , or the sophiscated canonical operator method of Maslov (Maslov and Fedoriuk12). Moreover, our proof leads t o the approximation of the derivatives in L2(R). We should point out that if we use the WKB expansion near the turning points although the first term in the WKB expansion is in L2(R), the second term is definitely not in L2(R). This makes i t difficult to use the WKB expansion as a method of approximating the exact eigenfunction in L2(R). The WKB-Langer expansion does not suffer from this difficulty. We prove that the WKB-Langer semi-classical approximate wave function (which is well defined at the turning points) actually gives the correct approximation to the exact eigenfunction in L2(R). See Simon17 for the low lying eigenvalue case. We should like to point out that the pointwise approximation of the WKB-Langer semi-classical solution (which reduces to WKB for z being sufficiently far away from the turning points) is not mathematically justified. To see this we refer to the asymptotic expansion formula (2.41) of this paper. As the first term cos(k J,”b(z)dz- 2) has many zeros, g110711).
b=
(XI
334
4 sin( is," b(z)dz- :)&(z)h is the dominant term at these zeros, not b y ) 1 cos(i s," b(z)dz - 2). However, the first term in the WKB-Langer
bz (T) semiclassical asymptotic expansion approximates both the eigenfunction and its derivative in L2(R), the natural norm of quantum mechanics. There will be many applications of the results of this paper, e.g. quantum probability, quantum tunnelling problems (Jona-Lasinio, Martinelli and Scoppola ', Simon l a ) etc. The Hopf-Cole transformation applied in this setting should also yield new results for Burgers equation and its inviscid limit. This is not so surprising since the Hamilton-Jacobi continuity equations first arose in this context. We do not include these results here due to the length of this paper. But we aim to study some of these applications in our future publications.
2. WKB-Langer asymptotic expansions
Let Qg be the eigenfunction of H with eigenvalue Eh. Then satisfy the following time-independent Schrodinger equation: d2 h2-Q;(z) dx2
Qg and E h
+ Q 2 ( ~ ) Q k =( z0).
Here Q2(z)= 2(E - V(z)). Let the real function be defined by
( d 2 ( E - V(z)) if V(z) < El b(z) = - E ) if V ( z )> E.
First we consider the simplest case where we have only one turning point in order to obtain some useful formulae. In the next section we will apply these formulae to more complicated but practical and useful situations. Let T denote a turning point and for simplicity here we suppose r is a simple zero of Q 2 ( z ) .In this section we consider the case when V ( z )> E if z < T and V ( x )< E if x > 7- for a T E R1.Then Q ( x )is simply
Define a complex single-valued function [(z) by
335
and a real valued function @(z)by
The following simple proposition tells us about the smoothness of @(z) and the dependence of @ E on E .
C" and 0 < IV'(x)I < 00 for z E [a,b] a small neighbourhood containing r , then the function @(x) is C" on [a,b] and for any fixed z, I@E(z) - @fi(z)l= O(IE - El) and I@&(z)@k(z)I= O(IE - 81) for any E , E E V ( [ a 61). , In particular I@(z)I> 0 for z E [r - 6, r 61 for some 6 > 0 . Furthermore, if V ( z )is analytic for z E [r- 6,r + 61,then @(z)is analytic on [r- 6, r + 61. Lemma 2.1. Suppose that V ( z ) is
+
Proof. For y E [ V ( b ) , V ( a ) define ], z = V-'(y) V(z) = y , assumed unique for z E [a, b]. Set
to be the solution of
G-dy) = V - l ( y ) , Gn(y)
=
G k - l ( ~ )n, = 0 , 1 , 2 , . . .,
and & E ( Y ) = @E(V-'(y)).
Then for y E [ E ,V ( a ) ]changing , the integration variable and integrating by parts lead to
=
1-
--Go(y) 3
+ -31.
Ly
(2(y-E))+
By induction we can prove for each n 2 0,
Gl(y)(2(y- E ) ) Q d y .
336
Similarly, for y E [V(b),El, for each n
2 0,
I t follows that & ~ ( yis) smooth in y and Lipschitz continuous in E. Then the smoothness of @.E(x)in x and Lipschitz continuity in E follow from the identity @ E ( x )= & E ( V ( X ) ) The . Lipschitz continuity of S b ( x ) in E follows similarly due to the derivative formulae (2.4) and (2.5). From the definition of G o ( y ) , IGo(y)l > 0 for y E [V(b),V ( a ) ] .Therefore from (2.4) and (2.5) for n = 0, I6(y)I > 0 for y sufficiently close to E , which implies I@(z)I> O for x sufficiently close to T , say x E [T - 6,r S ] for a S > 0.
*
+
In the following we always write r* = T 6,for a S > 0 such that 0 < IV'(x)I < 00 and I@(x)I> 0 for x E [ r - , r f ] Denote . S(z) = @-'(z)@''(z) which is smooth for x E [ T - , T + ] by Lemma 2.1. Moreover e ( x ) is analytic for z E [r-,r+]if V ( x )is analytic. Define
Lemma 2.2. O n [ r - , r f ]aj , E CW,@E C", are bounded. Therefore, the power series a j ( x ) h 2 j ,1 ,6j(x)h2j, and their derivatives C,"=, ai(x)h2j, ,L?: (x)h2j are asymptotic expansions as ii -+ 0 f o r x E [r-,r+]in the sense of Poincare'. And if V ( x )is analytic, then a j ( x ) , P j ( x )are also analytic for x E [ T - , T + ] . Furthermore, a j ( x ) , &(z) and & ~ j ( x ) , &.Pj(x) are Lipschitz continuous with respect t o E uniformly in x f o r x in [r-,r f ] .
cj"=,
for x E [r-,7'1,
Cj"=,
Proof. Define for any function
+ cj"=,
e,
Suppose for x E [r-,T + ] , fi(x)is smooth and V ( x )# 0. Define V - ' ( y ) to be the solution of V ( x )= y for y E [ V ( r f )V, ( r - ) ]and
G-l(x) =
lx
&x)dx.
337
Let G-l(y), Go(y), G l ( y ) , . . . be defined by
~ - I ( Y )= G-i(v-'(y)) G n ( y ) = en-l(y), n
= O , I , 2 , . . .,
and G(Y) = c.(v-'(Y)).
Similar to (2.4), using the integration by parts formula and induction principle, we have for y E [ E ,V ( T - ) ]and , each n = 0 , 1 , 2 , . . .,
and for y E [V(T+), El, and each n
= 0 , 1 , 2 , . . .,
Applying (2.8) and (2.9) to al, a2,. . . , we have the smoothness of aj and pj in z and Lipschitz continuity in E . The rest of the lemma follows from van der Corput's fundamental theorem on asymptotic series and its consequence on asymptotic series with a parameter (Theorem 4.1 and P391, van der Corput 2z). See also 0lverl3>l4. We follow Langer to define wave functions using Bessel functions near the turning point. Alternatively one can use Airy functions (Olver 13,14,15,Heading 6 ) . Define 9110111
K"4 = C-ESJ-&) 1
=I
6
+c+[+J+(,) 6
e$i(Jz," b(z)dx)i
x(C-J-g(-fi J,'b(z)dz)
(2.10)
+ C + J + ( - i J , ' b ( s ) d z ) ) , if x < 7 ,
(J,"b(z)dz)i x ( C - J - + ( i J , " b ( z ) d z ) + C + J + ( k S , " b ( z ) d z ) ) , if z > 7 ,
where J - + ( - ) and J + ( - ) are two Bessel functions and C - and C+ are some constants, and then, Langer's approximate wave function is defined by !P~(x)= @ ( ~ ) K ' ( Z ) . (2.11)
338
The following celebrated result was given in Langer
g>lOzll.
Proposition 2.3. (Langer) For any E , suppose V ( x ) is smooth near the turning point T : V ( T )= El and V ' ( T )# 0 , then there exists 6 > 0 such that 0 < IV'(z)I< 00 and I@(z)I> 0 for x E [ T - , T + ] , T* = T f 6 . T h e function Qo(x)defined by (2.11) satisfies the following differential equation for z E
[T-,T+]
d2 1 (2.12) -Qo(x) jgQ2(x)Qo(x)= O(x)Qo(x). dx2 Furthermore, f o r x E [ T - , T + ] the solution of the equation (2.1) has the following representation
+
Q'(Z)
= Q;(x)A'(x)
+ Qo(x)B'(x).
(2.13)
Here A ( x ) ,B ( z ) satisfy the differential equations f o r x E
(2ti28-2Q2)A'+(ti20'-2QQ')A+h2B"+ti20B
= 0,
[T-, T+]
BA+A" +2B'
And moreover, A ( x ) ,B ( z ) have the asymptotic expansions f o r x E as h 4 0 , 00
M
j=1
j=1
= 0.
[T-, T+],
where c ~ j , p jgiven b y (2.6) are smooth and bounded functions for x
E
[T-, T'].
For x < T , recall some standard results about Bessel functions (c.f. e.g. Whittaker and Watson 25)
-e2" i J L ( - -; L ' b ( z ) d x ) = e : ' J ~ ( ~ / ' b ( x ) d x )= I + (1 z L' b ( x ) d x ) , 3 t i x
and
Then for x
< T . we have
339
(lT
1 2 a
b(z)dz)s(-C-K1(7r
=
3
Notice that I ; (
;lr
b(z)dz)
b(x)dz)- (C+ - C - ) I ; ( i s,'b(z)ds)).
-
exp{ JzT b(z)dz}when fi is small for z E [T-,T In order to have a L2(R) solution, we have to choose C- = C+. Recall for z E [ T - , T - $61,when ti is small, JZr
is].
with M h ( x )= 1
C+ =
&,
+ O ( h ) having an asymptotic expansion. If we take C-
=
then
and so when tL is small, (2.16)
For x E
[T
+ ? ~ S , T + ] ,when fi is small,
(2.17)
with
j=1
j=1
and
(2.19)
Notice that L1 and L2 in (2.19) for J-; are the same as those in (2.17) for J; . So for z E [T $5, T + ] , using the same C- and C+ as in the region
+
xE
[T-,T -
$71, i.e. C-
= C+ =
&,we have from
(2.10) and (2.11), as
340
h is small. Qo(x) =
1 ( x ) d x- -.rr)LF(x) 4
+
sTx
(2.20) 1 b(x)dx - -.rr)Lg(x)). 4
It is noted that the term sin( b(x)dx- ;7r)Lg(x) = O ( h ) should not be neglected. For simplicity in the following we only consider bound states where we take C-
= Cf =
6.So
for r- 5 x
From P366 in Whittaker and Watson
< r,
25,
341
From P354 in Whittaker and Watson
25,
it follows that
By similar argument we have
I t follows that d -K"z) dx
So for
T
< rc 5 T + , (2.22)
and the term ( ~ ~ , " b ( z ) d z ) ~ ( J - ~ ( ~ ~-~Jb+(( zi J) :db (z z)) d z ) ) is bounded. I t is evident that for z E [ T - , T + ] , for any ti > 0, 9 0 ,h 9 & ,h29&' are bounded. But we need some uniform (in h) estimate. We prove the following lemma which will be used in the next section. Lemma 2 . 4 . Assume conditions of Lemma 2.1. Then for any interval [a,b ] , J'(90(x))2dz, Jab(hQ;(z))2dz and Jab(h2Qg(z))2dz are bounded uniformly in ti.
Proof. We only need to prove that (90(x))' is integrable uniformly in ti with respect to z if the turning point T E [a,b]. Note z + K + ( z )z, i J ; ( z )
342
and , z i J - + ( z ) are bounded, so (I 'J , b ( y ) d y l ) k K h ( z )is bounded uniformly in k. So by the definition of \ko and Lemma 2.1, we know
for a constant M > 0 wich is independent of h and z. But T is a simple b zero of V(z) - E , so the improper integral dx is convergent.
s, (I J,r
b(Y)dYO
That is to say s , b ( \ k ~ ( x ) ) ~is d xbounded uniformly in h. Then results for J:(h\kb(z))2ddz and J : ( h " @ & ' ( ~ ) ) ~follows dx from (2.21), (2.22) and (2.12) respectively. Away from the turning point, Langer's construction turns out t o be a simple formula. We first study the asymptotics of (2.13) for z E [ T - , T - $1 and z E [T $5, ~ f ] .Then we will extend the solution t o the whole line R1. From the asymptotics (2.15) and K ; for large argument (P367, Whittaker and Watson 2 5 ) 1 for x E [ T - , T - +6]for small ti,
+
+
with R f i ( z )= 1 O(h) and having an asymptotic expansion. It turns out from (2.16), (2.21) and (2.23) that for z E [ T - , T - 461 (XI
(2.25) where
F ( z , h) = M"z)B(x)+h-M @'(x)
@(x)
fL (x)T+b(z)Ryz)A(%)
A ( x ) = l+O(h). h
It is obvious that P - ( x ) has an asymptotic expansion in powers of ti, say Dc)
P - ( z , h)
N
1
+=-pi)j$j(z), j=1
(2.26)
343
for some smooth and bounded functions $j on [ T - , T - ;6] which are combinations of aj and ,Bj and the asymptotic expansions of M h ( x ) and Rh(x). In particular we have fked values of $j ( T - ) . Note if we take
rv1 aj(z)h2jand Bm(z)= 1
A, =
+
WI ,Bj(z)ti2j
and define
Then
P - ( z , ti) - P J z , ti) = O(tirn+1). For z E Watson 2 5 ,
[T+
$S,T+], when ti is small, again from P362 in Whittaker and
(2.27)
and
Similarly
(2.28)
Note that R1 and turns out that
R2
in (2.28) are the same as the ones in (2.27). It
344
From (2.20), (2.22) and (2.29)
Clearly we have
and
Now by (2.13), we have for any fixed r
+ i6 I x I r f , for small fi,
where
+
= 1 O(fi2)and Pz(x)= O(fi). Moreover, PI and and therefore PI(z) PZ have asymptotic expansions of even powers of h and odd powers of h respectively, say the following
+
for some smooth and bounded function 4j on [r i6,r+].In particular we have fixed values for $j (r+). We study the WKB asymptotic expansion outside [r-,r f ] .From Theorem 26.3 in Wasow 24, for x < r - :6, the Schrodinger equation (2.1) which can be reduced to a system of 2-dimensional singular perturbed differential equations possesses a solution of the form (2.33)
345
and for x form
>r
+ i6,the Schrodinger equation possesses a solution of the (2.34)
and P - ( x ) , P k i ( x ) have asymptotic expansions in tr. for x < r - i6 and x > r i6 respectively. It is easy to prove the following lemma.
+
Lemma 2.5. For x < r dafferentaal equation,
>r+
$6, the function P - ( x ) satisfies the following
$.-
+
(x) = - 1fi- d2 (b- (x)P- (x)), 2 dx2 i6,P*Z(x) satisfy
b+ (x) and f o r x
-
b+(X)-&P*'(X) d
= &-hz-(b-qx)P*Z(x)). 1 d2 I
2
dx2
(2.35)
(2.36)
Proof. The proof is some simple elementary computations. We leave it to the reader. 0 Define $O(Z)= 1, and
+j(X)
=
i
;s,'
4 q - ) - b-~(y)~(b-3(y)$~-~(y))d ifx y , < 7-, q + ( ~ + ) $+ J : b - + ( y ) ~ ( b - q ( y ) ~ ~ - ~ ( y ) ) difx y , > r+, (2.37) j = 1,2,...,
+
where $ j ( r * ) are defined in (2.26) and (2.32). It is evident that 4 j ( x , E ) is Lipschitz continuous with respect to E as r* is Lipschitz continuous with respect t o E . It turns out that $j ( j = 0 , 1 , . . .) satisfy the following iterated time-independent Hamilton Jacobi continuity equations (Truman and Zhao 20), for x < r- and x > r + ,
d 1 d2 1 = --(b-qZ)&I(x)),j = 0,l; (2.38) dx 2 dx2 with convention that 4-1 = 0. Note that $j(x) are bounded for any x < r- and x > r+ and Lipschitz continuous with respect to E . Therefore Cj"=,$j(x)(-fi)j, for x < r - , and Cj"=o$j(x)(ffii)j, for LC > r+, are asymptotic expansions in the sense of Poincare, as fi -+ 0 by the van der Corput Theorem. It is a simple exercise to check that formally Cj"=oq5j(x)(-h)j,for x < r - , and Cj"=04j(z)(ffii)j,for x > r+, satisfy (2.35) and (2.36) respectively. However, since P- and P*i should have unique asymptotic expansions, bf(Z)-$j(X)
00
P-(x)
N
C $j(x)(-fi)j, j =O
for x
< r-,
346 00
~ * z ( z )N
C ( ~ ~ ( z ) ( * h for i ) Jz, > T + .
(2.39)
j=O
As P - ( z ) has the asymptotic expansion (2.26) for z E [ T - , T - ;a] and satisfies (2.35), so it is easy to check that $j ( j = 0 , 1 , 2 , . . .) satisfy (2.38) Similarly (2.38) is satisfied for z E [T it?, 7 + ] . for z E [ T - , T Therefore $j ( j = 0 , 1 , 2 , . . .) are smooth for z < 7 - $5 and z > 7 $5. In particular, $ j are smooth a t x = T * . To see the asymptotic behaviour of P*i(z) for z a t infinity, we can easily show that l$j(x)I 5 cj1zl-j and l$>(z)l 5 cjlx1-j for a cj. Therefore for large z, by the van der Corput
is].
+
00
oc)
Theorem,
C $j(z)(-h)Jforz < 7 - ,
+
and
C $j(z)(fhi)jforz > T +
are also
j=O
j=O
asymptotic expansions of powers of z for large 1x1. In particular P - ( z ) is uniformly bounded and P - ( z ) - 1 = O(fi) uniformly in z for z < T - . The same conclusion about P*i is true for z > T + . It is noted that $j(z) are analytic for z < 7- and z > T + if V ( z )is analytic. We will only need this in the next section. It is important to note that linear combinations of 9+2 and 9-2 are also solutions of the Schrodinger equation (2.1). We have to choose the appropriate combination of V iand V ifor x > T + to match the solution from Langer’s construction. For the bound state we set 9 = 9- for z < r - , i.e. *‘(z~) = b-+(z)exp{--
k LT
(2.40)
b(y)dy} x ~ - ( z ) ,
+
where P-(x)= 1 O(fi) for small fi uniformly in z for z 9 = exp{-$i}Q+i + e x p ( $ i } W for z > T + , i.e.
<
7-,
and
(2.41)
where
+
P l ( z ) 1 Cj”=1(-l)j$~j(z)h2j and P2(x) Cj”=1(-l)j$2j-l(z)h2j-1 and Pl(z)= 1 O(h2)and P2(z) = O ( h ) for small h uniformly in z for z > T + . And for z near 7, 9 ’ ( z ) = Qh(z)A(z) * o ( z ) B ( z )as in Proposition 2.3. From our construction we know that 9 is smooth on R’. Similar combinations were also used in Furry 4 , Heading and Berry t o exploit the Stokes’ phenomenon in physics literature. We formulate a proposition. N
+
N
+
347
Proposition 2.6. Assume all conditions of Proposition 2.3. Then @ ( x ) which is given by: (2.40) for x < T - ; (2.13) for r- I x 5 r f ; (2.41) for x > I-+, is a smooth solution of the Schrodinger equation (2.1). Remarks. (i) The asymptotic expansions (2.39) only make sense for fixed x # T . They give a pointwise WKB asymptotic expansion ((2.4O), (2.41) for x < I- and x > T respectively) of the wave function for x # r . Although the first term is in L2(R), the second and higher terms are not due to higher order singularities of q5j (j = 1 , 2 , . . .) at x = r . The key to solve this problem is to use WKB-Langer semi-classical asymptotic expansions presented in this paper.
4
(ii) Equation (2.1) possesses another solution @+ which for x < r - S is of the form
:L
9 + ( x )= b-$(x)exp{ -
(2.42)
b(y)dy}P+(x),
where P+ has an asymptotic expansion in powers of ti. We can choose appropriate C- and C+ different from before so that for x E [r-,r !P;(x) given by Langer’s formula (2.13) has asymptotic expansion (2.42). A smooth extension to the whole interval (--00, m) can be done in the same way as before and by exploiting the asymptotic properties of the Bessel functions. This solution is linearly independent of 9 given in Proposition 2.6, but is exponentially large for x < I-.
is],
3. Semi-classical approximation of eigenfunctions and their derivatives in L2 Consider a smooth well-shaped potential V ( x ) bounded below with limlzl+mV(x) = +00. Then by the limit point criteria H = V ( x ) is a self-adjoint operator with discrete spectrum {E(N,h ) } ~ = ..., ~,l, E ( N , Ti) -+ +m as N -+ +00 with corresponding orthonormal eigenfunctions @ k ( x )for any fixed ti > 0 (see Reed and Simon 16). Consider the N-th eigenvalue E(N , h) and corresponding eigenfunction @ k ( x ) .Suppose there are only two classical turning points r l ( E ) and r2(E), the only two roots of V ( x ( E ) )= E. Assume V ( x )is smooth near q ( E ) and r2(E) and V‘(r1)# 0 and V ‘ ( T ~#) 0. Therefore we can apply Langer’s construction of the wave function near both r1 and 7 2 . First by Lemma 2.1, there exists 6 > 0 such that if writing rf = ~j f6, J] and 0 < j = 1,2, then 0 < IV’(x)I < 00, lQT1(x)l > 0 for x E [rL,r1 IV’(x)I < 00, 1QT2(x)I> 0 for x E [ r ; , r z ] . Let 6 > 0 be small enough such that r,’ < r p . We construct the wave function Jrk(x) by Proposition
-& :
+
348
2.6. In the following, Qk,m(z)denotes the first m terms of WKB-Langer semi-classical asymptotic expansions in 5 different regions respectively. For x _< 71,take
m
= QE,O(Z)(l+
X(-h)j$j(Z)
+ 0(hrn+l))
(3.1)
j=1
=
Qk,,(x) + QE,o(z)0(hm+l),
with a uniform 0(hm+')for x E ( - w , T ~ ] . Hence Q;(x),Qk,,(x) and Q;,,(x) are exponentially small for z < 71. For 71 5 x 5 T:, take Langer's construction and apply Lemma 2.4,
Qk(4= Q E , O b ) % 4 + QL,,(x)A(x) IYl = QE,&)(l+
c rw1 c
Pj(Z)h2j
+U ( P + l ) )
(3.2)
j=1
+(hQk,o(x))(
+ 0(hrn+l))
CYj(z)h2j-l
j=1
=
For
71' < x <
T;,
Qk,,(z)
+ QE,o(x)O(II'"+').
set
(3.3)
=
Qk,,(x)
+ 0(hrn+l),
where $ j are defined by (2.37) with conditions, i.e. $j(x) = $ j ( ~ ; ) f $ other hand we should also have
$j(r:)
derived from (3.2) as initial
JTy+ b4(~/>A(b-+(y)$j-l(y))dy.On the
349
=
Gk,&E)
where
$j
+ o(hm+l)'
(3.4)
are defined by (2.37) with JJ(r;) derived from (3.5) below as
szTz -
initial conditions, i.e. $ j ( z ) = $ j ( r F ) - $ b:(y)A(b-i(y)Jj-1(y))dy. And for 72 5 x 5 r2$,take Langer's construction
G k ( x ) = G,,o(x)B(x) + G.',,,(z)A(z) r-1 = GE,O(Z)(l+
c c
&(z)h2j
+O(P+l))
j=1
IF1 +(hGL,o(x))( clj(z)h2j-1
+ O(h"f1))
(3.5)
j=1
=
For x
Gk,,(z)
+ GE,o(x)o(hm+l).
2 r z , take ' k k ( x )= b-i(z)exp{--
:1:
b(y)dy} x p - ( x )
c m
=
!i&,o(x)(l+
$j(Z)(-h)j
+ O(hrn+l))
j=1
= Gk,,(z)
+ GE,O(z)O(hm+l)l
with a uniform O(hm+l)in z for x > 7-2'. Here +k(x) and G;,,(x) are both exponentially small when x > r2$ is large. Here P- in (3.1) and P in (3.6) are defined as in Section 2. From Section 2, we know that 9 is smooth for z E (-00,r;] and ?t is smooth for x E [r:, 00). Remarks. (i) From Section 2, especially Remark (ii) following Proposition 2.6, Equation (2.1) also possesses a solution Q+ # L2(R) given by (2.42), i e . 9 i ( x ) = b-*(x)exp{i b ( y ) d y } P + ( s ) , for x E ( - m l ~ , - ] .
szT1
350
The smooth extension of the solution to the whole space (-00, m) can be done by using the same method as (3.1)-(3*6) and (3.8). The solution is linearly independent of the L2 solution 9 given in (3.1)-(3.6) and (3.8). Furthermore any solution 9 1 of (2.1) is of the f o r m 9 1 = c19- +c29'+ for constants c1 and c2. But for a L2 solution 91,c2 = 0 is satisfied, whence 91 = c1Q. That means any L2 solution Q1 of (2.1) is linearly dependent on 9, which is equivalent to the vanishing Wronskian determinant property for any x , d d -d@ x 1 ( x ) 9 ( x ) - 9 1 ( 2 ) d- Q x ( x ) = 0. I n particular, we can choose c1 = 1/11911 so that 11Q111 = I. Therefore 9 is the unique L2 solution up to normalization. Here we state our results for the WKB-Langer solution 9 . One can give our results for the normalized wave function if one likes. (ii) The semi-classical WKB-Langer approximate solution ~ E , ~ ( isx given by the first m terms of the series in (3.1)-(3.6) in five different regions respectively. Note
)
m
lim
~ E , ~ ( z )
ZTT;
j=l
and lim 9 ~ , ~ = ( x )
b(y)dy}P;(r;, fi)
ZIT;
from (2.26). These two limits are different, but the difference is 0(fim+') as P - ( z , f i ) - PG(x,fi) = 0(hm+'), so are limxfT; hQk,m(z) and limxLT; k9b,m(x).A similar remark applies for x = I-:, 72, I-;. However, the discontinuity of Q E , ~ ( Xand ) ti9&,m(x) at only four discrete points r1 ,r t , 1-2, r$ does not give rise to any dificulty in L2(R) as ~ E , ~ ( isx ) differentiable on (-m, T T ) ,(I-;, I(T?, :) I-;), , (I-;, I-,') and (I-:, m) respectively. One can choose a continuous or even differentiable 9 ~ ,But~ this . is not necessary here and is not the point of the paper. I n the following, Q E , ~ ( Xand ) f i 9 k , m ( x ) at I -&~ (j = 1 , 2 ) are not necessarily defined. But one can define them by either the left limits or the right limits as these two limits are asymptotically close as fi -+ 0 . The semi-boundedness of V guarantees that the Schrodinger operator H has a unique L2(R*)eigenfunction up to normalization. As V is assumed to be smooth, this eigenfunction is also smooth. For the validity of the formula for the exact wave function @ E , 9~ and &, must be linearly dependent
351
in x E [r:, r;]. That is to say the following Wronskian determinant must vanish for any ti > 0, x E [T:, r;]:
(-Q;(x))@;(x) d dx
-
*;(x)-QE(x) d -ii dx
= 0.
(3.7)
We will see soon that quantization condition gives the exact eigenvalue E , i.e. { E ( N ,~ ) } N = o , I.... , Now we transform (3.7) to an explicit equation of E. For this we first differentiate (3.3) and (3.4),
and
It is crucial here that the leading term in (3.8) has a different sign from the leading term in (3.9). Substituting (3.3)-(3.4) and (3.8) and (3.9) into (3.7) we obtain
= -H"x).
(3.10)
352
Here the formula of H can be given explicitly if one wants to. We note here that H " ( z ) is bounded for all z E [ T ~ , T ; ] uniformly in h. It turns out that 7r
sin(;
b(y)dy - -) = 4hHh(z). 2
(3.11)
Recall that the Wronskian determinant (3.7) is vanishing for all z E [T:, T;] is equivalent to that the Wronskian determinant (3.7) is vanishing at a particular point ( e g . see Hartman 5). Therefore (3.11) is equivalent to
6
J2(E
-
V(y))dy = ( N
1 + -)7rh + harcsin(4hHfi(M)), 2
(3.12)
for N = 0 , 1 , . . .. Here M E [T?, T;] is the minima of V(z). The solution E = E ( N ,h) of the above equation gives the exact N-th eigenvalue of the Schrodinger operator. We take the first m terms in the asmptotics expansions of Q i ( z ) and *;(x), denoted by 9;,,(x) and *;,,(x). We require the Wronskian determinant vanishes a t x = M I i.e.
(y5,7r(4)*t,m(4 d - Q 5 , 7 J ( ~d) ~-ii Q E , , ( ~ ) / s == M0. Similar
to
(3.13)
we will see that this gives discrete values as follows. First we go through all the calculations of (3.8)-(3.12) for Q;,(z) and *;,,(x), then we derive
{E,(N,
(3.7),
~ ) } N = o , J... ,
7r
b(y)dy - -) = 4hHk(M). 2
(3.14)
Similar t o H ( z ) , H k ( M ) is also bounded uniformly in ti. It is followed from (3.14) that
/
72 (
71(
E)
E)
/ 2 ( E - V(y))dy = ( N
1 + -)7rh + harcsin(4hHL(M)), 2
(3.15)
for N = 0,1, . . .. The solution of the above equation depends on m, denoted by E,(N, h). That is to say we have 1 d 2 ( E m ( N ) - V(y))dy = ( N + -)nh+h arcsin (4hHk (Ad)) ,(3.16) 2
for N = 0,1, . . .. It will be seen that that E,(N, h) is an approximation to E ( N ,h) for each N (see (3.26)). Recall the Bohr-Sommerfeld quantization condition
353
Here TI(&) and 72(E0) are the only two roots of V(a:)= Eo. We will analyze the solution in ascending order, setting E = Eo(N,h) for N = 0,1, . . .. The following result is uniform for all N if E ( N ,h) is in a compact subset of { E : J V ( 2 ) < E da: < +m}. For low lying eigenvalues J -
2(E-V(z))
( N is fixed), similar estimate was obtained by Simon (1983). Lemma 3.2. Suppose the same conditions as in Lemma 3.1 and V i s analytic. Assume the E ( N , h ) and E r n ( N I h )satisfies following travel T z ( E (N, ti)) 1 time inequality < ST1(E(N,h))J 2 ( E ( N , f i ) - q y ) ) d y < +GO and 0 < 72 (Em( N ,h)) JTl
( E m( N J 3 )
1 J2( Em( N , h )- V(y))
d y < +aand Eo(N,ti) is the solution of the
Bohr-Sommerfeld quantization equation (3.17) and 0 < (V’(x)(< IV”(a:)I< 00 f o r 2 between
00
and
min{n(E(N, h ) ) ,71(Em(N,h ) ) ,7i(Eo(N, h ) ) )
and
and between
and
Then E ( N ,h) = Eo(N,h)
+0(h2),
(3.18)
354
is O ( h z ) uniformly in N . Without losing generality we assume that E ( N ,h) > Eo(N,h). Then the above gives
That is
(3.21)
355
> 0. Together with (3.20), we have
s
72(E(N,ft))
1
dY(E(N, ti) - Eo(N,ti))= 0 ( h 2 ) &(E(N, ti) - v(Y)) Then (3.18) follows. The proof of (3.19) is similar. TI
(E(N,W)
0
We need a spectral gap result. This can be proved by using the BohrSommerfeld quantization rule and Lemmas 3 . 2 . We first prove the following lemma.
Lemma 3.2. I f 0 < JV'(x))< 00 f o r x E [71(E0(N+l,ti)),71(Eo(N,h))]U [72(EO", ti)), 72(EO(N + 1, and 0 < S7 E';:" d mdx < 00 for
E
= Eo(N,Ti)) and
E = Eo(N + 1,h ) ) , then for suficiently small ti > 0,
356
I Eo(N + 1,ti) - Eo(N, 27rh
Proof. By the Bohr-Sommerfeld quantization rule we know
But also
(3.22)
357
And
for sufficiently small ti. Here M2 in above is a constant. The lower bound 0 of E o ( N 1, ti) - E o ( N ,ti) in (3.22) follows.
+
358
It follows that there exist constants C1 sufficiently small h > 0
Clh 5 E ( N
> 0 and C2 > 0 such that for
+ 1,h) - E ( N ,h) 5 Czh,
and Clh 5 Em(N
+ 1,h) - E m ( N ,h) 5 Czh.
Therefore there exists a neighbourhood I N of Eo(N,h) of which the length is O ( h 2 ) ,there exists one and only one E and Em which are E ( N ,h) and E,(N, h) respectively. But it is easy to see that
d
d -
I
h(~QE(N,fi),m(x)\E(N,~),m ).(
+O(Pfl)
Recall d d x QE , = 0.
- ~\E(N,fi),m(x)\~(~,fi),m)(x)I~=M
= 0.
(3.23)
I
w,ft),m ( x ) Q E( ~N , R ) (x> ,~ -
d z
L=M
QE, ( ~ , f i ) (x) , m %,,, ( N , R ) (x) ,~
(3.24)
But from the construction of Q m we know that there exist constants L1 > 0 and L2 > 0 such that
d d h2I (-QE,m dx (x)*E,m(x) - z Q E , m ( Z ) Q E , m(x))~ z = M d dx
d
- ( z Q E , , n ( Z ) * E , , , , m ( x ) - -QEm,m
(z)8~m
, m ) ( T )Iz=M
I
2 (LI - Lzh)(E- Em(.
(3.25)
This can be seen from the fact that
/
n(Ei)
TI( E I )
1
Q(Ez)
d2(E1- V ( x ) ) d x-
~ ( E- Vz ( z ) ) d x 2 CIES- Ezl,
71(Ez
for a constant C > 0 and Lipschitz continuity of $ j in E . This can be seen easily from the proof of Lemma 3.2. It follows from (3.23)-(3.25)that
E ( N ,h ) - E m ( N ,h) = 0(hm+').
(3.26)
We are now in the position to prove the following lemma. Let QE,,~ be the WKB-Langer approximate eigenfunction corresponding to the approximate eigenvalue E , ( N ,h). Lemma 3.3. Assume conditions in Lemma 3.1. Then for small h IIQ'E,m - S E m , m l l ~ 2 ( W )= 0(hm+'),
(3.27)
359
and
Proof. Without any loss of generality we assume E (N , h) 2 Em ( N ,h). We begin by estimating ' & ( N , h ) ( Z ) - EE,(N,h)(z)
360
From the Lipschitz continuity of @ ~ ( x )in E in Lemma 2.1 we know @ E ( N , A )(x)- @ E , ( N , A ) ( ” )
= O(E(N1h) - Em(N,
o(tim+’)
But by the definition G ( N , h )).(
(4J-S
= C-Ei(N,h)
1 ( $ E ( N , h ) (XI)
1
-tC+Ee(N,h)(x)’JQ(ilEE(N,h) (z)),
1 G m ( N , t i ) ( 4= c-E~_(N,h)jZ)J-;(~EE,,(N,h)(~))
1
+C+
constant M
> 0 and E* between E ( N ,ti)
Therefore there exists M I
and E,(N, ti) such that
>0
And similarly
for x E [T;, ‘,.I So from the Lipschitz continuity of cxj and to E l we know for x E [T,,T,’],
with respect
36 1
> 0.
for a constant M2
Similar to the proof of Lemma 2.4,
(3.32)
The same estimate is true for x E [T,, ~ $ 1 . For z < 71 and x > T$ we know ~ QE,,~ are exponentially that the approximate wave functions Q E , and small. Thus the L2 estimate (3.27) follows immediately. The derivative estimate (3.28) can be proved by a similar argument. Here, similar to (3.29), for < z < 72,
rt
h % ( N , h ) , m ( X ) - hQL,(N,h),rn(X)
= 0(hrn+')
which can be easily proved by straightforward calculations. For 71 < z < r:, recall (3.31) and (2.12), and the Lipschitz continuity of a j , ,Bj,a;, in E , similar to (3.32), we also have (3.33)
Lemma 3.4. Suppose that V E C" and bounded below, and liml,l+oo V ( x )= +m. Assume E is an exact eigenvalue of the Schrodinger operator H and 7 1 and 72 are the only two classical turning points, with V'(rj) # 0 , j = 1 , 2 . Then (3.7) is satisfied for any x E [7,f,72]which gives the exact eigenvalue E ( N , ti) in ascending order and the exact L 2 ( R ) wave function Q E i s approximated by the corresponding semi-classical approximate wave function Q E , in ~ L 2 ( R ) up t o m - t h order in h, i.e. ll*E
(3.34)
- QE,mllL2 = o ( h m f l ) ,
uniformly for E in a compact set of { E
: JV(,)<E
d-2 ( E - - V ( z ) )
Furthermore, if V is analytic, then the derivatives of IIhQllE:- hQ'E,mllL2
*E
and
< $00).
@ E , satisfy ~
= 0(hrn+l).
Proof. We have shown (3.7) holds for any z E
totics
@E
dz
(7-1,7-2).
(3.35) From the asymp-
in (3.1)-(3.6)in the different regions respectively, we have -
ll*E
- QE,mI($ =
(~E,o(z)~(h~+'))~dz
(3.36)
362
(9E , O
1:
( x ) o(
(Q E,O (z)0 (ti"f1 )>2dx.
72
Note that J?: (QE,O(x)0 (hmfl ) ) dx and JT' (Q E ,0 (x)0 (ti"+')) dx are exponentially small because of exponentially small integrand. Then using Lemma 2.4, (3.34) follows easily. To prove (3.35), calculate the derivative of Qk(z) in different regions respectively. For x 5 71, d hQ.lE(X) = hQb,o(x)P-(x) h @ E , o ( z ) z P - ( s ) .
+
But P - ( x , h) is analytic in z and fL for x 5 71 and h # 0 as it satisfies the differential equation (2.35) with analytic coefficients. Therefore we can differentiate its asymptotic expansion term by term (Wasow 24 and van der
-C 00
Corput
22),
i.e. & P - ( x )
q!((x)(-h)j, therefore
j=1
+ QE,O(Z)O(h"+2).
hQh(2)- hQ&,"(2) = (k9/,,,(z))O(ti"+l)
(3.37)
71 5 x 5 ~ f ,
For
+
hQ L(z) = Ti@ k, ( x ) B(x) f i 9 s , o (5)B'(X) Here similarly] B ' ( z ) fore,
- C,"=,
+fiQ z,o(x)A(X) +fLQL,,(x)A'( X) .
/3;(x)h2j,and A'(z) N
Cj"=, a i ( z ) h 2 j lthere-
fiQ'lE(2)- hQ&&(Z) = (h";,,(x))O(h'+')
+
+ h Q / , , o ( ~ ) o ( h ~ ~+~~, )o ( x ) o ( h ~ + ~ ) . ( 3 . 3 8 ) For
7-1'
< x < 72,
Here similarly]
363
and M
j=1
therefore
~Q&(x )F L Q ~ , ~ ( X=) b+(x)O(hm+') d 1
1 +(-(r) ,-))o(hm+2). (3.39) dX bT(x) bT(x)
+
72 5 x 6 T$,
For
similarly we have
hQ'lE(2)- hQ&,,(x) = ( h 2 Q ~ , o ( x ) ) o ( h m + ~ )
+hQ&,o(z)O (hm+' )
+ QE,O(x)O(h m f 2.()3.40)
And for x 2 r;,
+
~Q&(x )h\IIL,,(x) = h\IIL,o(x))0(hm+') Q E , O ( X ) O ( ~ ~ (3.41) +~). Then (3.35) follows from (3.37)-(3.41)and similar argument as (3.36). Here we use Lemma 2.4. 0
Remark. The quantity T
= JV(x)lE
d-2 ( E -'- V ( x ) )dx
in Lemma 3.4 i s the
classical travel time between turning points. If {x : V ( x )= E } consists of two simple zeros, then the classical travel time T is finite. The main result of this paper is the following result.
Theorem 3.5. Assume conditions of Lemma 9.1, then the exact N-th eigenvalue E ( N ,ti) of the Schrodinger operator H is approximated by the m-th order approximate N-th eigenvalue E m ( N ,h) which satisfies the mth order quantization condition in the sense that E ( N , h ) - E m ( N , h ) = O(hm+2),and the corresponding exact L 2 ( R )wave function Q E ( N , ~ )and its derivative T L Q & ( ~ , ~are ) approximated by the WKB-Langer semi-classical approximate wave function Q E , ( N , ~ L ) , ~associated with E m ( N ,h) and its derivative hQ",,(N,h),m in L2(R), i.e. l l Q E ( N , h ) - Q E m ( N , h ) , m l I L 2 ( R )= o ( h m f l ) i
(3.42)
and
I I% '(
N,h) -
hQkm( N , f i ),m 1 ILz(a) =
).
(3.43)
TZ(EO) In particular, set Eo(F) to be the solution of JTl(Eo) (2(Eo- V ( y ) ) i d y = 7rF for any given F > 0, then the exact eigenvalue E ( N , h ) has semi-
364
classical limit Eo(F) in the sense that
lim
E ( N ,h) = Eo(F), and has
h-0 N-CC
(N+$)h=F
asymptotic expansion Em up to m-th order in the sense that 1
lim h-0 N - m.~ .
(3.44)
-ti"( E ( N , ti) - E m ( N ,h ) ) = 0 ,
(N+ i)h=F
and the exact L2(R) eigenfunction 9 ~ ( ~ , has h ) the semi-classical asymptotic expansion I ; I I E ~ ( F ) , ~up to m-th order in the sense that
lim h-0
1
km IIQE(N, W
- QE,(N,fi),mIILz(R)
(3.45)
= 0,
N-CC
(N+$)h=F
Prooj By the triangle inequality IIQE(N,h) - qE,(N,h),mIIL2(R)
5
IIq'E(N,h)
-
9E(N,h),mllL2(R)
+I I Q E (N,FL) ,m - @ E ,
II
( N , h ),m L2(R)
and applying Lemma 3.4 and Lemma 3.3 we have (3.42). Similarly we have (3.43) by using
ll"L(N,/i)
-
hQLm(N,ti),mIIL2(R)
5
ll"L(N,h)
- "L(N,h),mIIL2(R)
+ll'Qk(N,h),m
-
hQL, ( N , h ),m I ILZ(R).
The rest of the theorem follows immediately.
0
We have the following simple, but interesting corollary.
Theorem 3.6. Suppose that V is analytic and bounded below, and liml,l+oo V ( x )= +oo. For any constant EO > minwl V ( x ) ,let 71 and 7 2 be TZ(EO) the only two classical turning points and define F ( E ~=) JT,(Eo) (~(EO-
+
V(y)) i d y . If V'(rj) # 0, j = 1 , 2 , and the following travel time inequal7 2 (Eo1 Zty O < J n ( ~ o ) , / w j d y < t o o holds, then as h + 0 , N -+ oo and
+
( N i ) h = F(Eo), the exact eigenvalue E ( N ,h) of the Schriidinger operator H = -$h2A V ( x ) has the semi-classical limit EO in the sense that E ( N , h ) Eo, as ti -+ 0 , N -+ 00 but ( N $)Ti = F(E0) and the
+
-+
+
365
exact L2(W) eigenfunction Qk(N,hlhas the semi-classical limit Qko,o, the WKB-Langer solution, in L 2 ( R ) , lim
IIQE(N,h) - QEo,011L2(R) = 0,
(3.47)
h-0 N-CC (N+ t r = ~( E ~ )
4
and
(3.48)
Acknowledgement
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13. F.W.J. Olver (1958), Uniform asymptotic expansions of solutions of linear second-order differential equations for large values of a parameter, Phil. Dans. R. SOC.Lond., Series A, Vol. 250,479-517. 14. F.W.J. Olver (1959), Linear differential equations of the second order with a large parameter, J . SOC.Indust. Appl. Math., Vol. 7, 306-310. 15. F.W.J. Olver (1974), Asymptotics and Special Functions, Academic Press, New York, London. 16. M. Reed and B. Simon (1975), Methods of modern mathematical physics 11: Fourier analysis, self-adjointness, Academic Press, New York. 17. B. Simon (1983), Semiclassical analysis of low lying eigenvalues, I, nondegenerate minima: asymptotic expansions, Ann. Inst. Henri Poincare, Vol. 38,295-307. 18. B. Simon (1984), Semiclassical analysis of low lying eigenvalues, 11. Tunnelling, Annals of Mathematics, Vol. 120,89-118. 19. M. Sirugue, M. Sirugue-Collin and A. Truman (1984), Semi-classical approximation and microcanonical ensemble, Annales De L Institut Henri PoincarePhysique Theorique, Vol. 41,429-444. 20. A. Truman and H.Z. Zhao (1996), Quantum mechanics of charged particles in random electromagnetic fields, J. Math. Phys., Vol. 37,3180-97. 21. A. Truman and H.Z. Zhao (2000), Semi-classical limit of wave functions, Proceedings of the American Mathematical Society, Vol. 128,1003-1009. 22. J. G. van der Corput (1956), Asymptotic developments I. Fundamental theorems of asymptotics, Journal D'analyse Mathkmatique, Vol. 4, 341-418. 23. N. Ya. Vilenkin et a1 (1972), Functional analysis, Wolters-Noordhoff Publishing, Croningen, The Netherlands. 24. W. Wasow (1965), Asymptotic expansions for ordinary differential equations, Interscience Publishers. 25. E.T. Whittaker and G.N. Watson (1915), A course of modern analysis, Cambridge University Press, Cambridge, Second Edition.
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