Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions: Lectures Given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.)Held ... 1, 1998 (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris

Subseries: Fondazione C. I. M. E., Firenze Adviser: Roberto Conti

1715

Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo

N.V. Krylov M. Rtckner J. Zabczyk

Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Cetraro, Italy, August 24 - September 1, 1998 Editor: G. Da Prato

Fonda. ione

C.I.M.E.

Springer

Authors

Editor

Nikolai A. Krylov Department of Mathematics, Computer Science & Statistics University of Illinois at Chicago M/C 249, 851 Morgan Street Chicago, IL 60607, USA

Giueppe Da Prato Scuola Normale Superiore Piazza Cavalieri, 7 56126 Pisa, Italy

Jerzy Zabczyk lnstytut Matematyczny Polskiej Akademii Nauk ul. Sniadeckich 8 00-950 Warszawa, Poland Michael R6ckner Department of Mathematics University of Bielefeld Universit~itsstrasse 25 33615 Bielefeld, Germany Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufllahme

Stochastic PDE's and Kolmogorov equations in infinite dimensions : h e l d i n C e t r a r o , Italy, A u g u s t 2 4 - S e p t e m b e r 1, 1 9 9 8 / N. V. K r y l o v ... E d . : G . D a P r a t o . - B e r l i n ; H e i d e l b e r g ; N e w Y o r k ; Barcelona ; Hong Kong ; London ; Milan ; Paris ; Singapore ; Tokyo : Springer, 1999 ( I x c t u r e s g i v e n at the ... s e s s i o n o f the C e n t r o I n t e r n a z i o n a l e M a t e m a t i c o E s t i v o ( C I M E ) ... ; 1 9 9 8 , 2 ) ( L e c t u r e n o t e s in m a t h e m a t i c s ; Vol. 1715 : S u b s e r i e s : F o n d a z i o n e C I M E ) ISBN 3-540-66545-5

Mathematics Subject Classification (1991): 60H10, 60H15, 60G15, 31C25, 60J60 ISSN 0075-8434 ISBN 3-540-66545-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. O Springer-Verlag Berlin Heidelberg 1999 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors/editors SPIN: 10700262 41/3143-543210 - Printed on acid-free paper

Preface

Kolmogorov equations are second order parabolic equations with a finite or an infinite number of variables. They are deeply connected with stochastic differential equations in finite or infinite dimensional spaces. They arise in many fields as Mathematical Physics, Chemistry and Mathematical Finance. These equations can be studied both by probabilistic and by analytic methods, using such tools as Gaussian measures, Dirichlet Forms, and stochastic calculus. Three courses, of eight hours each, have been delivered. N. V. Krylov of the University of di Minnesota presented Kolmogorov equations coming from finite-dimensional equations, giving existence, uniqueness and regularity results. M. RSckner has presented an approach to Kolmogorov equations in infinite dimensions which is based on an LP-analysis of the corresponding diffusion operators with respect to suitably chosen measures, extending the classical Dirichlet form approach. Finally, J. Zabczyk, started from classical results of L. Gross, on the heat equation in infinite dimension, and discussed some recent results, including applications to control theory and mathematical finance. Afternoon sessions have been devoted to research seminars delivered by the participants. I wish to thank the lecturers and all the participants for their contribution to the success of the School. I thank the CIME scientific committee for giving me the opportunity to organize the Meeting and the CIME staff for their continuous help. Scuola Normale Superiore (Pisa) March 1999

Giuseppe Da Prato

Table of C o n t e n t s

Preface ....................................................... O n K o l m o g o r o v ' s e q u a t i o n s for f i n i t e d i m e n s i o n a l N.V. K r y l o v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V diffusions

1. Solvability of It&'s stochastic e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . 2. M a r k o v p r o p e r t y of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 R e g u l a r e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some p r o p e r t i e s of E u l e r ' s a p p r o x i m a t i o n s . . . . . . . . . . . . . . . . . . 2.3 M a r k o v p r o p e r t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o n d i t i o n a l version of K o l m o g o r o v ' s e q u a t i o n . . . . . . . . . . . . . . . . . . . D i f f e r e n t i a b i l i t y of solutions of stochastic e q u a t i o n s with respect to 4. initial data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 E s t i m a t i n g m o m e n t s of solutions of It5's e q u a t i o n s . . . . . . . . . . . 4.2 S m o o t h n e s s of solutions d e p e n d i n g on a p a r a m e t e r . . . . . . . . . . . 4.3 E s t i m a t i n g m o m e n t s of derivatives of solutions . . . . . . . . . . . . . . . 4.4 T h e n o t i o n s of L - c o n t i n u i t y a n d L - d i f f e r e n t i a b i l i t y . . . . . . . . . . . 4.5 Differentiability of c e r t a i n e x p e c t a t i o n s d e p e n d i n g on a p a r a m e t e r 5. K o l m o g o r o v ' s e q u a t i o n in the whole space . . . . . . . . . . . . . . . . . . . . . . . 5.1 Stratified e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Sufficient c o n d i t i o n s for r e g u l a r i t y . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 K o l m o g o r o v ' s e q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Some i n t e g r a l a p p r o x i m a t i o n s of differential operators . . . . . . . . . . . . 7. K o l m o g o r o v ' s e q u a t i o n s in d o m a i n s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 22 26 31 33 36 42 43 46 48 53 58

LP-analysis of finite and infinite dimensional diffusion operators Michael RSckner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. S o l u t i o n of K o l m o g o r o v e q u a t i o n s via sectorial forms . . . . . . . . . . . . . 2.1 P r e l i m i n a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Sectorial forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Sectorial forms on L 2(E; m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 E x a m p l e s a n d A p p l i c a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 66 66 68 70 72

.

1 8 8 11 15 16

Table of Contents 3.

4.

5.

6.

7.

Symmetrizing measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 T h e classical finite d i m e n s i o n a l case . . . . . . . . . . . . . . . . . . . . . . . 3.2 R e p r e s e n t a t i o n of s y m m e t r i c diffusion operators . . . . . . . . . . . . . 3.3 O r n s t e i n - U h l e n b e c k t y p e operators . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 O p e r a t o r s with n o n - l i n e a r drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . N o n - s e c t o r i a l cases: p e r t u r b a t i o n s by divergence free vector f i e l d s . . 4.1 Diffusion o p e r a t o r s on LP(E; m) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 S o l u t i o n of K o l m o g o r o v e q u a t i o n s on L 1 (E; m) . . . . . . . . . . . . . . . 4.3 U n i q u e n e s s p r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 C o n c l u d i n g r e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I n v a r i a n t measures: regularity, existence a n d u n i q u e n e s s . . . . . . . . . . 5.1 Sectorial case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 N o n - s e c t o r i a l cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C o r r e s p o n d i n g diffusions a n d r e l a t i o n to M a r t i n g a l e p r o b l e m s . . . . . 6.1 E x i s t e n c e of associated diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 S o l u t i o n of t h e m a r t i n g a l e p r o b l e m . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 U n i q u e n e s s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix ................................................... 7.1 K o l m o g o r o v e q u a t i o n s in L 2 (E; #) for infinite d i m e n s i o n a l m a n ifolds E : a case s t u d y from c o n t i n u u m statistical mechanics . . . 7.2 E r g o d i c i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII 78 78 80 81 83 86 86 88 92 95 96 96 99 103 103 105 105 106 106 110

Parabolic equations on Hilbert spaces J. Z a b c z y k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

1. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. P r e l i m i n a r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 L i n e a r o p e r a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Measures a n d r a n d o m variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 W i e n e r process a n d stochastic e q u a t i o n s . . . . . . . . . . . . . . . . . . . . 3. Heat E q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 R e g u l a r i n i t i a l f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Gross L a p l a c i a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Heat e q u a t i o n with general i n i t i a l f u n c t i o n s . . . . . . . . . . . . . . . . . 3.5 G e n e r a t o r s of the heat semigroups . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 N o n p a r a b o l i c i t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. T r a n s i t i o n s e m i g r o u p s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 T r a n s i t i o n s e m i g r o u p s in the space of c o n t i n u o u s f u n c t i o n s . . . . 4.2 T r a n s i t i o n semigroups in spaces of square s u m m a b l e f u n c t i o n s . 5. Heat e q u a t i o n with a first order t e r m . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 R e g u l a r i n i t i a l f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 G e n e r a l i n i t i a l f u n c t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 R a n g e c o n d i t i o n a n d examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

117 119 119 123 127 130 131 135 137 139 143 147 149 150 154 157 158 159 163 170

VIII 6.

7.

8.

9.

10.

Table of Contents

G e n e r a l p a r a b o l i c equations. R e g u l a r i t y . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 C o n v o l u t i o n t y p e a n d e v a l u a t i o n m a p s . . . . . . . . . . . . . . . . . . . . . . 6.2 Solutions of stochastic e q u a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Space a n d t i m e r e g u l a r i t y of generalized solutions . . . . . . . . . . . . 6.4 S t r o n g Feller p r o p e r t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G e n e r a l p a r a b o l i c equations. U n i q u e n e s s . . . . . . . . . . . . . . . . . . . . . . . . 7.1 U n i q u e n e s s for the heat e q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 U n i q u e n e s s in the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P a r a b o l i c e q u a t i o n s in open sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 M a i n t h e o r e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 E s t i m a t e s of the exit p r o b a b i l i t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . Applications ................................................. 9.1 H J B e q u a t i o n of stochastic control . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Solvability of H J B e q u a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 K o l m o g o r o v ' s e q u a t i o n in m a t h e m a t i c a l finance . . . . . . . . . . . . . . Appendix ................................................... 10.1 I m p l i c i t f u n c t i o n t h e o r e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174 174 178 179 181 186 186 187 191 191 192 195 198 198 202 204 206 206

On Kolmogorov's equations for finite dimensional diffusions N.V. Krylov School of Mathematics, University of Minnesota, Minneapolis, MN, 55455 The purpose of these lectures is to present a more or less self contained exposition of Kolmogorov's equations for finite dimensional diffusion processes which are treated as solutions of It6's equations. Kolmogorov's equations provide probabilistic solutions for elliptic and parabolic second order partial differential equations. On the one hand, this allows one to prove the solvability of such equations and, on the other hand, quite often one can get a substantial information about probabilistic quantities by solving these equations. We start with proving the solvability of It6's stochastic equations with r a n d o m coefficients. This is done by using Euler's method, which turns out to be very powerful in m a n y situations. Then we mainly concentrate on equations with nonrandom coefficients and prove the Markov property of solutions. Next step is deriving Kolmogorov's equations under the assumption that the probabilistic solutions are smooth functions of initial data. After this we give sufficient conditions for the smoothness in the case of equations in the whole space. Final sections are devoted to Kolmogorov's equations in domains when the probabilistic solutions are not necessarily continuous let alone differentiable. In this case Kolmogorov's equations are understood in the sense of generalized functions. All material of these lectures, perhaps apart from the last sections, is classical and can be found in m a n y articles and books. For this reason we only cite the original article [KO] by Kolmogorov himself which opened up the new chapter in probability theory we will be talking about. It is a great pleasure to thank Director of CIME R. Coati and Scientific Director of the CIME Session on Stochastic P D E ' s and Kolmogorov Equations in Infinite Dimensions G. Da Prato for inviting me to deliver these lectures and for excellent organization of the session in such an exquisite place as Cetraro (Cosenza).

1. S o l v a b i l i t y

of It6's

stochastic

equations

Let ( Y / , S , P ) be a complete probability space and let (w~,~-t) be a dldimensional Wiener process on this space defined for t E [0, c~), with c~algebras ~t being complete with respect to F , P. Assume that, for any w ~ ~2, 1991 Mathematics Subject Classification. 58G32,60J60,60J65 Key words and phrases. Stochastic equations, Kolmogorov's equations The work was partially supported by NSF Grant DMS-9625483

2

N. Krylov

t > 0, and x E ~ , we are given a d • d:-dimensional m a t r i x a(t, x) and a d-vector b(t, x). We assume t h a t ~ and b are continuous in x for any w, t, measurable in (w, t) for any x, and Jrt-measurable in • for any t and x. We also assume that, for any finite T and R and w E ~ , we have T

f0

s u p {I[a(t' x)II2 + [b(t'x)l}dt < txl_
co.

(1.1)

Notice that, for a m a t r i x a, by ]Jail 2 we mean ~ i j ]
Remark 1.1. It is shown in [KR1] that, under the monotonicity condition (1.2) and the coercivity condition (1.3) below, condition (1.1) is satisfied if

~o

T{l[a(t,x)lt 2 + tb(t,x)]} dt < ~

Vx C ~d.

T h e o r e m 1.2. In addition to the above assumptions, for all R, t E [0, ~ ) , IxI, [Yl <- R, and w, let

2(x-y,b(t,x)-b(t,y))+ila(t,x

) - a ( t , y ) ] l 2 < K t ( R ) ] x - y I 2,

2(x,b(t,x)) + ]I~(t,x)lI ~ _< Kt(1)(1 + Ix[2),

(1.2) (1.3)

where K t ( R ) are certain .Tt-adapted nonnegative processes satisfying p

T

aT(R) : : Jo Kt(R) dt < for all R, T E [0, co), and w. Also let Xo be an Y:o-measurable d-dimensional vector. Then It6's equation dxt = a(t, xt) dwt + b(t, xt) dt,

t >_ 0

(1.4)

with initial condition Xo has a solution, which is, moreover, unique up to indistinguishability. Remark 1.3. It follows from inequalities like [b(t, x)l <_ ]b(t, 0)] + ]b(t, x) b(t, 0)[ t h a t assumptions (1.1), (1.2), and (1.3) are satisfied if T

fo

[]la(t'0)II2 + ]b(t,0)]] dt < :xD

and the Lipschitz condition []a(t,x) - a(t,y)]] + ]b(t,x) - b(t,y)l < KIx - y[ holds for all t E [0, T] and x, y C I~d, where K is a constant. However (1.2) and (1.3) are also satisfied if, for instance, d = 1, : - 0, and b(t, x) is any decreasing function of x equal zero at x = 0. Before proving the t h e o r e m we deduce two lemmas.

Kolmogorov's equations

3

L e m m a 1.4. Let Yt be a continuous nonnegative ~ - a d a p t e d process, ~/ a finite stopping time, N a constant, and assume EyT <_ N ]or any stopping time ~- <_ 7. Then P { s u p Yt >_ ~} ~ N / ~ for any ~ > O.

For the proof, it suffices to take ~- = ~/A inf{t >_ 0 : Yt ?_ ~}, to notice t h a t 7- < ~, and {w: supyt ~ c} = {w: y~ >_ E}, t_<~ and to use Chebyshev's inequality. L e m m a 1.5. Suppose that, for n = 1,2,..., we are given continuous ddimensional ~t-adapted processes x~ on ~2 • [0, co) such that x'~ = xo and dx~=a(t,x~

+p~)dwt+b(t,x~

+p~)dt

t>_O

for some .Tt-adapted processes p~ which are measurable in (w,t). For n = 1,2, ... and R >_ O, let ~'n(R) be stopping times such that (i) Ix~l + Ip~l <- R if O < t <_ T~(R), (ii) fTA~(R) ~imE]o Ip'}ldt=O VR, T E [ 0 , ~ ) , (1.5) (iii) there exists a nonrandom function r ( R ) such that r ( R ) --+ oc as R ~ co and

lim lim P{~-'~(R) _< T, sup Ix~l ~ r ( R ) } = 0 R --+c~ n --+c~ t <~r~ (R )

VT E [0, co).

(1.6)

Then, for any T E [0, (x~), we have

sup Ix~ t
-

-

x~nl P 0

(1.7)

as n, rn--~ cx~.

Proof. W i t h o u t loss of generality (see (1.1)), we assume t h a t Ib(t, x)l <_ K t ( R ) for Ixl _< R. Furthermore, it is easy to find stopping times T(R) such P t h a t T(R) -+ oo as R -+ oo and atAT(R)(R) is a bounded process for any R > 0. For instance, it suffices to take T ( R , u ) := inf{t _ 0 a t ( R ) > u}, to observe that, obviously, P{~-(R, u) < R} -+ 0 as u -+ oe, to find u ( R ) so that

u(R)) < R} _< l/R,

4

N. Krylov

and to put T ( R ) = T ( R , u ( R ) ) . For such T(R), the transition from T n ( R ) to T"(R) A T(R) preserves all the assumptions of the l e m m a and, therefore, without loss of generality, we may and will assume that a t ( R ) < u ( R ) for t < Tn(R), where u ( R ) are some finite constants. Observe t h a t now (,TAT~ (R)

lim E I n--~ J0

Ip~lKt(R) dt=O

VR, TE

[0, oo).

(1.8)

Indeed, to see this, we merely split the integrand in (1.8) into a sum using 1 = I K , ( R ) k i + IK~(R)
X n~,

y~ = x m~ ,

n p~ = p~,

q~ = p~m ,

r

= r

= exp(-2~(R)-ix0I).

According to It6's formula,

+ Ilk(t, x~ + p~) - ~ ( t , y~ + q~)ll 2 - 2 K ~ ( m I ~ - Y~ 12] at + dZ~,

where ~t = /3~~m is certain local martingale with 130 = 0. To transform the right-hand side, we represent xt - Yt as (xt + Pt) - (Yt + qt) - Pt + qt. Then we use the monotonicity condition (1.2) and the inequalities 2txt - yt[ 2 > I(xt + Pt) - (Yt + qt)l 2 - 2Ipt - qt] 2,

Ipt - qtl 2 < 2R(IptI + lqtl),

the last of which holds at least for 0 < t < 7 n ~ ( R ) := r~(R)AT'~(R). Finally, we also use ]b(t, x)] < K t ( R ) for Ixl < R and find that

I~

-

x~l

r

_< ~ ( R )

+ ~F(R)

+ Z~"~ (R)

t _< ~ ( R ) ,

(1.9)

where

Ip~iK~(R) ds.

a~(R) := 4 ( R + 1)

From (1.9) it follows at once that, for any R, T E [0, oo) and any stopping time ~- _< T A 7~'~(R),

ml~ - ~V~I=~,(R) _< Sa~A~.(~)(m + EA[,~A,-,(/{)(R).

(1.10)

Owing to (1.8),the right-hand side in (1.10) tends to zero, and since T is arbitrary, by L e m m a 1.4 we get sup

[x~ _

m

2

Kolmogorov's equations

5

as n , m -+ co. The factor CA(R) can be dropped since it is positive and independent of n , m . We now see that, to prove (1.7), it only remains to show that lim lim P{~-'~(R) < T} = 0 VT 9 [0, oc). (1.11) R-+oo

n - + o4~

So far we have not used the growth condition (1.3). By using this condition and assuming K t ( R ) >__Kt(1) for R ___ 1, which does not restrict generality, similarly to (1.9) we get that, for t <_ T~(R),

___ (1 + Ix012)e -'xol + a~(R) +Z~,

(1 + Ix;~12)r

where/3t is a local martingale starting at zero. Then similarly to (1.10) we obtain that, for any stopping time ~- < T A rn(R),

ElXnTI2~)T(1) ~ S -~- E/~ATn(R)(R), where N is the least upper bound of (1 + ]x]2)e -Ix]. From the above stated properties of A~(R) and from Lemma 1.4 we further obtain that lim sup lim P { c--+oo R~0

n--+oo

sup

Ix~n I2r

_> c} = 0.

t
Again since CA(l) is independent of n and R, this term can be dropped. It follows that, as R -+ oo, lim P {

n--+~z

sup

t<_~-~(R)

Ix[~]_>r(R),T~(R) < _ T } = O .

Combining this with the assumption (1.6) we get (1.11). The lemma is proved.

Remark 1.6. In the above proof we did not use the continuity of cr and b in X.

P r o o f o f T h e o r e m 1.2. We are going to use Euler's method. Define the processes x~ so that x~ = x0 and

dx~~ = a(t, x;/n) dwt + b(t, x~/~) dt

for t 9 [k/~, (~ + 1)/~]. Clearly ~' satisfy dx~~ = a(t, x~(,,,t)) dwt + b(t, x~(,,,t)) dr,

(1.12)

d ~ = ~(t, ~ + p~) d ~ + b(t, ~ + p~) dr, where ~(n, t) = [tn]/n, p[~ := x'~(,~,t) - x An. Also define T'~(R) as the first exit time of x~ from the ball BR/3 = {x 9 I~d :Ix[ < R/3}, and let r(R) := R/4. Then, for 0 < t _< T ' ( R ) , we have Ip[~l _< 2R/3, ]x~l <_ R/3.In particular, the event entering (1.6) is empty and this condition is satisfied. Moreover,

-p[~ =

f

f~

(~,t)

a(s,x~(n,t))dws +

f

7~

(n,t)

b(s,x~(n,t))ds.

6

N. Krylov

Hence according to well known inequalities for stochastic integrals, for any ~, ~ > 0, we have P{[p~[ > 2e, t < Tn(R)} < P{

f

t

sup [b(s,x)]ds > ~}

J ,~(,~,t) I=l<_n

ft

sup ilo(s,~)ll2ds>~}+5/c =. (1.13)

-F P{Ja(n,t) txl<_R

In view of (1.1), this shows that the left-hand side goes to zero as n --+ oo. In other words, the product of ]p~] and the indicator of {t < T~(R)} tends to zero in probability. Since this product is less than 2R/3, its expectation is bounded in t, n and tends to zero as n --+ oo. An application of ~ b i n i ' s theorem and the dominated convergence theorem proves t h a t (1.5) holds. Now by L e m m a 1.5 and the completeness of the space of processes with respect to the uniform convergence in probability, we get that there exists a continuous process xt such that suplx~'-xtl-~0

V T e [ 0 , oo).

t
Here one m a y replace t in x~ and xt with n(n,t). Moreover, one can do this only for x~ since x~ is continuous in t. Then, using the continuity of a and b in x and passing to the limit in (1.12), we see that xt is indeed a solution of equation (1.4) with the initial condition x0. Its uniqueness and even continuous dependence on the initial data we prove in the following theorem. The theorem is proved. 1.7. Let the assumptions of Theorem 1.2 apart from assumption (1.3) be satisfied. Let xo,x~ be ~d-valued J:o-measurable vectors such that P - l i m n ~ o o x~ = xo. Assume that equation (1.4) has solutions xt and x~ with initial data Xo and x'~, respectively. Then

Theorem

P - lim suplx~' - x~l

0

=

VT 9 [ 0 , ~ ) .

(1.14)

n--+o~ t<_T

Proof. Without loss of generality, we may and will assume that x~' --+ x0 almost surely. Then ~t(R) := e x p ( - a ~ ( R ) - sup lx~t) n

satisfies Ct > 0. Next, for 7~(R) = inf{t _ 0: ]x~l + [x~ I _> R ) A T as in the proof of L e m m a 1.5 we find that

lXtA.r~(n)~

-- XtA'r~(R)12r

< lx ~ _ xol2r

+ m~'(R),

Kolmogorov's equations

7

where m~(R) is a local martingale starting at zero. Since [x~ - xo[ -+ 0 in probability, we have E]x~ - xo]2r -+ 0, and L e m m a 1.4 implies t h a t P - lim sup x~ . . . . .

O,

-XtA~,,(R)[2exp(--r

P - lim sup IX~A~R~ -- XtA~n(R)] = 0

(1.15)

for any finite T, R. Add to this t h a t P{V'~(R) < T} < P{sup(]xtnA~(R)] + [XtA~(R)[) >_ R} t
< P { s u p ]xtnA.r~(R) -- XtA~(R)] >_ 1} + P { 2 s u p [xt] > R - 1}. t
t
T h e n from (1.15) we get lim

lim P { 7 ~ ( R ) < T} = 0,

R--+~ n-+~

which along with (1.15) implies (1.14). T h e theorem is proved. Above we have only used L e m m a 1.5 when the probability in (1.6) is zero. Later on we will need the following result in the proof of which the verification of (1.6) is not t h a t trivial. T h e o r e m 1.8. Let the assumptions of Theorem 1.2 be satisfied. Let c~n(t, x) and bn(t, x) be some functions satisfying the conditions imposed on ~ and b before Remark 1.I and let processes y~ be such that y~ -+ xo in probability and (1.16) dYt = a n '[t , Y n' t ) d w t + b~ (t, y ~t ) d t t > O .

Define Vn(R) -- inf{t > 0 : ]y~] ~ R}

and assume that, for all R, T E [0, co), we have oTA'rn(R) [][~ _ ~[]2 + ]bn _ b[] (t, y~) dt --~ O.

(1.17)

Then y~ --+ xt uniformly on [0, T] in probability for any T E [0, cx~), where xt is the process from Theorem 1.2. Proof. Let xt2'~ = y~ - Pt2 n , where

p~ =

/0

(~'~-a)(s,y2)dw, +

/0

(b~-b)(s, y 2 ) d s - y g + zo,

~2n+l and let xt2n+1 -- xt, Pt = 0. Also let vn(R) be the first exit time of ]x~l + [P~I from ( - 1 , R) and r(R) = R/2. Clearly T2n(R) < vn(R) and [P~I, [Y~I <- R for 0 < t < T2~(R). From this and (1.17) similarly to (1.13), we obtain that, as n --+ ~ ,

8

N. Krylov

# n ( T , R ) := sup Ip'~lIo
This certainly yields (1.5) and since the probability in (1.6) is less t h a n

P { # ~ ( T , R) >_ R / 2 } , condition (1.6) is fulfilled as well. Other conditions of L e m m a 1.5 are also satisfied and we conclude t h a t x~ n = y~ - p~n __+ xt uniformly on [0, T] in probability for any T C [0, oo). It only remains to notice that, for R, ~ > 0, we have P { s u p Pc2n >_ e} < P { p Z n ( T , R ) >_ e} + P{r2'~(R) < T } t
_< 2P{p2'~(T, R) _> g} + P { s u p Ixfnl _> R - g}, t
which after letting first n --+ oo and then R --+ ec implies t h a t pt2.~ -+ 0 uniformly on [0, T] in probability for any T E [0, co). T h e t h e o r e m is proved. E x e r c i s e 1.9. Prove t h a t for d = 1, the equation xt = - f2 xa ds + wt has a unique solution defined for all times. E x e r c i s e 1.10. Prove t h a t for d = 1, the equation xt = f2 x~ ds + wt has a unique solution defined before it blows up to cx~ or - o o .

2. M a r k o v

property

of solutions

We saw t h a t under some conditions equation (1.4) with r a n d o m coefficients can be solved by using Euler's method. It turns out t h a t it is useful to forrealize this p r o p e r t y of equations. In this section we assume t h a t the coefficients a and b of our equations are n o n r a n d o m functions satisfying the conditions stated before R e m a r k 1.1. 2.1 R e g u l a r

equations

We will be dealing with the equation

x

=x+

/0

/0 b(s+r,x

)dr t_>0,

(2.1)

where s and x are some n o n r a n d o m data. To introduce general Euler's approximations, let us call I = {t(k)}~__ o a partition of [0, oo) if t(0) = 0, t(k) T oo and A ( I ) : = supk>0 (t(k + 1) - t(k)) < oo. For every partition I = {t(k)} of [0, oo) we define Euler's approximation 8~X for (2.1) recurrently as a process xt, denoted by x t (I), such t h a t x0 = x,

xt = x t ( k ) +

(k)

c~(s + r, xt(k))dw~ +

(k)

b(s + r, xttk))dr

(2.2)

Kolmogorov's equations

9

if t E [t(k),t(k + 1)], k = 0, 1 , 2 , . . . . It is useful to note t h a t x~#(I) is well defined under the above hypotheses and satisfies the equation

fo'

xt = x +

cr(s + r,x~(~))dwr +

fo

b(s + r,x~(r))dr,

(2.3)

where ~(r) = t(k) for r E [t(k),t(k + 1)), k = 0 , 1 , 2 , . . . . Denote C = C([0, oo), IRd) the space of all continuous I~d-valued functions with metric

Ix. - Y.Ic = ~

2-'~ arctan sup Ixt - Ytl t
rt=l

under which C is a complete separable metric space. Observe t h a t xP -+ x. in C if and only if suPt 0 and x EII{ d equation (2.1) has a unique solution on [0, T] for every T E (0, oo). Denote this solution by xt #. Moreover, suppose t h a t for every R E [0, co) as A ( I ) --+ 0, we have x~ ' x ( I ) --+ xs 'x (convergence in C) in probability, uniformly with respect to s E [0, T] and x E BR, i.e. for every R , T E [0, cxD), and : > 0 sup sup P { sup Ix~ '~ - x t # ( I ) l sE[0,T] I~I
> :} --+ 0.

(2.4)

T h e n we say t h a t equation (2.1) and its solution xt '~ are regular. Sufficient conditions for the regularity are given in Sec. 5.2. Regular processes have m a n y nice properties. In this subsection we prove only one of them. To save some space we will put indices s, x at the symbols of probability and e x p e c t a t i o n m e a n i n g t h a t they should be put inside at a p p r o p r i a t e places. For instance (2.4) will now read sup sup Ps,z{ sup Izt sE[O,T] Izl_
-

xt(I)l

>_ :}--+

O.

2.2. Suppose that equation (2.1) is regular. Then for every T, R > O, the processes x~ 'x and x~ '~ (I) are uniformly in t E [0, T] bounded in probability, uniformly with respect to s E [0, T], Ix] < R, and partition I, i.e.

Lemma

lim

sup

sup s u p P s # {

c~oo sE[0,T] I~I_
lim

sup

sup

sup Ps,~{ sup

c--+oo sE[0,T] I*I
Ix~(/)l

c} = o,

(2.5)

tE[0,T]

tE[0,T]

Ixel >_ c} = O.

(2.6)

10

N. Krylov

Proof. Take a partition I = {t(k)} and use (2.2). Then for any increasing sequence of constants ck >_ 0 and xt := xt '~(I) by Doob's inequality we get

[xt]2Ilx,(,)l<_c~<_

sup

E

[t(k),t(k+l)]

_< 3c~ + 3E

sup

[

[t(k),t(k+l)]

+3E(

f

t(k+l)

Jr(k)

f

Itxt(k)]<_ckO'(S+ r, Xt(k) ) dwr[ ~ (k)

Ib(s + r,x~(k))l dr) 2 Ii~(k)l_
ft(k+l)

< 3c~+12E /

Ila(s+r, xt(k))ll2 drZl~,(k)l
Jr(k)

<_3c~ +12jt(k )

I~l_<~ksup [Icr(s+r,x)ll2 dr + 3(jt(k)

t(k)

I~l<~

Ixl<_ckSUpIb(s+r,x)]dr) 2

=: ~(ck,s + t(k),s + t(k + 1)) <_((ck,O,T + t(k + 1)) =: ~(ck,t(k + 1)). (2.7) By assumption (1.1) the function ~(ck, t(k + 1)) is finite, and by Chebyshev's inequality P{

sup

IXtI_>Ck+l}_<

t<_t(k+l)

< P { sup ]xtl >_ck}+P{ sup [xt[2 II~(k)l<_ck >_ck+l} t__Ck}+Ck:l~(Ck,t(k+ 1)). t_
(2.s)

By induction we conclude that for every T, R, 5 > 0 P~,x{sup [xt(I)[ > c} --+ 0

(2.9)

t
as c -+ oe uniformly with respect to s _< T, Ix] _< R, and I such that A(I) >_5. Furthermore, condition (2.4) implies that for every V > 0 we can find 5(7) > 0 such that if A(I) <_5(7), then Ps,x{sup [xt - xt(I)[ > 1} < 7

(2.10)

t
for all s < T, ]x[ < R. This fact and the fact that (2.9) holds uniformly with respect to s _< T, [x I _< R for any I immediately prove (2.6). We also see from (2.9) that the supi in (2.5) can be replaced by sup over {I : A(I) <_ 5(7)} without changing the left-hand side of (2.5). By (2.10) and (2.6) this side of (2.5) is less than V- Since V is arbitrary, the lemma is proved.

Kolmogorov's equations

11

2.2 S o m e p r o p e r t i e s o f E u l e r ' s a p p r o x i m a t i o n s The first purpose of this subsection is to investigate the continuity of Euler's approximations xt'X(I) with respect to x for any fixed partition I. This is an easy task since Euler's approximations are defined by induction and we get the continuity on [t(k),t(k + 1)] from the explicit formula for x~'~(I) expressing it as a function of x t(k)(I) s'~ 9 We are also dealing with conditional expectations of g(x ts,~ (I)) given "~'t(k)" The results are intuitively obvious and just amount to fixing all which depends on w[0,t(k)] and then averaging. T h r o u g h o u t this subsection we fix a partition I = {t(k)} of [0, co) and do not assume that the process is regular. 2.3. The C-valued function x~ 'x (I) is continuous in probability with respect to (s, x).

Lemma

Proof. For the sake of simplicity of notation, for s , r E [0, ec), x , y E N d, denote

xt := x ts'x (I~ ~ j,

Yt : = x [ 'y(I).

In the same way as in the proof of L e m m a 2.2 from the definition of Euler's approximation for any increasing sequence ek > 0 and R _> 0 we get P{

ytl > Ek+l} < P{ sup Ixt - ytl > ~k}

sup Ix~ t<_t(k+l)

t<_t(k)

+ P { sup Ixt

y~l ~ ~k, sup (Ix~l + ly~l) ~ R}

-

t<_t(k)

-2 [ 3 c ~ + 1 2 -1- gk+l

+3(

t<_t(k)

ft(k+i) ] lul,lvl<_R Jt(k) I~-vl_<~k sup

sup f t t(k+l) lul,lvl_
IIc~(s+p'u)-~(r+p'v)ll2dp

(2.11)

Ib(s + p , u ) - b(r + p,v)l dr)2].

I~,-vl_
t(k+l)

F(s,r,u,v) :=

fJr(k)

IIG(s + p,u) - G ( r + p,v)ll2 dP

is a continuous function of ( s , r , u , v ) . To prove this, let (sn,r,~,un, vn) ( s , r , u , v ) . Then, bearing in mind the inequality la 2 - b2[ < ([a I + [bD[a - bl, Hhlder's inequality, and condition (1.1), we get t h a t

12

N. Krylov

- F(s,r,u,v)l 2

lim


ft(k+1) lim I II~(r,~+p,v~)-a(r+p,v)ll2@:=Nl~+N4, n~oc Jr(k) _

+N

t(k+l) / Ila(sn+p,u~)-cr(s+p,u)[12dp n--+c~Jt(k) lim

_

where, for instance,

ft(k+l) ltc~(s,~+p,u~)-a(s,~+p,u)ll2dp

/ 1 _ < 2 lim /

n--+oo Jt(k) +2

ft(k+~) I Ilo(s Jr(k)

'~

+p,u)-

(s+p,u)ll

dp.

(2.12)

T h e first limit in (2.12) is zero because of continuity of cr in x and because it is less t h a n

l~m f T ][a(P, Un) _ c r ( P , U ) ] [ 2 dp, n--+oo gO where T is an a p p r o p r i a t e constant. The second limit in (2.12) is zero because of L2-eontinuity of square integrable functions. In the same way, 12 = 0, so t h a t F is continuous indeed. Similarly one treats the second integral in (2.11). Consequently, if if we suppose t h a t for a k and every ek > 0 Jk(ek):=!imP{

s--+r

sup ] x t - y t l > c k } = O ,

t
(2.13)

then passing to the limit in (2.11) first as (s, x) -+ (r, y) and ck $ 0 and finally letting R -+ oc and using L e m m a 2.2, we conclude t h a t J k + l ( e k + l ) = 0 for any ek+l > 0. For k = 0 we have (2.13) since x0 = x and Y0 = Y, and this proves the lemma. C o r o l l a r y 2.4. If g = g(x) is a bounded continuous function on N d, then E~,~g(xt(I)) is a continuous function of (s, x, t) for any partition I. Further we use the o p e r a t o r Qs,t defined for 0 < s < t on Borel functions

g ( x ) , x E IRd, by the formula

Os, g(x) -- Eg( L), where ~ , t is an R a - v a l u e d n o r m a l l y distributed vector with parameters

(x+

fsL

dr,2

let s

dr),

1 *

Kolmogorov's equations

13

Remark 2.5. If g is a b o u n d e d continuous function, then obviously Q,,tg(x) is well defined for 0 < s < t, x E IRd and is continuous in (s, t, x). It follows by s t a n d a r d a r g u m e n t s t h a t Q~,tg(x) is well defined and Borel in (s, t, x) if g is Borel, positive or bounded. L e m m a 2.6. Let g be a Borel, positive or bounded function on Nd. Then for every s _> 0, ~ c n~d, t(n) _< t _< t(n + 1) we have ,

Eg%(,~),t)lz=~;i:

)(~)

8~2~ = O~+~(~/,~+~g(x~(~/(I))

(a.s.),

(2.14)

where ~t(n),t

= Z -[-

(n)

~(S + r, z) dw~ +

(n)

b(s + r, z) dr.

Proof. For simplicity let

xt = x~ '~(I),

S~X

y = xt(,, )(I),

and for p E ]R, and m = 1, 2, ... let am(p) = 2--r~[2"~p] and for z C R d let

~m(z) = ( ~ ( z ~ ) , . . . , ~ ( ~ ) ) . By F ( m ) we denote the countable set of all values of ~,~(z), z E N d . By using the continuity of a and b with respect to x, for t(n) <_ t <_ t ( n + l ) , we get xt (m)

P) Xt as m --+ ec, where

9 ~(~) := ~m(y) +

(~)

~,(~ + r, ,~,~(y)) aw~ +

(n)

b(~ + r, ,~m(y)) dr.

If we m u l t i p l y b o t h parts of this equality by the indicator of the set {w : ~,,~(y) = z} with n o n r a n d o m z, then we can bring this indicator inside the integrals (a.s.). After this the integrands will not change if in the a r g u m e n t s of ~, b we replace ~,~(y) by z. Having m a d e this substitution we can pull the indicator back, thus obtaining t h a t on every set {n,~(y) = z} (a.s.) we have x t ( m ) = r~t(~),t" ~'z

8~Z

Note also t h a t Vt(~),t is independent of 5ct(~) and has a n o r m a l distribution, n a m e l y t h a t of ~z+t(n),s_t_t. All these a r g u m e n t s show t h a t for a b o u n d e d continuous and, say positive function g the l e f t - h a n d side of (2.14) is (a.s.) equal to lim

?~q,--+ o o

E{g(xt(m))lFt(n)} zCFm =

lim

E

jr

8~Z

(.~/=~E{g(~(n).JIA(n)}

z C Fm

=

lim Q~+t(,),s+tg(~m(Y)) = Qs+t(,),s+tg(y),

m - - ~ cx3

14

N. Krylov

where in the last step we have used Remark 2.5. We have proved (2.14) for bounded continuous g > 0. The passage from these functions to all Borel, positive or bounded ones is standard, and the lemma is proved. By taking expectations in (2.14) and using induction we get the following. C o r o l l a r y 2.7. We have E , , ~ g ( x t ( I ) ) = Q~+t(o),s+t(1) "..." Q~+t(n),~+tg(x).

(2.15)

Let us derive from Lemma 2.6 the last result of this subsection. L e m m a 2.8. Let g be a Borel, positive or bounded f u n c t i o n on ff{d. For an integer n >_ 0 define (2.16)

9 ~(~, ~ + ~, x) = Es,~g(x~(O,(.)I)), ~.here O~(,~)I = {t(k) - t(n) : k >_ n}. F i x n

andt

>_ t(n). Then for every

s > O and x E N a E~,~{g(xt(I))l~t(,~)}

= ~,~(s + t(n), s + t, xt(,~)(I))

(a.s.).

(2.17)

Proof. Denote by t ( m ) the element of I nearest to t from the left. Use (2.14) with t ( m ) instead of t(n), take conditional expectations with respect to ~t(,~-l), and use again (2.14) to transform the conditional expectation. After this take conditional expectations with respect to 5rt(m_2) of the terms of thus obtained formula, apply again (2.14) and proceed further in an obvious manner. For r C [t(k) - t ( n ) , t ( k + 1) - t(n)), k = n , n + 1 , . . . , also denote

~(u,u+r,x) = Qu,~+t(,~+l)-t(n) Q~+t(,~+l)-t(~),~+t(n+e)-t(n)

"'''"

Q~,+t(k)-t(n),~,+~g(x).

We see that the left-hand side of (2.17) (a.s.) is equal to s x Qs+t(,~),s+t(,~+l)Q~+t(,~+l),s+t(n+2) " - . . 9 Qs+t(,~),s+t9( X ti,~) (I ) ) =

= ~ n ( S + t(~), S + t, % ~ ) ( 5 ) -

If we apply the same argument taking r, Ot(n)l, 0 instead of t, I, t(n), then we immediately obtain that the right-hand side of (2.16) is equal to ~,~ (s, s + r, x), that is ~ = 9 and the lemma is proved.

Kolmogorov's equations

15

2.3 M a r k o v p r o p e r t y In this subsection we apply the results of Subsec. 2.2 assuming that equation (2.1) is regular. We will need a very important operator. For 0 < s < t define

a,,,~(x) = E , , ~ ( ~ _ , ) . This definition make sense, for instance, if u is Borel and bounded. L e m m a 2.9. For every T E (0, oo) (i) the C-valued function x~ 'x is continuous in probability with respect to

(s,x); (ii) if u is bounded and continuous, the function Gs,tU(X) is bounded and continuous function of (s, t, x) on the set 0 ~ s ~ t, x E II~~. In particular, the process x ts,z is a Feller process; (iii) if u is Borel, bounded or positive, then Gs,tu(x) is a Borel function

of (s,t,x) Actually, there is nothing to prove since the first two statements follow immediately from Lemma 2.3 and Definition 2.1, and the third one is a standard consequence of (ii). Now let us prove the main results of this section. T h e o r e m 2.10 ( M a r k o v p r o p e r t y ) . Let u be a Borel, positive or bounded, function on ~d and x E I~d, T, s > 0, t E [0, T]. Then

Es,z {U(XT) I/Ft}

:

G,+~,s+Tu(x~s,x ) (a.s.).

(2.18)

Proof. We only need to consider bounded and continuous u. In this case (2.18) follows from Lemma 2.8 if we take a sequence of partitions I(n) such that A ( I ( n ) ) --+ 0 and, in addition, t E I(n) for all n. The theorem is proved. C o r o l l a r y 2.11. If u is bounded and continuous, then, for any T E [0, c~), Gs+t,s+TU(X~ 'z) is a bounded continuous martingale for t E [0, T]. C o r o l l a r y 2.12. Equation (2.18) implies that, for r,s > O, and t > r,

es,s_Ftu

=

Gs,s_FrGs_Fr,s_FtU.

T h e o r e m 2.13 ( S t r o n g M a r k o v p r o p e r t y ) . Let u be a Borel, positive or bounded, function on ]~d and x E ~ d , T , s > O. Then equation (2.18) holds for any stopping time t ~ T. Indeed, for bounded continuous u the assertion follows from Corollary 2.11 and Doob's optional sampling theorem. For general u, it is derived from this particular case by a standard measure theoretic argument.

16

N. Krylov

3. C o n d i t i o n a l

version

of Kolmogorov's

equation

In this section we keep the assumption that ~r and b are nonrandom and replace condition (1.1) with the following stronger condition: For any R E [0, oo) there exists a finite constant K(R) such that ]l~(t,x)l 2 + l b ( t , x ) l <_K(R)

Vt>_O, Ix I <_R.

(3.1)

In this section we also assume that, for any s _> 0 and x E R d equation (2.1) has a solution, which we denote x~ ''~ as usual. First we want to say several words about forward Kolmogorov's equation. Take a test function (, that is take a C C C ~ (Ra), fix s > 0 and x E IRa, and define xt = xt '~ and

~(t) = Es,~(x~) = f JR d

<(y)p(s, x, s + t, y) dr,

(3.2)

where p(s, x, s + t, y) is the generalized function of y acting on test function by formula (3.2). Observe t h a t p(s, x, s + t, y) acts as a measure. Therefore, expressions like

s

f(y)p(s,

X, S

+ t, y) dy

d

are well defined for any bounded Borel f. By It6's formula

C(x~) = ~(~) +

zr

+ r,x~)d~ +

r

(3.3)

1 * where, with a = gacr and Dtu = Ou/Ot,

Lu(t,y) := Dtu(t,y) + aiJ(t,y)u~J(t,y) + bi(t,y)u~,(t,y). The second integral in (3.2) is a martingale, which follows from (3.1) and from compactness of the support of ~. Hence

u(t) =

dp(s,x,s + r,y)L~(s + r,y)dydr,

which after being multiplied by X'(s + t), where X(s + ") E C~(O, oo), and integrating by parts implies that

f0

p(s,

s + y)x'(s + p(s, x, s + t, y)x(s + t)L~(s + t, y) dy dt, .Io[~176 J~feP(S'X' s + t, y)L[xC](s + t, y) dy dt = O.

Kolmogorov's equations

17

It is easy to u n d e r s t a n d t h a t one can replace here X~ with any function of class C~((O, oo) • ]~d). Once this has been done, we see t h a t by definition

L*p(s,x,.,.)(t,y)=O

on

(s, oo) z]R d

in the sense of generalized functions on (s, oo) x ~d, where L* is the formal adjoint to L applied with respect to the variables (t, y). Thus we got w h a t is called the/orward Kolmogorov's equation. This equation has usual sense if we a s s u m e t h a t p(s, x, t, y) is once differentiable in t and twice differentiable in y with the derivatives been s u m m a b l e (or b e t t e r say been Sobolev derivatives), a(t, y) is twice differentiable in y and b(t, y) is once differentiable in x with the derivatives been bounded. Indeed, in this case L*p is a usual, r a t h e r t h a n generalized, function. The backward Kolmogorov's equation is o b t a i n e d in the following way. Let T E [0, oo) be a fixed n u m b e r and g a Borel function on ]Rd. Under certain conditions on g the function

v(s, x) := G~,Tg(x)

(3.4)

is well defined for s E [0, T] and x E IRd. If we assume t h a t v is a s m o o t h function, then, u p o n a p p l y i n g ItS's formula, we get t h a t

v ( s + t , x ts,~ ) = v ( s , x ) +

Lv(s + r, x~s,x ) d r + m r

t<_T - s,

(3.5)

where m t is a local m a r t i n g a l e starting at zero. S o m e t i m e s (see Corollary 2.11) for any s E [0, T], the process v(s + t,x~ '~) itself is a m a r t i n g a l e in t E [0, T - s]. T h e n the integral in (3.5) is a continuous local m a r t i n g a l e and since it has b o u n d e d variation, it is zero. Hence, Lv(s + r, x~ '~) = 0 for almost all r E [0, T - s], and this n a t u r a l l y leads to the guess t h a t

Lv(s, x) = 0

(3.6)

for s < T, x E IR4 9 E q u a t i o n (3.6) is called (backward) K o l m o g o r o v ' s equation. Below we formalize the last a r g u m e n t assuming certain s m o o t h n e s s of v. First, it turns out to be useful to formalize the p r o p e r t y which says t h a t v(s + t, x~ '~) is a martingale. D e f i n i t i o n 3.1. Let Q be a b o u n d e d d o m a i n in (0, T) • ]Rd and u be a b o u n d e d Borel function defined in Q. For (s, x) E Q let ~-s'~ = T~'~ = i n f { t >_ 0: (s + t, xt '~) r Q} be the first exit t i m e of (s + t, x~ 'x) from Q. (i) We say t h a t u is superharmonic (relative to Q) at a given point (so, x0) E Q if u(s0, x0) > Eso,~oU(So + ~- A t, •

+ o(t)

(3.7)

18

N. Krylov

as t $ 0 (which is true with o(t) - 0 for u - v by Theorem 2.13). (ii) We say that u is subharmonic at a given point (so, xo) E Q if U(S0, X0) (_ Eso,zoU(So 4- r A t, XTAt) 4- o(t)

(3.8)

as t $ 0 (which is true with o(t) - 0 for u - v again by Theorem 2.13). (iii) We say that u is superharmonic (subharmonic) in Q if it is superharmonic (subharmonic) at any point (so, xo) C Q and o(t) in (3.7) (respectively in (3.8)) is uniform with respect to (so, Xo) lying in any closed subset of Q. (iv) We say that u is harmonic if it is superharmonic and subharmonic. The most important example of superharmonic and subharmonic functions is the function v (Theorem 2.13), and the reader may save some memory if he/she always thinks of v when we state and prove various assertions about superharmonic or subharmonic functions. It turns out that the above notions are local properties. T h e o r e m 3.2. (i) If a function u is superharmonic relative to Q at (so, xo) c Q, then for any subdomain QI c Q containing (s0,xo) it is superharmonic relative to Q' at (so,xo). (ii) If a bounded Borel function u is defined in Q and (so,xo) C Q and there is a subdomain Q~ c Q containing (so, xo) such that u is superharmonic relative to QI at (so, xo), then it is superharmonic relative to Q at (so, xo). (iii) Similar statements hold for functions subharmonic at (so, xo) or superharmonic or subharmonic in Q. To prove the theorem we need the following lemma. L e m m a 3.3. For s )_ 0 and x C I~g define 7~'~(5) = inf{t >_ 0 : Ixt '~ - xl >__ 5}.

Then for every R > O, s > O, t E [0,1], Ix[ ( R , have Ps,x{sup Ix~ - xl _> 5} _< 5-nE~,~ sup r(_t

(3.9)

n >_ 1, a n d 5 > 0 we

Ix~ - xl ~ _< N 5 -ntn/2,

(3.10)

r(tAq~(5)

where N = N ( n , K ( R + 5)). Proof. Note that in the first expression in (3.10) we can replace t by t A 7(5). Therefore the first inequality follows by Chebyshev's inequality. Then obviously the middle expression times 5 n is less than Es,x[sup I r
~o~I~<.y(6Vr(s + u , x ~ ) d w ~ l + ~o~A'Y(~)[b(s + r,x~)ldul'k

It remains only to apply the inequalities (a + b) n ~_ 2n(a n + bn) and t ~ ~ t ~/2 and the Burkholder-Davis-Gundy inequality which shows that

Kolmogorov's equations Es,~ sup [ r
_< NE~,~(

19

/or

I~,<~(~)~r(s + u, x~) dul '~

C_<~(~/ll~,(s +u,x~)ll2duP/2 ___NK'~/2(R+5)t ~/2.

The lemma is proved. P r o o f of T h e o r e m 3.2. Let (So, x0) E Q~ c Q and let 5 be the distance from (so, x0) to the boundary of Q'. By Lemma 3.3 for 0 < t < 1 A (5/2) it holds that Pso,~o{~-Q A t r 7Q, A t} ~ Pso,~o{~-Q, < t} _~ P,o,~o{t 2 + suplxt - Xo]2 _> 52} _~ N S - 4 t 2, r
where and N is independent of (So, Xo) and Qr. This and boundedness of u obviously imply all our assertions. The theorem is proved. As yet another application of Lemma 3.3 we derive the following. T h e o r e m 3.4. If u is subharmonic at (So,Xo) and v is a bounded Borel f u n c t i o n in Q such that u(s, x) - v(s, x) = o(Is - sol + Ix - xol 2) as Is - sol + Ix - xo] 2 --+ 0, then v is subharmonic at (so,xo).

For the proof, it suffices to denote u( s, x) - v ( s , x) = ~( s - so, x - Xo )(I s sol + Ix - Xo]2) and observe that in the inequality

IE~o,~oU(S0 + < (E~o,~o~(~ A t , x ~

T A t,x~At) --

-- E~o,~oV(So + T / ~ t , x ~ A t ) l

Xo))~/~(E~o,~0(~ A t + I x ~ - x01~)) ~/~

the first factor on the right goes to zero as t $ 0 because (~- A t, x~At - xo) --~ 0 and r is bounded, whereas the second one is less than N t by Lemma 3.3 (applied with 5 bigger than the diameter of Q). Theorem 3.2 allows us to localize investigating differentiability properties of sub- and superharmonic functions by reducing general domains to cylinders lying inside. The following result is applicable to any cylinder and not only to QR :---- (0,T) x BR. T h e o r e m 3.5. Let Q = QR and let u be a function in Q such that, for any s E (O,T), u(s, x) is twice continuously differentiable in BR. Also assume that the first and second order derivatives of u with respect to x are bounded on Q. Then there exists a constant N such that, for any x E BR, (i) u(s, x) + N s is an increasing function of s E (0, T ) provided that u is subharmonic in Q, (ii) u(s, x) - N s is a decreasing function of s E (0, T) provided that u is superharmonic in Q, (iii) [u(t,x) - u ( s , x ) [ < N i t - s[ provided that u is harmonic in Q.

20

N. Krylov

Proof. Assertions (ii) and (iii) trivially follow from (i). To prove (i), apply It6's formula to u(s + t,x~ ) as a function of r. Then it is seen that, for s C (0, T) a n d t E [ 0 , T - s ) a n d l x [ < R ,

~(s, x) <_ E~,~(s + t, x~A.) + o(t) =u(s+t,x)+E~,~

f

~A~-

L[u(s+t,-)](s+r,x~)dr+o(t)

JO

< u(s + t,x) + Nt + o(t). Now assertion (i) follows easily due to uniformity of o(t) with respect to s. The theorem is proved. We will be using this theorem while checking the assumptions of the following theorem. Denote by BI'2(QR) the set of all bounded continuous functions u(s, x) defined on QR which are Lipschitz continuous in s with constant independent of x, are twice differentiable with respect to x for any s with u~ and u~, being bounded. Remember that Lipschitz continuous functions are almost everywhere differentiable (Rademacher's theorem), so that if u E B 1'2 (QR), then D~u(s, x) exists in QR (a.e.). It turns out that, if u E BI'2(QR), then, for almost any point (s, x) E QR,

~(s+t,x+y)

= ~(~,~)+~(s,

x ) r + ~ u ~ i (s, x)y~y ~+D~u(s, x)t+o(It I+ ]yl 2 )

as Itl + lyl 2 -+ 0. This fact is very easy to prove if, for instance, ux and uxx are continuous as functions of (s, x). A more general result can be found in Appendix of [KR]. The following is our conditional version of Kolmogorov's equation. T h e o r e m 3.6. Let u C BI'2(QR). Then

(i) Lu > 0 (a.e.) in QR if u is subharmonic in QR, (ii) Lu < 0 (a.e.) in QR if u is superharmonic in QR, (iii) Lu = 0 (a.e.) in QR if u is harmonic in QR. Proof. Again we only need to prove (i). Under the assumptions in (i), by Theorem 3.4, for almost any (s, x) E Qm

~(s, x) _< ~(s, x) + ~xr

x)E~,x(x~A~ - x ~)

i + ~x~x~(s, x)E~,x(x~A. - x~)(x~A~ - xJ) + D ~ ( ~ , x)t + 1

9

o(t).

Therefore, it only remains to prove that, for almost all (s, x) E QR,

H~(s, x, t) := E~,, (x~A~ - x i) = bi(s, z)t + o(t), ~ J (~, ~, t) :=

E.,~ ( ~ A . - x ~) ( x ~ - .~) = 2~ ~ (~, x)t + o(t).

By ItS's formula and Lemma 3.3, for any 5 > 0,

(3.11)

(3.12)

Kolmogorov's equations

Igi(s,x,t) -

21

bi(s + r,x) dr I = IEs,x

f

tA'r

[bi(s+r, X r ) - b i ( s + r , x ) ] d r l + o ( t )

J0

<_

h(5, s + r, x) dr + o(t),

(a.13)

where

h(~,t,x) :=

sup

ly-xl<_5

Ib(t,y)-b(t,x)l.

Since b is continuous in x, h(5, t, x) is Borel in t and h(5, t, x) -+ 0 as 5 $ 0. Furthermore, by using the Lebesgue differentiation theorem we get that, for almost all (s, x),

lifo

f

s+t

r, x) dr =

s, x)

r

for all rational 5 > 0. Thus, for almost all (s, x) and all rational 5 > 0,

1-~mllg~(s,x,t) t$O

g

/o

b~(s + r , x ) drl <_ h(5, s,x).

Here the left-hand side is independent of 5 and the right-hand side goes to zero as 5 $ 0. Therefore, the last limit is zero, and to get (3.11) it only remains to invoke again the Lebesgue differentiation theorem. Next,

H ij (s, x, t) : Es,~

f

tAr

[2aij (s+r, xr)+(xir-xi)b j (s+r, xr)+(xJ-xJ)bi(s+r, x~)] dr

J0

= 2Es,x

f

tAT

+ r,x ) + o(t)

JO

and one gets (3.12) similarly to (3.11). The theorem is proved.

Remark 3. 7. It is known that the equation Lu = 0 (a.e.) in QR with boundary conditions u = g on ([0, T] • SR) U (T x BR) (SR := { x : Ix] = R}) can only have one solution in BI'2(QR) A C((~R) and even in a much wider class of functions (see, for instance, [KR]).

4. Differentiability of solutions of stochastic equations w i t h respect to initial data Here we want to derive some results which would guarantee that under some smoothness assumptions on a and b the function v (see (3.4)) is twice continuously differentiable in x.

22

N. Krylov

4.1 E s t i m a t i n g m o m e n t s o f s o l u t i o n s o f It6's e q u a t i o n s

The only assumptions in this subsection are those stated before Remark 1.1. Fix some real-valued, ~ct-adapted, and j r | B([0, c~))-measurable processes ct, f t , g t such that ft >_ 0, gt >_ 0, and assume that

~t :=

/0

ca ds,

/0

(f~ + g~) ds

are finite for any ~o and finite t. Also, fix some numbers 7 E (0, 1) and K > 0. Denote by 9)I the the set of all finite stopping times. For two cadlag processes ~t and Vt we write d(t <_ drh if and only if the process ~t - Ot is decreasing. The following result which we give without proof turns out to be extremely useful in many applications L e m m a 4.1. Let ~t and rJt be nonnegative processes. (i) Let m t be a local martingale starting at zero and let ~t <_ ~t + m t for all t. Then for any T E ffJ[

E{~[~-0} < E{suput[9~0}. t
(ii) Let ~t and 77t be continuous and :;t-adapted, and let

E{~,lJ:o} _< E { ~ l f 0 }

(a.s.)

(4.1)

for any 7- E 9)I. Then for any ~- E 92i[ we have

E{sup(71$ro} <_ 12_-~E{sup~7[~ro}

(a.s.).

(4.2)

L e m m a 4.2. Let ~t be a nonnegative cadlag process and let (t be a onedimensional process having stochastic differential d~t = bt dt -k (zt dwt, where bt and crt are JZt-adapted and jz | B([0, c~))-measurable processes of appropriate dimensions. A s s u m e that for all t > O, d~t ~_ bt dt + ot dwt. and bt ~_ Ct~t -~- ft for all w and t. Then for all T C if2

E ( , e -~" _< E[(o +

I"

e-~ftdt],

E s u p ~ e -~'~ ___ ~ E [ ( o

+

~-~~

t
/o"

dt]~.

(4.3)

Kolmogorov's equations

23

Proof. To prove (4.3), notice that by It6's formula

Gte-~'~ <-- GO+

/0'

e - ~ f s ds +

/0'

e-~"as dws =: ~t-

(4.4)

Since ~, f are nonnegative, by Lemma 4.1 (i) for any "r E 9)I, we get E{~Jjr0} < E{G o +

/o

e - ~ f s dsJgVo}

(4.5)

and this and (4.4) prove the first inequality in (4.3). To prove the second one we apply Lemma 4.1 (ii) to (4.5) and again use (4.4). The lemma is proved. E x e r c i s e 4.3. Prove (4.4).

Remark ~.~. Assertions like in Lemma 4.2 are equivalent to their "~-0-conditional versions". By this we mean that under the condition of Lemma 4.2 it holds that (a.s.) E{Ge-~'12:o} <_ Go + E{

/0" I"

e-~'~ftdtJ:Zo},

Indeed, this conditional statements obviously imply the corresponding statements of Lemma 4.2. On the other hand, one can apply Lemma 4.2 to the processes ~tIA, ftIA, 9tIA where A is any element of ~'o. Then in (4.3) one gets the factor In inside all expectations, which due to the arbitrariness of A is equivalent to the above "conditional" inequalities. L e m m a 4.5. Let Gt be a process as in the beginning of Lemma 4.2 and let ct >_ ctI(,r (for instance, ct > 0). Take a numberp > 0 and assume that for all w and t pGtbt + 89 - 1)Jcrt[2 _< ctG2.

Also assume that bt <_ ftGt and JatJ2 <_ ftG~. Then for any r E 9)I EG~e -~" < EG~,

E sup Gt~Pe-~ <_ N(7)EG~ p. t

Proof. Since Gt = ~t - At where At is an increasing process and Gt > 0, we can apply It6's formula to ((t + e) p for any ~ > 0. Then we get

d(Gt + e) p <_ (Gt + ~)p-2 (P(Gt + r

+ 89

+p(Gt + ~);-~,t d~t <_ (Gt + r

- 1)[at[ 2) dt

dt

+(Gt + ~)P-2epbt dt + P(Gt + e)P-l(Tt dwt < h,#o(~t + e)P[rl2tct + pftzl~te/(Gt + e)] dt + P(Gt + e)Pnetl,~teoGtlo-t dwt,

24

N. Krylov

where ~ t = ~t/(~t +E). By Ietting r $ 0 and using the d o m i n a t e d convergence t h e o r e m and the fact t h a t II~,#o~tlatl 2 < ft, which is integrable, we obtain

d~ <_~c, dt + pr

dw,.

This inequality and L e m m a 4.2 (with f - 0 there) i m m e d i a t e l y i m p l y our assertion. T h e l e m m a is proved. Theorem

4.6. Let xt be a solution of equation (1.4). Assume that

Ilo(t,~,)]l + Ib(t, ~,)1 _< t~(1 + Ix, I) for all w G ~ , t G [0, T]. Then for any p > 0 and t G [0, T]

Esup Iz~ - zol' _< Nt'/2eNt( 1 + ElxolP), ~_
(4.6)

s
where N = N(p, K). Proof. By It6's formula for ~t := xt - xo and ~t := I~tArl 2 + S(IX01~ + 1), where e > 0 is a n u m b e r , we have d~t = {2(~t, b(t, x~)) + lta(t, xt)ll2}It<_T dt + 2 ~ ; a ( t , xt)It<w dwt. Here for g E (0, 1) and t < T 1 + Ixtl 2 _~ 1 + 21~t] 2 + 21Xol2 _~ 4~-1~t,

I~*(t,~)v,I: < N(1 + Ix, l~)lv, I: < N(1 + I~1:)~, < N ~ - ~ : 2(~, b(t, ~ ) ) + ]l~(t,~)ll ~ <_ I~1 ~ + [b(t,x~)l ~ + II~(t,~)ll ~ _< N ~ - l ~ . where N = N ( K ) . Hence for ct = Nc -1 from the first assertion of L e m m a 4.5 we get

E s u p ~ p / 2 e - N c - t t ~ N~P/2(1 -k ElxolP), t

where N = N(p, K). It follows t h a t E s u p [x~ - Xo[p <_ geP/~eY~-~t(1 + Elxo[V).

(4.7)

s(t

If t < 1, we take ~ = t and if t _> 1 we take ~ = 1. In b o t h cases from (4.7) we o b t a i n the first inequality in (4.6). Since the second one is an obvious corollary of the first one, the t h e o r e m is proved. T h e second inequality in (4.6) can be generalized.

Kolmogorov's equations

25

T h e o r e m 4.7. Let xt be a solution of equation (1.~). Assume that

IIc,(t, xt)ll <_Klxtl + gt,

Ib(t,xt)l < KIxtl + ft

for all w E ~2, t C [0, T]. Then for any p > 0 and t E [0, T]

Esup

l~I ~

<

s
NeNtE(IxoI p § (

/o'

fs de) p + (

/o'

g2 ds)p/2),

(4.8)

where N = N(p, K). Proof. For any integer n > 2 by It6's formula

dlXtATI'~ = {nlxtl'~-2(xt, b(t, xt)) +

§

~[(n - 2)lxtln-4la*(t,

xt)xtl 2

xdll2]} h
Here

nlxtl'~-2(xt, b(t, xt)) +

~[(n

-

2)Ixtl=-4l~*(t, x~)xtl 2 + Ixtl"-2ll~(t, xt)II 2]

_< m([xtln + ixtln-l ft + ix ~ n-29t2,), where the constant m > 0 depends only on n and K. Hence by Lemma 4.2 for any ~- C E sup [XtATin/2e -mt/2 5 gEIxol "/2 + N E ( ~0 AT Xs n--lrjs e-msds) l/2

t
+NE(

~

7"AT

xs

n--2 2e_mS 1/2. ds) gs

Furthermore, by Young's inequality

E ( / i ~AT Ixsln-l fse -m" ds) 1/2

<- EsuPlXtATl(n--1)12e--'~(n--1)tl(2n)( <_ 88

E(

fTAT

t <'r

pTAT

f~ds)ll2

-~t/2 + NE( f

TAT

fsds) ~/2,

JO

xs ,~-2 gse2 -ms ds) l/2

JO 1 sup IXtATin/2e -mr~2 -~-N E ( < ~E t
It follows that

f "rAT 2 ds) ~/4 gs JO

26

N. Krylov E s u p [XtAT[n/2e -rnt/2 t
< NE[x01~/2 + XE( --

F

/~ ds) ~/~ + NE(

do

f ,AT

g2s d s ) n / 4

(4.9)

J0

with N = N ( n , K ) if the l e f t - h a n d side is finite. Since the process xt is continuous and defined for all t, any 7 E 9~ can be a p p r o x i m a t e d by stopping times for which the l e f t - h a n d side is finite. Therefore, (4.9) holds for all ~- E 9)t This by R e m a r k 4.4 and L e m m a 4.1 implies t h a t for any 7 ~ (0, 1) a n d ~- 6

E sup

IXtATIn'~/2e-mt'~/2

t
<_Nm~ol~/~ + NE( f

dO

TAT

/~ as) n~/2 + wE(

FATg~as) ~/~.

dO

By choosing n 7 = 2p and ~- = T, we easily get (4.8). T h e t h e o r e m is proved. E x e r c i s e 4.8. By considering xt - x0 instead of T h e o r e m 4.7. 4.2 S m o o t h n e s s

of solutions depending

xt derive T h e o r e m 4.6 from

on a parameter

Here we apply the results of Sec. 1 to equations depending on a p a r a m e t e r and show t h a t the solutions are s m o o t h as functions of the p a r a m e t e r provided the coefficients are smooth. Let Q be a d o m a i n in a Euclidean space IRk and let functions a(t, x, q) and b(t, x, q) be defined on F2 • [0, cx~) • I~d X Q. We assume t h a t a takes values in the set of d • dl matrices and b is I~d-Valued. We also assume t h a t t h e y are f t - a d a p t e d and 9v | B([0, o o ) ) - m e a s u r a b l e for any x, q. As far as their dependence on x , q is concerned we fix an integer n > 1 and assume t h a t a and b are n times continuously differentiable with respect to (x, q) for any (w, t) and for any finite T, R > 0 T

~--~f

sup {[]Dicr(t,x,q)[[2+]D%(t,x,q)l}dt
(4.10)

where D i stands for a r b i t r a r y derivative of order i with respect to (x, q) and the s u m m a t i o n is taken over all derivatives of all orders i < n. We will be dealing with solutions of the following equation

xt=xo(q)+

~(s, xs,q) dws+

b(s, xs,q) ds t>_O.

(4.11)

If there is no stochastic t e r m in the equation, then from the t h e o r y of O D E one knows t h a t the solutions are differentiable with respect to q if this is true for the initial data. Also the partial derivatives satisfy a linear system which

Kolmogorov's equations

27

is o b t a i n e d by formal differentiation of (4.11). Here we show t h a t the s a m e is true in the general situation as well. In particular, the first and the second directional derivatives in the direction of a vector ~ C Nk of the solution of (4.11) satisfy

d~t = O(~t, t~)a(t, xt, q) dwt + O(~t, n)b(t, xt, q) dt,

d,7~ = [O(v~, O)~(t, x,, q) + 0~(~,, ,~)o(t, x~, q)] d~,, + [O(rh, O)b(t, zt, q) + 02(~t, ~)a(t, ~t, q)] dr,

(4.12)

(4.13)

where for any function u(x,q), ~ E R d, and ~ E If{k

0(~, ~)u(x, q) = us, (x, q)(i + uq, (x, q)~i, 02(~, n)u(x, q) = Ux~x~ (x, q)~i~j + 2ux~q~ (x, q)~it~J + Uq~q~(x, q)t~it~ j. D e f i n i t i o n 4.9. Given a r a n d o m process xt(r) defined for t _> 0 and r E ( r l , r 2 ) , where rt < r2, and a point r0 E ( r l , r 2 ) , we say t h a t xt(r) is t uniformly continuous at point r = ro in probability if P - lim sup Ix~(r) - x~(ro)l = 0 r----~ro t ~ T

VT e [0, oo).

Also, if a r a n d o m process ~t is defined for t E [0, oo), we call it a t-uniform

derivative of xt (r) in probability with respect to r at r = r0 if P - lim s u p I ~ - l [ x t ( r 0 e--->o t < T

+ ~) - x t ( r o ) ] - ~t I = 0

V T e [0, oo).

We write

it = P-Oxt(r)/Orl~=~o and notice t h a t two different processes it and ~ can satisfy this equality only if they are indistinguishable. In an obvious way one defines t - u n i f o r m continuity (on ( r l , r 2 ) ) , higher order t - u n i f o r m derivatives in probability and also partial derivatives and directional derivatives if xt depends on a multidimensional p a r a m e t e r . In the case of processes like solutions xt = xt(q) of (4.11), if ~ E ]Rk and q E Q, we denote

xt(~)(q) = g-Oxt(q + r~)/0rl,=o,

xt(~)(~)(q) = P-Oxt(~)(q + r~)/Orl,=o.

Notice t h a t it m a k e s sense to speak a b o u t t - u n i f o r m continuity in probability of t - u n i f o r m derivatives in probability. T h e above definitions m a k e sense and will be also used in the case of r a n d o m vectors rather t h a n r a n d o m processes depending on q.

28

N. Krylov

T h e o r e m 4.10. Let xt(q) be a process which satisfies (3.11) for any q E Q. Assume that xo(q) has all derivatives in probability of all orders ~ n in Q. Then the process xt (q) has all t-uniform derivatives in probability of all orders <_n i n Q . Also, for any q E Q and ~ E ~k, the process xt(~)(q) satisfies (3.12) (where xt = xt(q)) and xt(~)(~)(q) satisfies (3.13) if n >_ 2. Finally, if the derivatives of xo(q) are continuous in probability, the derivatives of xt(q) are t-uniformly continuous in probability. Proof. First of all notice that the functions 0(~, ~)cr and 0(~, ~)b are affine functions of ~. Furthermore, for any T, w, i, j T

o {ll~,(t,x~(q),q)ll ~ + II~(t,~(q),q)ll ~

+ ]b~(t, xt(q),q)[ + [bqj(t, xt(q),q)]}dt < cc which follows from (4.10) and the fact that any trajectory of xt(q) on [0, T] is bounded. By Theorem 1.2 for any fixed q E Q and ~ E I~k, there exists a unique solution (t of (4.12) with initial data (0 = x0(~)(q). Next, for r small enough so that q + r~ E Q let ~(~) t : (~) (q) : r - l [ x t ( q

-}- r~) -- xt(q)],

~(') (t, ~) : r -~[o(t, x~(q + r~), q + r~) - ~(t, x~(q), q)] and introduce b(~) (t, ~) similarly. Also let

7(r)(R) --- inf{t _> 0: I~r)l _> R } A T . It is not hard to see that ~ ) satisfies the following analogue of (1.16) =

(t, ~t ) dwt -~ b(r) (t, ~ r ) ) dt.

Moreover, for t < 7 (~)(R) obviously Ixt(q + r~) - xt(q)l < rR which in view of (4.10) and the continuity of the first derivatives of a and b by Hadamard's formula and by the dominated convergence theorem implies that, for (xt (q, s, r), q(s, r) ) := (sxt (q + r~) + (1 - s)xt (q), s(q + r~) + (1 - s)q), we have

0#')(R) [la (r) (t, ~r)) - 0 ( ~ ~), ~)a(t, xt(q), q)[I2 dt =

II

O(~}~),~)~(t, xt(q,s,r),q(s,r)) ds

dO

- 0 ( ~ ~), ~)o(t, x~(q), q)ll 2 dt

Kolmogorov's equations

.r(~)(R)

[

<

29

sup

Jo

sup

~c(o,1) Ir

IIO(i,

x (q, - O(C, ~)a(t, xt(q), q)]ll e dt -+ 0

as v --+ 0 for every R, T, and w. The same is true for b. By Theorem 1.8 this implies ~t = xt(~) (q) and proves the first assertion of our theorem for n = 1. Notice t h a t equation (4.12) combined with (4.11) composes a system which can be considered as a stochastic equation relative to the couple xt, ~t. By the previous result the solution of this system-equation is t-uniformly differentiable if n > 2 and as above one can write an equation for its derivative. This equation turns out to be system (4.12)-(4.13). One can obviously keep going to higher values of n, so that the first two assertions of the theorem are proved. To prove the last assertion, we first prove that xt (q) is t-uniformly continuous in probability. Fix q0 E Q and let a cut-off function ~ E C ~ ( Q ) be such t h a t ~ = 1 in a neighborhood of q0. Observe that for q close to q0 the couple (xt (q), q) satisfies the system

dxt = a(t, xt, qt)~(qt) dwt + b(t, xt, qt)~(qt) dt,

dqt = O.

The coefficients of this system satisfy (1.2) with Kt(R) equal to a constant, depending only on d and k, times sup

(][crx, (t, x, q)((q)]l 2 + I](~(t, x, q)((q))qj I]2

.. Izl
"t~J

+ Ibx~(t,x,q)((q)[ + [(b(t,x,q)((q))qj 1), which is integrable in t owing to condition (4.10). By Theorem 1.7 we conclude t h a t xt (q) is t-uniformly continuous in probability indeed. Now let qn -+ qo E Q. Observe that ~ := xt(~)(q~) and ~t := xt(~)(qo) satisfy

d(~ = a'~(t,{'~)dwt + b'~(t,(2) dt,

d~t = a(t,~t)dwt + b(t,(t)dt

where

crn(t,~) = O((,~r

xt(qn),qn),

or(t, ~) = 0(~, n)a(t, xt(qo), qo),

bn(t,~) = O((,n)b(t, xt(qn),qn), b(t, ~) = 0(~, ~)b(t, xt(qo), qo).

Define 7n(R) = inf{t >_ 0 : I ~ 1 > R} A T, and notice t h a t by virtue of t-uniform continuity of xt(q) (and condition (4.10)) we have

30

N. Krylov

~oo"r~(n) [llan - c~ll2 + Ibn - bl](t,~'~) dt <_ N R E f T ~ (n) [IIcr. ~(t, xt (qn), qn) - e~ (t, xt (qo), qo)ll 2 9 . JO "t,j

~- IIcrqj (t, xt(qn), qn) -- O'qj (t, xt(qo), qo)112 + Ib~ (t, xt(qn), %) - b~, (t, xt(qo), qo)] + IbqJ (t,

- bqj (t, x (q0), q0)l] at

O,

where N depends only on d and k. Also ~ --4 ~0 by assumption. Hence, by Theorem 1.8 we get t h a t ~ --+ ~t t-uniformly in probability, which means that the first t-uniform derivatives of xt(q) are tuniformly continuous in probability. By applying this conclusion to the couple (xt(q), xt(~)(q)), which satisfies (4.11)-(4.12), we see that its first t-uniform derivatives are t-uniformly continuous in probability if n >_ 2. This means t h a t the second t-uniform derivatives of xt (q) are t-uniformly continuous in probability if n _> 2. We apply again this result to the couple (xt(q), xt(a) (q)) and in an obvious way by induction we get that the nth t-uniform derivatives of xt (q) are t-uniformly continuous in probability. The theorem is proved.

Remark 4.11. Without assuming that the nth derivatives of xo(q) are continuous in probability we still have that the n - 1st t-uniform derivatives of xt (q) are t-uniformly continuous in probability. Indeed, above we have seen that xt(q) is t-uniformly continuous in probability (for any n _> 1). By the same reason the couple (xt(q),xt(~)(q)) is t-uniformly continuous in probability if n ~ 2. Hence, the first t-uniform derivatives of xt(q) are t-uniformly continuous in probability if n ~ 2. By applying this conclusion to the couple (xt(q), xt(n)(q)), we see that its first t-uniform derivatives are t-uniformly continuous in probability if n _ 3. This means t h a t the second t-uniform derivatives of xt (q) are t-uniformly continuous in probability if n _> 3. By induction we get our conclusion. E x e r c i s e 4.12. Under the conditions of Theorem 4.10 take a real-valued function f(t, x, q) satisfying the same conditions as a(t, x, q) and b(t, x, q). Prove that the process

~o t f (s, x~(q), q) ds is n times t-uniformly differentiable in probability with respect to q. (Hint: denote this process by xtd+l (q) and consider (x~,..., xd+l~ t J.J E x e r c i s e 4.13. In the above exercise also take a real-valued function ct (x, q) satisfying the same conditions as a(t, x, q) and b(t, x, q). Prove that the process

/o

f(S, Xs(q),q)exp(

fo

c(r,x~(q),q)dr)ds

is n times t-uniformly differentiable in probability with respect to q.

Kolmogorov's equations

31

R e m a r k 4.14. In [KR92] a version of Theorem 4.10 is given in which equation has solutions which stay in a domain for all times and the coefficients may blow up when the process approaches the boundary.

4.3 E s t i m a t i n g m o m e n t s o f d e r i v a t i v e s o f s o l u t i o n s Let xt(q) be a solution of equation (4.11). It is convenient in this subsection to assume that the domain Q c I~k (where q varies) is convex. Fix some constants T , K , p , m > 0. In Subsec. 4.2 we found some conditions under which xt (q) is differentiable in probability. In the future we need xt (q) to be differentiable in the mean because we are going to differentiate expectations. Here we give some sufficient conditions for that to happen. These conditions also provide some estimates of derivatives. T h e o r e m 4.15. A s s u m e that ][cr(t, xl, ql) - a(t, x2, q2)][ + [b(t, x l , ql) - b(t, x2, q2)l <__K[Ixl - x2 ] + ]ql - q2 [] for all t E [0, T], xi C ]~d qi E Q, and w. Then for any t E [0, T], q E Q, r > O, and unit ~ c ~k such that q + rn E Q we have Esupr-Plxs(q

+ r~c) - Xs(q)l p <_ N e N t ( 1 + E r - P l x o ( q + rt~) - xo(q)lP),

s
where N = N ( p , K ) .

Proof. Denote ~ ) = r - l [ x t ( q + rn) - xt(q)]. Obviously ~r) satisfies d ~ ") = a (~) (t, ~ ' ) ) dwt + b (~) (t, ~ ) ) dt,

where a (r) (t, ~r)) :----r _ l [or(t, xt (q + rn), q + rt~) -- ~(t, xt (q), q)] and b(r)((~ ~)) is defined similarly. Since

+

C ) ) [ < K(1 +

I),

the assertion of our theorem follows from Theorem 4.6. The theorem is proved. C o r o l l a r y 4.16. If Elxo(q + rn) - xo(q)] p --+ 0 as r -+ O, then

E s u p Ixs(q +

- xs(q)L p -+ 0.

s
By combining Theorem 4.15 with Theorem 4.10 and well-known results bearing on convergence in probability under boundedness of moments, we arrive at the following corollary.

32

N. Krylov

Corollary 4.17. If a ( t , x , q ) and b(t,x,q) are continuously differentiable with respect to (x,q) and if xo(q) is differentiable in the mean of order p in Q, then for any 0 < Pl < p

lim Esup [xs(,~)(q) - r-l[xs(q + ra) - xs(q)]l m = O, r-+O

s
sup ]xt(,)(q)l p < g e N T ( 1 + E[Xo(,)(q)IP). s
T h e o r e m 4.18. Let the assumption of Theorem ~.15 be satisfied. Let xo(q) be differentiable in the mean of order 2p5 in Q, where 5 is a constant,

5 E (1, oc). Moreover, let a ( t , x , q ) and b(t,x,q) be twice continuously differentiable with respect to (x,q) and the absolute values of any their second order partial derivative with respect to (x, q) be less than K(1 + Ix[m). Then for any t E [0, T], q E Q, r > 0, and unit a E I~k such that q + ra E Q we have E sup r-P[xs(,~) (q + ra) - xs(,c) (q)] p s
<_ NeNt(Er-PiXo(,~)( q + ra) -- Xo(~)(q)[ p + J),

(4.14)

where N = N ( 5 , p , K , d , d l , k , m ) , J = JiJ2, and forS' := 5 / ( 5 - 1)

J~ := 1 + Er-2VSixo(q + ra) - xo(q)l 2p5 + E[xo(,)(q)[ 2v~, J25' : = l + E x 0 ( q )

Pm~ + E ] x o ( q + r a ) [ pm~'.

Proof. By Theorem 4.10 the process ~t(q) := xt(~)(q) satisfies (4.12). Hence, for 71~r) := r-l[~t(q + rt~) - ~t(q)] we have d ~t(r) = r-1 [O(~t (q+ra), a)~(t, xt (q+ra), q+ra)-c3((t (q), tc)a(t, xt (q), q)] dwt +r-l[O((t(q + ra), a)b(t, xt(q + rn), q + ra) - cO(~t(q), a)b(t, xt(q), q)] dt,

d~}~) = [a(~)(t,~} ~)) + g}~)]dwt + [b(~)(t, ~}~)) + f(~)ldt,

(4.15)

where a(r) (t, ~}~)) = O'(Vt(~) , O)a(t, xt(q + ra), q + ra), b(~)(t, ~}~)) = O'(~t(~), O)b(t, xt(q + ra), q + ra), g}~) = r-~[O(~t(q), t@y(t, xt(q + rt~), q + rt~) - O(~t(q), t@y(t, xt(q), q)],

f(t r)

---- r - 1

[0(( t (q), n)b(t, xt (q + ra), q + rtr - c9(~t (q), tc)b(t, xt (q), q)].

Furthermore, _

N

(~)

[ft(~)l + 19}~)[ _< N(1 + [~t(q)l)(1 + [(}~)l)(1 + [xt(q)[ m + ]xt(q+ra)im). Hence from (4.15) by Theorem 4.7 we get

Kohnogorov's equations

33

E sup [rt~~) Ip < g e Nt ( E I ~ ~) Iv s
+ E s u p { ( 1 + I~t(q)[P)(1 + I~r)]v)(1 + [xt(q)[ pm + [xt(q + ra)]vm}). s
Finally, we use the inequality

EaPbPc m p < S ( a 2p + b2P)c"~p < i ( E ( a 2p5 + b2V~))l/5 (EcmVS')l/5', T h e o r e m s 4.6 and 4.15 and Corollary 4.17 and we arrive at (4.14). T h e theo r e m is proved. Similarly to Corollary 4.17 we get the following s t a t e m e n t . 4 . 1 9 . Under the assumptions of Theorem 4.18, if xo(q) is continuous in the mean of order proS' and differentiable in the mean of order 2p5 and if Xo(~) (q) is differentiable in the mean of order p in Q, then for any t E [0, T] a n d O < p l < P Corollary

lim E sup Ix~(~)(~)(q) - r-l[xs(~)(q + ra) - x~(q)]l pl = O, r-+O

s
E sup ]xt(~)(~ ) (q)l p

< N e NT

(Elxo(~)(~)(q)lp

s
+ (1 + Elxo(~)(q)lUP~) x/5 (1 + Elxo(q)lPmr162 4.4 T h e n o t i o n s o f E - c o n t i n u i t y

and E-differentiability

In this subsection T is a fixed n u m b e r from (0, cr D e f i n i t i o n 4.20. Let a real-valued r a n d o m process xt = xt(w) be defined for t C [0, T]. We write xt e s if the process xt(w) is m e a s u r a b l e in (w, t) and for all p > 0

E We write xt C s

IxtlPdt < oo.

if xt is separable and for all p > 0

E sup lxtl p < oo. t
T h e convergence in spaces s and s

is defined in the following way.

D e f i n i t i o n 4.21. Let x~t~ E s (E s n = 0, 1,2 .... We say t h a t / : - l i m i t (s of x~~ equals x ~ and we write s

lim x ~ = x ~ n--+oo

(s

lim x ~ = x ~ n--+oo

if for all p > 0

Ix~-x~

lim E n - + c~

JO

(respectively,

lim n-+ ~

Esuplx~-zt~ t
By HSlder's inequality in b o t h definitions we can replace p > 0 with p > 1.

34

N. Krylov

Having introduced the notions o f / ; - l i m i t and s it is clear what we mean b y / ; - c o n t i n u i t y and s at a point q0 of a function xt (q) defined in a domain Q c IRk . D e f i n i t i o n 4.22. Let qo E Q, ~ c I~k, yt E /; (respectively s Suppose that for each q from some neighborhood of q0 a process xt(q) E E (E~) is given. We say that Yt is a C-derivative (EB-derivative) of xt(q) in the direction of ~ at q0 and we write

Yt = s

(Yt = EL3-xt(~) (qo ) )

if

Yt = s lim r -1 [xt (qo+r~c)- xt(qo)] r-+O

(Yt = s

7" - 1

[xt(qo Jr-rt~) - xt(qo)]) .

We say that the process xt (q) is once s (respectively,/;Bdifferentiable) at qo if xt(q) has E-derivatives (/;B-derivatives) in all directions at qo. The process xt(q) is said to be i times (i > 2) s (respectively, s at q0 if xt(q) is once /;-differentiable (respectively, s at any point of some neighborhood of qo and each first E-derivative (/;B-derivative) is i - 1 times s (s differentiable) at qo. Notice that, of course, s also are t-uniform derivatives in probability. Definitions 4.20, 4.21, and 4.22 have been given for real-valued processes. They are extended to vector- or matrix-valued processes in an obvious way. Further, as is commonly done in calculus we write Yt(q) = /;-xt(~)(q) if Yt(qo) = E-xt(~)(qo) for any qo, E-xt(~l)(~2)(q) = /;-Yt(~2)(q), where Yt(q) = s (q), etc. We say that xt (q) is i times/;-continuously s if all/;-derivatives of xt (q) up to and including order i are E-continuous. We will not dwell on explaining such obvious figures of speech in the future. In order to get familiar with the given definitions, we note a few simple properties these definitions imply. It is obvious that for random variables, the notion of E-continuity is equivalent to that of s Furthermore, [Ex(q) - Ex(qo)l <_ Elx(q) - x(qo)]. Hence the expectation of a E-continuous random variable is continuous. Since

]~-~(Ex(qo + r~) -

Ex(qo)) - E y I <_ E l r - l ( x ( q o + r~;) - x(qo)) - Yl,

the derivative of Ex(q) in the direction of ~; at qo is equal to the expectation of the E-derivative of x(p) if the latter exists. Therefore, the sign of the first derivative is interchangeable with the sign of the expectation. Combining the above properties, we deduce that (Ex(q))(,;) exists and is continuous at qo if x(q) is E-continuously s at q0 in the direction of ~. A similar situation occurs for higher order derivatives. Since for a random variable ~- < T

Kolmogorov's equations

35

Elx~(q) - x~(qo)I <_ E s u p lxt(q) - xt(qo)l, t
the variable xr (q) is /2-continuous if xt (q) i s / : B - c o n t i n u o u s and x , (q) is a measurable function of w. A similar inequality shows that for the same

~-x~(~)(q) = (~B-x~(~)(q))l~:~

(4.16)

if xt (q) is s in the direction of ~ and x~ (q) and the right-hand side of (4.16) are r a n d o m variables. This argument allows us to derive properties of s and s of the random variable x~ (q) from those of the process xt(q). In addition, (4.16) shows that substituting v for t is interchangeable with derivating. Next, suppose t h a t x(t,q) := xt(q) is continuous with respect to t and s at q0. Also, let 7(q) be random functions with values in [0, T] continuous in probability at q0. Then we assert that x(7(q), qo) and X(T(q), q) are s at q0. In fact, Ix(T(q),qo) -- x(T(qo),qo)l p --+ 0 in probability as q --~ q0 and in addition this quantity is bounded by 2p suPt
El~(~(q), q) - ~ff(qo), qo)l" < 2PEIx(v(q), q) - x(T(q), qo)l p + 2PEIx(T(q), qo) -- x(T(qo), qo)I p < 2PE sup Ix(t, q) - x(t, qo)]P + 2PElx(T(q), qo) -- x('c(qo), qo)]P. t
t
tfJo

t

f J0

t

~T

J0 P

T

<_ T ' - ' E ]n I~s(q) - x~(qo)p ds. Therefore, f~ x~(q)ds is an /:g-continuous process if the process xt(q) is s It is proved in a similar way t h a t this integral has an s derivative in the direction of ~ and this derivative coincides with the integral of t h e / : - d e r i v a t i v e of xt (q) in the direction of ~ if the latter derivative exists. Combining the assertions discussed above one can obtain numerous useful facts. We do not list them, because all they are trivial. Less trivial is the following fact from which we later derive continuity of compositions. L e m m a 4.23. Let x~ be d-dimensional processes measurable with respect to (w, t) given for n = 0, 1, 2 .... Assume that s

?2--4OO

x't~= x~

(4.17)

36

N. Krylov

Let f ( t , x) be a random function defined for x E IRa and t E [0, T]. A s s u m e that f is measurable with respect to (w, t) for any x and is continuous in x for any co, t. Also let I f ( t , x ) l < K(1 + i x r ~) (4.18) for all w, t, x where K , m are some constants. Then s

lim f ( t , x~) = f ( t , x~ n - - ~ (x)

Proof. It follows from (4.17) that x~ ~ x ~ in measure d P x dt. The convergence in measure is related in a very well-known way to the convergence of subsequences almost everywhere, whence and from the continuity of f ( t , x) in x we get that f ( t , x ~ ) -4 f(t,x~ in measure d P x dt. It only remains to show t h a t for any p >_ 1 T

#

lim E [ n--+ ~

I f ( t , x ~ ) l P d t < oo.

Jo

Owing to (4.18) it suffices to show supE n

//

Ix~lP dt < oo

for any p _> 1. But the last inequality follows obviously from the fact t h a t x~ -4 xt~ in Lp(J2 x [0, T]) (for all p). The l e m m a is proved. C o r o l l a r y 4.24. If we are given one-dimensional processes x~ and y~ defined f o r n = O, 1, 2... and such that s lim,~--,~ xp = x ~ a n d / ; - l i m n - 4 ~ y~ = yo then s lim Xt~Ytn = xtoYto 9 n --~ O0

Indeed, the two-dimensional process ( X tn , Y~) h a s / : - l i m i t equal to (x ~ yt~ In addition, the function f ( x , y) := x y satisfies the growth condition I f ( x , Y)I 0. L a m i n a 4.25. Let the assumptions of Lamina 3.23 be satisfied and let x~(u) be random d-vectors defined for n = 1, 2, 3.., u C [0, 1] t < T. A s s u m e that x~(u) are continuous in u, measurable in (w, t) and satisfy Ix

(u) - x~ <

- x~

Then s

lim

/o

f(t,x~(u))du

= f(t,x~

Kolmogorov's equations

37

Proof. By HSlder's inequality for p >_ 1

E

/o L/o 1f ( t , x ~ ( u ) ) d u - f ( t , x ~

] 1[ f ( t , x ~ ( u ) ) - f ( t , x ~ foTfo

1

<- E fo

If(t,x'~(u)) - f(t,x~

dudt.

The last expression goes to zero as n ~ oo. Indeed, as in the proof of L e m m a 4.23 we have f(t, x'~(u)) ~ f(t, x~ in measure dP • dt x du and sup E

If(t, x~(u)) - f(t, x~ 2p dudt

n

< N(1 + s u p E

Ixr(u)12"~Pdt+E

Ix~

?I~U

<_ N ( I +

supE

Ixrl2mP dt + E

Ix~

dt) < oo.

The l e m m a is proved. In the following few results we consider composite functions like f (t, x~ (q)) of functions f ( t , x ) and xt(q), so that f(t, x) does not depend explicitly on the p a r a m e t e r q, which varies in a domain Q c ~k. This case is not less general than when f = f(t, q, x) because one can always consider the couple (q, xt(q)) as one process. T h e o r e m 4.26. Let an ]~d-valued random process x~(q), t < T be defined for q C Q and let a real valued process f ( t , x ) , t < T be defined for any x C ]~d. Assume that f ( t , x ) is measurable with respect to (w,t) for any x. Take a point qo C Q. (i) Let f ( t , x ) be continuous with respect to x for any w,t and let (4.18) hold and let xt(q) be g-continuous at qo. Then the process f(t, xt(q)) is g continuous at qo. (ii) Let f ( t , x ) be i times continuously differentiable with respect to x for any ~o, t and let f and all its derivatives up to and including order i satisfy (4.18). Finally, let xt(q) be i times (C-continuously)g-differentiable at qo- Then the process f(t, xt(q)) is i times (respectively, g-continuously) g-differentiable at qo as well. In addition,

g - ( f (t, xt(q)))(,~) = f(y~(q))(t, xt(q)), g - ( f ( t , xt(q)))(~)(~) = ]'(z,(~))(t, xt(q)) + f(y~(~))(~,(q))(t, xt(q)), where y (q) :

z (q) =

whenever the existence of the left-hand sides in (~.19) is asserted.

(4.19)

38

N. Krylov Proof. To prove (i) it suffices to take any sequence q,~ --~ qo, let x~ =

xt(q,~), and to use L e m m a 4.23. To prove (ii) first assume t h a t i = 1. Observe t h a t a continuous function of (x, y) E I~2d and

f(t, x, y) :-- ft(y)(X) is

If(t,x,y)l <_ g ( 1 + ixr~)lyl ~_ N ( I + I(x,y)]'~+l). Next, take a C I~k and a sequence of numbers r~ --+ 0 and define

x?(u) = uxt(qo + r~a) + (1 - u)xt(qo),

y? = r;l(xt(qo + r,~a) - xt(qo)).

By the N e w t o n - L e i b n i z rule we have

r~ ~ [f(t, xt(qo + r,~)) - f(t, xt(qo))] =

/o

f(t,

= r~ 1

/o

~

y2)

Here

Ix~(u) - xt(qo)] ~ Ixt(qo + rn~) - xt(qo)J, so t h a t by L e m m a 4.25 applied to x'~(u) and y~(u) : = y~ we have E-

f01

f(t, xr(u),y?)du = f(t, xt(qo),yt(qo)).

Therefore, E- lim

r--+O

r -1 [f(t, xt(qo + rt~)) - f(t, q0)] = f(t, xt(qo), Yt(qo)).

Finally, by (i), f(t, xt(qo),yt(qo)) is E - c o n t i n u o u s with respect to q0 if xt (qo) is E - c o n t i n u o u s l y s with respect to q0. This proves assertion (ii) for i = 1 and also proves the first formula in (4.19) which we choose to write in the following way

E-(f(t, xt(q)))(,~) = f(t, xt(q), Yt(q)). For general i _> i we use the induction on i. Assume t h a t (ii) is proved for i _< j and for any processes f(t, x) and xt(q) satisfying the assumptions in (ii). Let our couple f (t, x), xt (q) satisfy the conditions in (ii) for i = j + 1. We take the derivative E-(f(t, xt(q)))(,~) and prove t h a t it is j times E-differentiable at q0. Let us write this derivative as f(t, xt(q), yt(q)). Notice t h a t the process (xt (q), Yt (q)) is j times E-differentiable at q0 by assumption and the function f ( t , x , y ) is j times continuously differentiable with respect to the couple (x, y). Also the absolute values of the partial derivatives of f(t, x, y) of orders up to and including j are less t h a n N ( 1 + I(x,y)lm+l). Therefore, by the induction assumption, f(t, xt(q), Yt(q)) is j times E-differentiable at qo. Since

Kolmogorov's equations

39

n is arbitrary, f(t, xt(q)) is, by definition (!), j + 1 times g-differentiable at q0. Similarly one proves g-continuity of C-derivatives of f(t, xt(q)) at q0 if E-derivatives of xt(q) are E-continuous at q0. Finally, in conjunction with the first formula in (4.19)

g-(f(t, xt (q)))(~)(~) = g - ( f ( t , xt(q), Yt(q)))(,0 = f(y,(q),zdq))(t, xt(q), Yt(q)), which after simple transformations yields the second formula in (4.19). The theorem is proved.

Remark 4.27. Theorem 4.26 allows one to prove the C-continuity and g differentiability of various expressions containing random processes. For example, arguing in the same way as in Corollary 4.24, one proves that if x~ (q) and Yt(q) are real-valued i times g-differentiable processes, the product xt(q)yt(q) is i times s as well. Another important example is given by the process exp(-xt(q)) which is i times s if xt (q) is i times s and xt(q) is nonnegative or just bounded from below. Indeed, notwithstanding that the function e x p ( - x ) grows more rapidly than any polynomial as x ~ - o o while considering nonnegative xt (q) we may write e x p ( - x t (q)) = f(xt (q)), where f(x) is any smooth function coinciding with e x p ( - x ) for x > 0 and vanishing for x < - 1 . Combining these examples with the known properties of integrals of g continuous and g-differentiable functions we arrive at the following results. T h e o r e m 4.28. Let the processes xt(q), fl(t,x), ft(t, x) satisfy the assump-

tions of Theorem 4.26 (i) (respectively, Theorem 4.26 (ii)) and let fl(t, x) > O. Then the process f2(t, xt(q))exp { -

fl(t, xs(q)) ds}

(4.20)

is E-continuous (respectively, i times (respectively, g-continuously) s ferentiable) at qo. T h e o r e m 4.29. Let the processes xt(q),f(t,x)

satisfy the assumptions of Theorem 4.26 (i) (respectively, Theorem 4.26 (ii)) and let f(t,x) >__O. Let the random variable -r(w) E [0, T] and let the time independent processes y(q),g(x) satisfy the assumptions of Theorem 4.26 (i) (respectively, Theorem 4.26 (ii)). Then the random variable g(y(q)) exp { - fo r f(s, xs(q)) ds}

(4.21)

is C-continuous (respectively, i times (respectively, g-continuously) g-differentiabte) at qo.

40

N. Krylov

Remark 4.30. Formulas 4.19 and our discussion concerning s of integrals and derivatives of expectations show that while computing g - d e r i v a tives of (4.20) and (4.21) or of their expectations we can apply usual formulas from calculus. Let us apply these results to the situation in which xt (q) is a solution of ItS's equation (4.11). 4.31. Let Q be convex and xo(q) be once (respectively twice) f-.continuously f-.-differentiable in Q. Fix a point qo E Q and assume that a(t, 0, q0) and b(t, 0, q0) are bounded. Also assume that the first order derivatires of a(t, x, q) and b(t, x, q) with respect to (x, q) are continuous in (x, q) for any w, t and are bounded functions of w, t, x, q. Let c( t, x, q), f ( t, x, q), and g( x, q) be real-valued functions defined on Y2 • [0, T] • I~d • Q. Assume that they are once (respectively, they and a(t, x, q) and b(t,x,q) are twice) continuously differentiable with respect to (x,q) for any w, t and their absolute values together with the absolute values of their partial derivatives (respectively, up to and including the second order derivatives) with respect to (x, q) are less than K(1 + Ixl m) for any w, t, x, q. Finally, let c(t, x, q) >_ 0 and let T be a stopping time, ~- ~_ T. Then xt(q) is once (twice) s s in Q and the function Theorem

F(q) := E

~0 T f(t, xt(q),q)e- fot c(s'x~(q)'q)dS dt+Eg(x~(q),q)e- foc(s'x~(q)'q)ds

is once (twice) continuously differentiable in Q. Proof. By T h e o r e m 4.6 we have xt(q) E f_.B for any q C Q. By Theorem 4.10 the process xt(q) has one (respectively, two) t-uniform derivatives in probability which are t-uniformly continuous in probability. By Corollaries 4.17 and 4.19 the t-uniform derivatives in probability of xt(q) also are its /:B-derivatives. From the estimates of these derivatives given in Corollaries 4.17 and 4.19 and from the above mentioned t-uniform continuity of the derivatives in probability, it follows that the derivatives are s This proves that xt (q) is once (respectively, twice) s s differentiable. Our assertion about the function F(q) follows now from Theorems 4.28 and 4.29 and R e m a r k 4.30 applied to the two-component process (xt (q), q). The theorem is proved. An i m p o r t a n t particular case of this theorem occurs when Q = ~d, xo(q) = q, cr(t,x,q) = ~(t,x) and b(t,x,q) = b(t,x) so that we are dealing with solutions xt(x) of the equation x, = x +

/o

xs) dws +

fo

b(s,x

) ds.

Kolmogorov's equations

41

T h e o r e m 4.32. Assume that a(t, x) and b( t, x) are once (respectively, twice) continuously differentiable with respect to x for any w, t and

lla(t,O)ll + Ib(t,O)l + ila,~(t,x)ll + Ib~(t,x)l <_ K for any w, t, x, i. Let c(t, x), f (t, x), and g(x) be real-valued functions on ~2 • [0, T] • ~d. Assume that they are once (respectively, they and cr(t,x) and b(t, x) are twice) continuously differentiable with respect to x for any w, t and their absolute values together with the absolute values of their derivatives (respectively, including the second order derivatives) with respect to x are less than K(1 + [xl m) for any w, t. Finally let c(t, x) > 0 and let T be a stopping time, "r ~_ T. Then xt (x) is once (twice) s s with respect to x and the function F(x) := g

~0 ~ f(t,

xt(x))e- f~o c(~,z,(~))

ds

dt + Eg(x~(x))e

-

fo ~(~'x'(~))d~

is once (twice) continuously differentiable in ~d. In addition, for a constant N = N(d, m, K) and all x we have lF(x)l <_ NeNT(1 + IXlm), tF~(x)I <_ NeNT(1 + Ixl2m), (respectively, IF~(x)[ ~ NeNT(1 + IX[3m)).

(4.22)

Proof. We only need to prove the last assertion related to (4.22). The first inequality in (4.22) follows from Theorem 4.6. To prove the second one notice that for any f E I~d and it := E-(xt(x))(~) = xt(~)(x) we have

F(r

= Es = E(

f(t, xt(x))e- fo c(~'xs(x))dS dt+g(z~(x))e- fl c(8,x~(~))ds (~)

/o

[f(~,)(t, xt) - f(t, xt)

+ [g(~)(x~) - g ( x ~ )

/o

e(~)(s,x~) ds]e-fl ~(s'x~)ds dt

c(~)(s,x~)ds]e-fo~(~'~)ds),

(4.23)

where for simplicity the argument x is dropped. Hence [F(~) (x)l _< N(1 + T2)E sup ]it[ sup(1 + [xtl 2"~) t
t
_< N(1 + T2)(E sup [(t[2)l/2(E sup(1 + Ixtl4m)) 1/2 t<:T

To

t<:T

finish the proof of the second inequality in (4.22) it only remains to use Theorem 4.6 and Corollary 4.17, where Xo(~)(q) is to be replaced with fo -- f. While considering second order derivatives we use the fact that, for Vt = xt(~)(~) (x),

42

N. Krylov

F(~)(~)(x) = E (

/or[f(~)(t, xt) + f(~)(~)(t, x t ) - f(~,)(t, xt) /0 c(~s)(s,xs)ds -f(t, xt) ~0t {c(,l~)(s,x~) + c((~)((~)(s,x~)}ds

+

~0T

-

-g(xJ

{c(v.)(s,x j + c((~)((.)(s, xJ} ds

0T

~0T (4.24)

We estimate [F(~)(~)(x)l from above through N(1 + T 3)(Esup([~t[ + ](t[ 2) sup(1 + t
[xt[2m) + E s u p

t
[~t]2 sup(1 +

t
[xt[3m))

t
and then as above we use H51der's inequality, Theorem 4.6 and Corollary 4.17 along with Corollary 4.19, which in our situation says that (notice that ~0 = 0) E s u p I~tl2 _~ NeNT(1

t
+ IXl2m).

This easily gives the desired result. The theorem is proved.

5. K o l m o g o r o v ' s

equation

in the whole

space

Here we want to apply the results of Subsec. 4.5 in order to give conditions in terms of the given data which could allow us to "uncondition" the results of Sec. 3.. As in Sec. 3., we consider equation (2.1) with nonrandom coefficients assuming that the conditions stated before Remark 1.1 are satisfied. Our first goal is to give sufficient conditions for the regularity of (2.1).

Kolmogorov's equations

43

5.1 S t r a t i f i e d e q u a t i o n s In this subsection we prove a theorem which will be used in the following one for deriving more or less explicit conditions on a, b which are sufficient for (2.1) to be regular. It turns out that for further applications it is more useful to consider not equation (2.1) alone but append it by another equation. Definition 2.1 concerns (2.1) which, in fact, is a system of one-dimensional equations for each coordinate x~. Let us take a system for a (d + d0)-dimensional process zt = (xt, yt) whose first d coordinates we represent as xt and remaining do coordinates as Yr. We seek a process zt such that xt satisfies (2.1) and yt satisfies the equation

yt = y +

/0

5(s + r, zr) dwr +

/0

b(s + r, zr) dr, t >_ 0

(5.1)

For convenience we enumerate the following assumptions about 5 = 5(t, z) and b = b(t, z). A s s u m p t i o n 5.1. The functions ~, b are Borel functions defined on [0, c~) • I~d+d~ with values in the set of all do • dl-matrices and in I~d~ respectively. They are continuous with respect to z for every t >_ 0 and for every T, R > 0 ~0T sup [ll~(t,z)II 2 + Ib(t,z)l]dt < c~. Izl<_R

(5.2)

In other words, this assumption means that the coefficients of system (2.1), (5.1) considered as a single equation with respect to zt = (xt, Yt) satisfy the conditions assumed for the coefficients of (2.1). Therefore, Definition 2.1 makes sense for this system. A s s u m p t i o n 5.2. The couple (5, b) is monotone and coercive with respect to y. More precisely, for every R E [0, c~), on [0, co) there exists a positive Borel locally integrable function K t ( R ) such that for all t E [ 0 , ~ ) , x C ~d,y,y(1),y(2) C ]~d~ E [0,~), satisfying IxI ~_ Ro, ly(1)i, ly (~)] ~ R, it holds that

2(y (1) - y(~), ~(t, x, y(1)) _ ~(t, z, y(~)))

+ 115(t,x,y(1)) -5(t,x,y(2))ll

2 ~

Kt(/~v R0)ly(~) -y(2)l~,

2(y, ~(t, x, y)) + 115(t, x, y)ll 2 _< K~(R0)(1 + lyl~).

(5.3) (5.4)

Condition (5.3) is called the monotonicity condition whereas (5.4) is called the coercivity condition. T h e o r e m 5.3. If equation (2.1) is regular, then, under Assumptions 5.1 and 5.2, system (2.1), (5.1) is regular too.

44

N. Krylov

Proof. First we prove that system (2.1), (5.1) has a unique solution zt = z: 'z = (xt'X,y~ 'z) on [0, T] for any s > 0, z = (x,y) 9 ~d • ll~do, and T 9 [0, c~). For equation (2.1) this is given by definition, and the first d components of zt represent x~ 'x indeed. For equation (5.1) we get existence and uniqueness from Theorem 1.2 (with K t ( R V sup~
P{suplzt(?2)'~(?2)(I(n)) - z;(~)'~(~)[ _> E} = 0

V~ > 0, T 9 [0, oc).

t
Since these relations hold for xt in place of zt by Definition 2.1, we only need to prove that

sup ly~(~),z(~)(i(~)) - y~(~)'z(~)l s 0.

(5.5)

t(T

Define _

~

~ = ~(~)'~(~)(I(~)),

p7

-~

~ = x~ (~)'~(~),

Y,(n,t) - Yt ,

.s(~),~(n)

y~ = y~

Yt = Yt

(I(~)),

- qt ,

n -n q-'~ = ~0 t [5(s(n) + r, xr,y~(n)'z('~) ) - 5(s(n) + r,x,(,~,r),y~(n)'z(n))]dw~

+

t [~(s(~) + ~, ~ , v~ (~),z(n)) - ~(s(n) + . ,~ ~z~n o.s(n),z(~)~lj j ~,~. , (~,~), ~

fo

~(R):inf{t>0:lx~l+l~[>R}, 7~(R)

"/~(R)--inf{t>0:

> n+2},

i n f { t >- - 0 : l Y t- nl + l q t l- ? 2

r

"yn(n)

_

= exp(-2

----

~(~) ~(n) Yt ' ] > R}

n R 7~(R) A~/~(R) A~/2( ),

K~(~)+~(R) a~),

and notice that

d~'~ = 5(s(n) + t, x~(,~,t),-n ~t ~

dy t

~,~ + Pt-n)dwt + b(s(n) + t, x~(n,t),-'~

--?2

n

-n

~rt

Yt-?2+ p~) dt, ?2

-r~

~(s(n) + t, x~(,m), Yt + qt ) dwt + b(s(n) + t, ~(n,t), Yt + qt ) dt.

As in the proof of Theorem 1.2, we get that TA~/~ ( R )

lim E

f0

I~ldt=O

VR, T E [O, cc).

To prove similar relation for Iq~l, observe that, from the fact that x.~-~.n __+ 0 in C in probability (which holds by assumption) and x.~ - x.~'~ --+ 0 in C in probability (Lemma 2.3), we get that

Kolmogorov's equations

45

sup lx~ - 2~(,,~) t _< sup lx~ - x~(n,~) l + sup lx~(n,~) - 2~(n,~) I r
r
r
_< 2 sup Ixrn - XSr'~l + sup Ix s'x r
xS~ x ~(n,r)

+ sup Ix~ -- xr-n I --+ 0

r
r
in probability, which along with the continuity of ~(t, z) in z implies that, for s(n) + t <_ T, f tA'r~(R)

. ~,~ o.$(n),z(n)~,12n . II~(~(n) + ~,*~,y;(~)'~'~)) - ~(~(,~) + - , ~ ( ~ , ~ ) , ~ j,, ~ -

a0

_<

f

sup IlS(s,x~,y) - ~(s,2n(~,~),y)ll2 ds --+ 0 ~<_T@t<_R

in probability. It follows t h a t the stochastic integral in the definition of ~t tends to zero in probability. In the same way one treats the usual integral in the definition of q~ and one concludes that

(5.6)

qYA~(R) ~ 0 in C in probability for all R. Since ]q~l - R for 0 < t <

~/n(R),we get

f T A " / ~ (R)

lim E/o

Iq~ldt=0

VR, T E [ 0 , c~).

Hence, as in the proof of L e m m a 1.5 sup

]9~-y~[ P 0

VR, T E [0, cx~).

t
Owing to (5.6), we have proved (5.5) with T A ~/~(R) in place of T:

sup

Is - s

5 0

VR, T e [0, ~ ) .

(5.7)

t
Furthermore, for ~-~(R) := inf{t _> 0 : [9~1 -> R + 1}, we obviously have P{7~(R) _< ~-n(R) A T} <_ P{

sup

Iq-~l >- 1},

where the last probability tends to zero as n -+ cc again due to (5.6). Hence, (5.7) implies t h a t sup lY~ - Yt(n)'z('~)l ~" 0 t_
VR, T C [0, o~).

By the way, if ~-n(R) = T A ~3(R) A ~r(R) A ~n(R), then I s - s at t = ~-~(R). This and (5.8) yield sup lY~ - y~(n),z(n) I P 0 ~_
VR, r e [0, co),

(5.8)

> 1

46

N. Krylov

and it only remains to prove that lim

R-+oo

lira P{7~(R) A 7~(R) _< T} = 0.

n--+ oo

(5.9)

Coming back to (5.1), applying ItS's formula to lyt(")'z('~)12r ) with a fixed R0, and using (5.4), we get that for any stopping time T _< 7~(Ro) AT

Ely~(~)'~(~)12r

) <_ [y'~[2, T

Ely;(,),~(,)l 2 _< ]y.]2 exp(2 f0 Ks(n)+t(Ro) dt). Taking here 7 = ~3(R) A ~?(R0) A T and using Chebyshev's inequality, we get p

T

P { 7 ? ( R ) _< 73(Ro) A T} < R-2]ynl2 exp(2 Jo Ks(n)+t(Ro) dt). Therefore, the left-hand side of (5.9) is less than lim

lira P{7~(R0) A 7~(R) _< T} = lim P{7~(Ro) _< T}

R-+OO n-+oo

n - + (:x~

for any Ro. Now to get (5.9) remember that sequences converging in probability are bounded in probability. Therefore, the above stated properties of x~ and 2~ imply that lim

lim P{7~(Ro) <_ T} _< lim supP{sup[[x~[ + [2~[] > Ro} = 0.

Ro--+~x) n - + o o

Ro-+Oo

n

s(T

The theorem is proved. 5.2 Sufficient c o n d i t i o n s for r e g u l a r i t y First we give the following corollary of Theorem 5.3. C o r o l l a r y 5.4. Under the hypotheses stated before Definition 2.1 assume

that for every R >_ 0 there exists a Borel function Kt(R) defined and locally integrable on [0, oc) and such that 2(x - y, b(t, x) - b(t, y)) + Ha(t, x) - a(t, y)[[ 2 _< Kt(R)lx - y[2,

2(x,b(t,x)) + Ilcr(t,x)ll 2 _< Kt(1)(1 + [xl 2)

(5.10) (5.11)

ift k O, x,y E I~d, Ixl, lyl < R, R C [0, oc). Then equation (2.1) is regular. Indeed, it suffices to replace the system (2.1), (5.1) by a system which has the first equation with coefficients identically equal to zero and the second one coinciding with (2.1) (and to modify Kt(R) so that they would become increasing in R and satisfying the inequality Kt(R) > Kt(1) for all R > 0).

Kolmogorov's equations

47

C o r o l l a r y 5.5. Under the hypotheses stated before Definition 2.1 assume that there exists a constant K > 0 such that lb(t,x) - b(t,v)l + Ilo(t,x) -

(t,y)ll <_ K I x - vl

if t > O, x , y E IRd. Then equation (2.1) is regular. Indeed, in this case the left-hand side of (5.10) is less than 2 I x - y[ [b(t,x) - b(t,y)] + g21x - yl 2 < (2 + g ) K I x -

yl 2,

whereas the left-hand side of (5.11) is less than 2Ix I [b(t,x) - b(t, 0)] + 2Ix I Ib(t,O)l + 211cr(t,x) - a(t,0)ll 2 + 211a(t,0)lJ 2 _< 2Nix[ 2 + (1 + Ixle)lb(t,O)] + 2K2Ixl 2 + 2ila(t,0)ll 2 <_ (1 + ]xl2)(2K + 2K 2 + Ib(t,0)l + 21Ia(t, 0)112). Remark 5.6. Theorem 5.3 says that the property of equations to be regular is inherited while (2.1) is supplemented by equation (5.1) whose coefficients satisfy some conditions. It is obvious that we can add to system (2.1), (5.1) yet another equation, and if its coefficients satisfy conditions like (5.3) and (5.4), the new system, thus enlarged, will still be regular. In the future we will see quite a few applications of this remark. Remark 5. 7. If a, b satisfy the assumptions of Corollary 5.4 and (~, t) satisfy Assumptions 5.1 and 5.2, it may happen nevertheless that coefficients of system (2.1), (5.1) written as a single equation with respect to zt do not satisfy conditions like (5.10) and (5.11). An example of such a situation is given by the system of one-dimensional equations: dxt = xtdt, dyt = xtytdt. If we consider this system as one equation with respect to the two-dimensional process (xt,yt), then for instance the left-hand side of (5.11) is 2(xl) 2 + 2xl(x2) 2 and this function cannot be estimated from above by K(1 + Ixl 2) for all x = (x 1, x 2) E ~2. E x e r c i s e 5.8 ( m e t h o d o f n e w variables). Assume that on [0, ec)x R a we are given functions f ( t , x) and c(t, x). Assume that f, c are Borel functions with respect to t for every x and are continuous in x for every t _> 0, and for every T, R E (0, oc) ~0T sup (If(t,x)[ + Ic(t,x)])dt < oc. I~]
(5.12)

This just means that f, c satisfy our basic assumptions on a, b. Finally, assume that (2.1) is regular. Consider the system of equations consisting of (2.1) and the following equation

48

N. Krylov X d+l ---- X d + l -[-

C(S + r, xr) dr.

(5.13)

Observe that by virtue of (5.12) this equation makes sense and prove that this system is regular. Also add to system (2.1), (5.13) the following equation

x d+2 = x d+2 +

f ( s + r, xr) e x p { - x d+l} dr.

(5.14)

Prove that system (2.1), (5.13), (5.14) is regular with respect to ( t,..., 5.3 K o l m o g o r o v ' s e q u a t i o n We consider the equation

xt = x +

/0

a(s+r,x~)dw~+

/0

b(s+r,x~)dr,

(5.15)

with usual assumptions on dimensionalities of a, b. We assume that ~ and b are Borel functions on [0, oe) x I1{d satisfying II(r(t,x)ll + Ib(t,x)l <_ g ( l + Ixl), IIa(t,x)-a(t,x)ll+lb(t,x)-b(t,y)l

< KIx-y[,

(5.16)

where K is a fixed constant. By Corollary 5.5 equation (5.15) is regular. Assume that, on [0, oc) • I~d , we are also given real-valued Borel functions c(t, x) > 0, f(t, x), and g(x) satisfying

[c(t,x)[ + [f(t,x)] + [g(x)[ _< K ( I + [x['~), where m is a nonnegative constant. Define

~t'~ = ~0 t c(s + r , x /s x )dr, T--s

v(s,x) = Es,~[ ~ ~

f ( s + t, x t ) e - ~ dt + g(xT-s)e-~r-s],

where we use the standard stipulation about indices.

Remark 5.9. Clearly, ~t and v(t, x) are well defined and by Theorem 4.6 [v(t,x)[ ~ N ( K , m , T ) ( 1 + Ixl'~). Furthermore, if c, f , and g are continuous in x, Lemma 2.9 easily implies that the random variable

f ( s + t,x t' )e

, dt + g(xT_s)e vr-~

is continuous in probability as a function of (s,x). Theorem 4.6 provides locally uniform estimates of its moments and this implies that v is continuous in [0, T] • I~d if c, f , and g are continuous in x. In the general case v is Borel in [0, T] • ~g as follows from the above argument by well known measuretheoretic results.

Kolmogorov's equations

49

5.10. For any t 9 [0, T], let the functions a , b , c , f , and g be twice continuously differentiable with respect to x and let the absolute values of all their first and second order derivatives be less than K ( 1 + Ixl m) in [0, T] x ]~a. Then (i) for any t 9 [0, T] the function v(t, x) is twice continuously differentiable with respect to x and

Lemma

I =(t,x)l _< YeN (1 +

< NeNr(1 + Ixl

in [0, T] • I~d, where N is independent of t, x, T; (ii) for any x 9 ~d the function v(t, x) is Lipschitz with respect to t 9

[0,T], (iii) in (0, T) • I~d, the generalized derivative Dtv exists and lDtv I < _ N ( K , T , R )

in

QR

(a.e.)

VR;

(iv) in (O,T) • ~d, the function v satisfies Kolmogorov's equation Lv + f = 0

(5.17)

(a.e.),

where Lu := L u ( t , x ) = D t u ( t , x ) + aiJ(t,x)u~,=j(t,x) +bi(t, x)u~, (t, x) - c(t, x)u(t, x),

i * a = ~aa ,

Proof. Assertion (i) follows directly from T h e o r e m 4.32. To prove other assertions, we use Exercise 5.8, by which system (2.1), (5.13), (5.14) is regular. Define 2 = (x, x d+l, x d+2) = (x 1, ..., x d+2) and

~(2) = x d+2 + g(x)exp(--xd+l),

~(s,2) = E~,e~(2T-~),

where, of course, 2~ '~ is a solution of (2.1), (5.13), (5.14). Obviously,

9(s, 2) = x ~+2 + v(s, x) e x p ( - x d + l ) .

(5.18)

In particular, 5 is a smooth function of 2. By T h e o r e m 2.13, ~ is harmonic relative to x t' in any b o u n d e d subdomain of (0, T) • I~d+2 By T h e o r e m 3.5 and by smoothness of v in x, we get 5 9 BI'2((0, T) • {121 < R}), which proves (ii) and (iii) again owing to (5.18). Finally, applying T h e o r e m 3.6, we can write L~ : 0 (a.e.) in (0, T) • R d+2, where L is the operator corresponding to x-s,~ t . A very easy c o m p u t a t i o n shows t h a t the last equation amounts to (5.17) by virtue of (5.18). T h e l e m m a is proved. s,~ and ~t '~'~ -- xt(~)(~ "~'~ ) . Prove t h a t under E x e r c i s e 5.11. Let ~t '~'~ -- xt(~) ( s,x ~s,~,~ s,z,~ the conditions of L e m m a 5.10, the triplet ~xt , ~t ,fit ) is a solution of a regular system, and from formulas (4.23) and (4.24) and L e m m a 2.9 (i) derive t h a t v= and v== are continuous in (s, x), where v is the function from Lemma 5. i0.

50

N. Krylov

In the future we wilt need to view equation (5.17) in the sense of generalized functions. L e m m a 5.12. For any t e [0, T], let the function a be continuously differentiable with respect to x and for any finite R > 0 let a~ satisfy Lipschitz condition with respect to x on QR. Also let c, f , and g be continuous in x for any t E [0, T]. Then Kolmogorov's equation (5.i7) holds in the sense of distributions which means that for any ~1 C C ~ ( ( O , T ) x ]~d) we have fo T L

[vL*~? + f~?] dx dt = 0,

(5.19)

d

where L*~] := ( a i J ~ ) ~ i - (bi~?)~, - c77 - Dt~? = a '93 ~ j + (2a~j - bi)rl~, + ( aij~ j

- b~, - c ) ~ - Dt~.

Proof. First, use the fact that Lipschitz continuous functions are almost everywhere differentiable and have generalized derivatives. Then we see that the expressions and bi~ are well defined and L*r] makes sense and is bounded in QR for any R. Next, if the conditions of Lamina 5.10 were satisfied, we would get (5.19) multiplying (5.17) by 77 and integrating by parts. In the general situation we make the passage to the limit in (5.19). Take a nonnegative function r e C ~ (]~d) with unit integral and for 6 # 0 define {~(x) = g - d { ( x / 6 ) , u(~)(t, x) = u(t, ") * C~(')(x). Also let u (~ := u. For any g, define x t8~X~C as a solution of the equation

axixjiJ

xt = x +

/o

a(~)(s+r,x~)dw~+

f0

b(~)(s+r,x~)dr

and let =

v(s,x,g)----

E I f T-8 f Jo

/o

....

( s+t , X.... t ~) e - v ; .... dt+,~(~)(xS'~'~e-V~':':l ~ k T-s] J" (5.20)

It turns out that, v(s, x, 6) is a continuous function of (s, x, 6). Indeed, the equation d~t = 0 is regular. In addition, obviously for any 6, the functions is a (E) and b(c) satisfy (5.16). Hence by Theorem 5.3, the couple s,z,E, regular. Furthermore, c (C), f(~), and g(e) as functions of (s, x, 6), satisfy the growth condition imposed on c, f , g . Therefore, the continuity of v(s, x,6) follows as in Remark 5.9. In particular, v ( s , x , 6 ) --+ v(s,x,O) = v ( s , x ) as c -+ 0 uniformly on QR for any finite R. Next, by Lemma 5.10 (for each a ~ 0) we have

Kolmogorov's equations

fT f

JO

[v(E)L~/+ f(~)~l]dxdt:O,

51 (5.21)

JR d

where L~ is the operator corresponding to a (~), b(~), and c (~). We let ~ -+ 0 using the convergence v(e) = v(t, x, ~) --+ v(t, x), which is uniform on bounded sets. We also observe that (b(~))~ = (b~,) (~) -+ b~:, (a.e.), where b~ is Sobolev's derivative of b. This derivative is bounded so that (b (~))~ are uniformly bounded. The same is true for other coefficients of L~, for instance, 2(a~)x~J = (a(~)a(E)*)x~J = (a~J)(~)a(E)*+(a~,)(~)(a*j)(~)+(a~J)(~)(a;,)

(~)

+a(e)(a ~xj) (e) -+ a~xja* + a~,crxj * + axja~ + aax~j : 2ax~xJ. Now we obtain (5.19) from (5.21) by the dominated convergence theorem. The lemma is proved. Two main results of this section follow. As always we suppose that the assumptions stated in the beginning of this subsection are satisfied. T h e o r e m 5.13. Assume that

If(t,x) - f(t,y)l

]c(t,x) - c(t,y)[ + + Ig(x) - g(y)l _< Klx

-

yl(1 + Ixl

+ lyl TM)

(5.22)

for all t, x, y. Then (i) the function v(s, x) has first-order generalized derivatives in (0, T) • N d with respect to x. Furthermore, (a.e.) Iv~(s,x)l <_ NeN(T-s)(1 +

Ixl~),

(5.23)

where N = N ( K , m ) ; (ii) Kolmogorov's equation (5.17) holds in the sense of distributions which means that for any 77 C C ~ ( ( 0 , T) • Rd) L r ~ d [vDt~?+(aiJ~)~'v~j+v(biv)~'+~Tcv-

~?f] d x d t = O .

(5.24)

Proof. We introduce the same objects as in the proof of Lemma 5.12 but in (5.21) in the term containing the second-order derivatives of 77we integrate by parts to find

jo T

JR.

[v(g)Dt,+(aiJ77),,v,j(c)+v(~)(b(~)i~)~,

+77e(')v(~)-,f (~)] dxdt = O.

(5.25) Notice that, due to conditions (5.22) and (5.16), and Theorem 4.32, the functions v(t, x, c) are continuously differentiable with respect to x for any t E

52

N. Krylov

[0, T] and c # 0. Furthermore, vx(t,x,a) satisfies (5.23) with N = N ( K , m ) if 0 # lal ~ 1. In the proof of L e m m a 5.12 we have seen t h a t v(~) --+ v as -+ 0. This easily proves assertion (i) and also that v~ (~) --+ v~ weakly in L2(QR) for any finite R > 0. Furthermore, as in the proof of L e m m a 5.12 we have (a~J)x~ -~ a~ (a.e.) and all these functions are bounded by the same constant on QR for any finite R > 0. In particular, this convergence is strong in L2(QR) for any finite R > 0. It follows that

;J)~ v~j (e) ax at -+

va~, v~j dz, dt. d

Similar conclusion for other terms in (5.25) is much simpler to prove, so that letting c --+ 0 in (5.25) we get (5.24) and the theorem is proved. The following theorem generalizes L e m m a 5.12. R e m e m b e r the assumptions stated in the beginning of this subsection before R e m a r k 5.9. Theorem

5.14. Let the generalized function d

E

ij

(5.26)

ax~xJ

i,j=l

be a locally finite measure in (0, T) • ~d. Then Kolmogorov's equation (5.17) holds in the sense of distributions which means that for any ~7 E C~((O, T) • I~d) equation (5.19) holds. Proof. If in addition condition (5.22) is satisfied, then we can integrate by parts in (5.24) to get (5.19). However, notice that if before our generalized derivatives were bounded, so that we were dealing with absolutely continuous functions, this time we deal with measures and perhaps some additional explanation is needed. W h a t we actually need to do is to approximate v on the support of • by functions vn from C ~ ( ( 0 , T) • II~d) so t h a t

jR d

x9

= lim

dx dt rla~J,vndxdt = - lim

JO

JR d

n--~oo JO

(~la~vn., + ~l.,a,vn) d x d t JR d

= -

(rla~v ~, + rl~a~v ) dx dt. d

One can do this by taking mollifiers of v multiplied by a cut-off function since in this case on the support of rl: vn ~ v at any point, these functions are uniformly bounded, and vnx ~ v:~ (a.e.) and these functions are uniformly bounded too. Once we get (5.19) under condition (5.22), we fix rl, o, and b and consider the set ~ of all (c, f , g ) (satisfying the conditions from the beginning of

Kolmogorov's equations

53

the subsection with the same K,m) for which (5.19) holds. Notice that if (c~, f~, g~) E ~ and (e~, f,~, g,0 --+ (c, f, g) pointwise, then, obviously, for v~ constructed on the basis of c~, fn,g~ we have Vn --+ V pointwise and all the functions cn, fn, gn, vn are uniformly bounded on the support of ~. This allows us to use the dominated convergence theorem and prove that (c, f , g ) E ff~ as well. This property of (5.19) and a standard measure-theoretic argument (see, for instance, Theorem 19, Sec. 2, Ch. 1 of [M]) finishes the proof of the theorem.

6. S o m e

integral

approximations

of differential

operators

Assume the conditions and notation from Subsec. 5.3. Here we start preparing an important tool which will allow us to derive Kolmogorov's equations for generally nonsmooth functions which are harmonic in domains. Let Q be a subdomain of (0, T) • ~d. For Borel functions h(t,x) and g(t, x) which are defined in Q and 0Q, respectively, and A _> 0 denote

R~ (Q)h(s, x) = Es,~ R(Q) = Ro(Q),

e-~-Xth(s + t, xt) dt, R~ = R~((0, T) • ~d), g(s +

=

=

where T = ~-Q = inf{t > 0: (s + t, xt) !~ Q}A useful point of view at R~(Q)h and ~ ( Q ) g is the following. Take a random variable 7, which is exponentially distributed with parameter 1 and independent of w.. Then, for A > 0, the variable "//A is exponentially distributed with parameter A. By using this, one proves easily that

AR~(V)h(s, x) = Es,~e-~/~h(s + ~//A, x.r/~)I.~/~<~, 7rx(Q)g(s, x) = Es,xe-~r g(s + T, x~)I~/~>T, AR~(Q)h(s, x) + ~;~(Q)h(s, x) = Es,~e-~(~/~)^~h(s + (~//A) A T, x(.y/~)A~) Since 7/A ~ 0 as A -+ c~, it is natural that )~R~,(Q)h --4 h under reasonable assumptions on h. Also Lemma 3.3 shows that )~n~(Q)g(s, x) --+ 0 for any n. Furthermore, if It5's formula is applicable, then

)~Rx(Q)h(s,x) + ~r~(Q)h(s,x) = h(s,x) + gs,~

e - ~ L h ( s + t, xt)dt dO

= h(s, x) + )~E~,~

fo ~

e -At

ftA~ JO

e - ~ L h ( s + r, x~) drdt

54

N. Krylov = h(s, x) + Es,~

//

e - ~ " - ~ r L h ( s + r, xr) dr : h(s, x) + R~ (Q)Lh(s, x).

Thus A(AR~(Q)h - h) = A R ~ ( Q ) L h - l~r~(Q)h, which m a k e s it n a t u r a l t h a t if Lh satisfies "reasonable" a s s u m p t i o n s so t h a t )~R~(Q)Lh --+ Lh, then A(

R

(Q)h - h)

Lh

as )~ -+ cx~ and leads us to "integral" a p p r o x i m a t i o n s of the o p e r a t o r L. In w h a t follows we justify the above conclusions. L e m m a 6.1. Let g(t, x) be a bounded Borel function defined on OQ. Then for any n >_ 0 we have AnTr~(Q)g -+ 0 as A ~ oo uniformly on any closed bounded subset of Q. Proof. Obviously, we m a y a s s u m e g _= 1. Take a 5 E (0, 1) and let Q5 be the set of all points (t, x) E Q such t h a t Ixl < 5 -1 and for the closest to (t, x) point ( s , y ) on the b o u n d a r y of Q it holds t h a t It - s I > 5 and Ix - Yl > 5. T h e n by L e m m a 3.3 for any ~ _< 5 7rx(Q)l(s,x) < Es ~ e - ~ I ~ < ~ + e - ~

< Ps,~{sup ]x~ - x] > 5} + e - ~

< N c 3n -k e -)~e with N depending only on 5, n and K ( 5 + 5-1). For E = ~-1/2 and ;~ > 5 -2 we find AnTr~(Q)l(t,x) <_ N A - n / 2 + )~ne-A1/2 , which shows t h a t A~Tr~(Q)I -+ 0 uniformly in Qs. T h e arbitrariness of 5 proves t h a t A'~rx(Q)l --+ 0 uniformly in any closed c o m p a c t subset of Q. T h e l e m m a is proved. L e r n m a 6.2. Let h be a bounded Borel function defined in Q. (i) Let Q' be a domain, F be a closed bounded set and let F C Q' c Q. Then for any n > 0 lim A~ sup IR~(Q')h - R~(Q)h[ = O. )~ ---~ cx~

F

(ii) A s s u m e that either (a) generalized function (5.26) is a locally finite measure on Q, or (b) the function h(t, x) is continuous in x for any t. Then AR~(Q)h -+ h (6.1) as A -+ oo in the sense of generalized functions on Q.

Kolmogorov's equations

55

Proof. Assertion (i) follows from L e m m a 6.1 and the formula

IRx(Q)h(s, x) - R~(Q')h(s, x)l -- Es,xe-cP~'Q' --A'rQ'

f~

h(s+t, xt)exp{-

Q,

f

c(s+r, x r ) d r - A ( t - ' r Q , ) } d t I Q,

_ T s u p IhiEs,~e-~TQ ' - ~ Q ' = T s u p IhlTr~(Q')l(s, x). To prove assertion (ii) first of all notice that proving that AR:~(Q)h -+ h in the sense of generalized functions on Q is equivalent to proving that this convergence holds in the sense of generalized functions on any subdomain of Q. Therefore, we need only to prove t h a t (6.1) holds in the sense of generalized functions on Q1 for any cylinder Q1 c Q1 c Q. By assertion (i) it suffices to show t h a t AR~(Q1)h --+ h (6.2) in the sense of generalized functions on Q1To proceed further assume condition (a) and notice that R~(Q1)h is uniquely determined by the values of o, b, c, and h in Q1 and condition (a) bears on Q. It follows that by multiplying or, b, c, and h by a cut-off function we will not change R~ (Q1)h and will have condition (a) satisfied in (0, T) • IRd . Now we make another twist in the above argument concluding that instead of proving (6.2) under condition (a) it suffices to prove (6.1) in the case Q = (0, T) • ~d assuming (a) for this domain Q. In this situation by Theorem 5.14 we have ARch - h = LR~h and obviously [R~h[ _< A-1 sup ]h[ ~ 0 uniformly on Q. The latter implies that LRxh -+ 0 in the sense of generalized functions on Q. We have proved (6.1) under condition (a). We will prove t h a t under condition (b) the convergence in (6.1) is even stronger, namely, in Lp for any p on any closed bounded subset of Q. In the same way as above we see that while proving this we may assume t h a t Q = (0, T) • ]~d and c is bounded. Also, the functions ARch are uniformly bounded, so that to prove the convergence in Lp on any closed bounded subset of Q it suffices to prove that (6.1) holds almost everywhere, for instance, at any point (s, x) such that s is a Lebesgue point of the functions h(t, x) and sup

y:l,-yl<_l/~

]h(t,x) -

h(t,y)l

as functions of t, where n -- 1, 2, .... Denote 6) the set of all such points (s, x). Observe, that if s is a Lebesgue point of an integrable function g(t) given on (0, T), then f ( t ) := t -1

Ig(s + r) - g(s)l dr

is a bounded continuous function on (0, T - s) which tends to zero as t $ 0 so t h a t by the dominated convergence theorem

56

N. Krylov

A

~0T-sIg(s+t)

- g ( s ) l e - ~ t d t = A2

~oT--S(~otI g ( s + r ) - g ( s ) l d r ) e - ~ d t =

/?

Itl.X
as A --+ c~. Bearing in mind this observation, take (s, x) E O and an integer n _> 1 and write

Rxh(s, x) = Es,~ ~0 T-s (e - ~ - 1)e-~th(s + t, xt) dt

/o

+E~,~ Here

e-~t[h(s+t, x t ) - h ( s + t , x)] dt+

I h l <_ N

/o

~0 T-s te-:~t dt < NA -2, P T-s

lim AI3 = h(s, x) lim A ] A -...+o o

h(s+t, x)e -~t dt = h + 6 + / 3 .

A --)"cx)

e -:~t dt = h(s, x),

J O

Ihl G fO T-s e-;'*G,~{Ix,- xl ~_ 1/n} dt + ~o

e-At

sup Ih(s+t,y)-h(s+t,x)ldt. y:l~-yl<_l/n

Furthermore, Ps,~{Ixt - x I >_ 1/n} <_ Nt, where N is independent of A. Hence lim

Alhl

X-.+oo

< lim A

r/r-~e_~ y:Jx--yl<_l/n sup Ih(s+t,y)-h(s+t,x)ldt

-- ~--+oo JO

--

sup

y:lx-yi
Ih(s,y)- h(s,x)l.

The last expression tends to zero as n -+ co. This along with the above equations implies that lim ARch(s, x) = h(s, x) and the l e m m a is proved. A---~ o o

The following theorem is the main result of this section. 6.3. Let h(t,x) be a bounded Borel functions defined in Q. Assume that either (i) the generalized function (5.26) is a locally integrable function on Q,

Theorem

or (ii) for any finite R > 0 there exists a constant N such that Ih(t,x)-h(t,y)l~glx-Yl,

Ic(t,x)-c(t,y)l<_glx-y

I

(6.3)

for all (t, x), (t, y) E Q fq QR. Then A(AR~(Q)h - h) -+ Lh as A --+ oo in the sense of generalized functions on Q.

(6.4)

Kolmogorov's equations

57

Proof. First, assume condition (i). Similarly to our arguments before and after (6.2) it suffices to consider the case Q = (0, T) • I~d. In this case by Theorem 5.14 we have

A(AR~h- h) = L(.~R~,h).

(6.5)

By definition for any 77 E C ~ (Q)

/Q~L()~R~h) dxdt = fQr

(6.6)

where r = L*~ is an integrable function with compact support lying in Q. Observe again that ~R~h are bounded uniformly in Q by 2sup ]h I and for any ~ E C~r (Q) write lim ~-4 oo

]I r

- h) dx dt[

JQ

___2supIh, f [ ~ - ~ , d x d t +

JQ

lira

f ~ ( A R ~ h - h ) dxdt. JQ

(6.7)

The latter limit is zero by Lemma 6.2. The first term on the right can be made arbitrary small by choice of ~. Hence,

lim /Q~R~hdxdt=/Q~hdxdt=/Q~Lhdxdt,

~--+ o o

which means that L(i~R~h) --+ Lb. Coming back to (6.5) we get our assertion under condition (i) and the assumption that Q = (0, T) • I~d. Under condition (ii) we follow the same scheme which shows that we only need to consider the case Q = (0, T) • ]~d and, in addition, we may assume that (6.3) holds with the same constant N for M1 t E [0, T] and x, y E I~d. In this case we can start as above. We have (6.5) again, this time by Theorem 5.13. However, now instead of (6.6) we have

/Q 7?L(~R~h) dx dt =/Q ~AR~h dx dt - /Q )~(R~h)x~i dx dt, where = -Dry

- (b~v)~, - c v ,

r

= (a%)~

are integrable functions with compact support lying in Q. Now the estimate ;qR~h I < sup 1hi, Lemma 6.2, and formulas like (6.7) imply that to finish the proof it suffices to show that for any R > 0 there is a constant N such that for all A > 1 on QR (a.e.)

;~l(R~h)xl < N.

In the same way as in the proof of Theorem 5.13 it suffices to prove that for any R > 0 there is a constant N such that for all )~ >_ 1 and 0 < I~1 _< 1 on QR

58

N. Krylov

4 ~ ( s , x , 4 1 _< N,

(6.8)

where v is taken from (5.20) with g _= 0 and c + A, h in place of c, f , respectively. To prove (6.8) we repeat some arguments from the proof of Theorem 4.32. By (4.23) we have

~(r

= E ~ fO T - s ~#~) (s + t , ~ ( 4 ) ,/

_ h*/(~ + t, x~(4)

< NE~,~ sup [~t[

t
~0T-s (1 + t)e -xt dt < 2NA-1Es,~

sup I~tl,

t
where the constant N depends only on h and c and (t is a solution of the equation

/0

t (~)(s+r, xr(e))dw~+

/o I;

b ))(s+r,x~(e))dr.

To finish the proof of (6.8) it only remains to use the same argument as in the proof of T h e o r e m 4.32 after (4.23). The theorem is proved.

Remark 6.~,. Observe that whenever (6.4) holds in the sense of generalized functions on Q, AR~(Q)h --~ h in the sense of generalized functions on Q. Remark 6.5. We need condition (5.16) satisfied only in Q.

7. K o l m o g o r o v ' s

equations

in domains

We continue investigating the objects from Subsec. 5.3 and Sec. 6.. Thus we consider the equation

xt = x +

/0

a(s + r, xT) dwr +

/0

b(s + r, xr) dr,

with usual assumptions on dimensionalities of a, b. We assume that a and b are Borel functions on [0, oc) x ]~d satisfying

IIo-(t, ~)ll + Ib(t, :~)l < K(1 + 14), ]la(t,x) - ~(t,x)[[ + Ib(t,x) - b(t,y)[ < K[x - y[, where K is a fixed constant.

Kolmogorov's equations

59

7.1. Assume that generalized function (5.26) is a locally finite measure on Q. Assume that we are given bounded Borel functions h(t, x) and g(t,x) defined in Q and 3Q, respectively. Then the functions ul := R(Q)h and u2 := 7r(Q)g satisfy Theorem

Lul + h = O,

Lu2 -- O

(7.1)

in the sense of generalized functions on Q. Proof. If (5.26) were a locally integrable function in Q, one could give quite a short proof of the theorem. To show this observe that if A > 0 and 7 is a r a n d o m variable exponentially distributed with p a r a m e t e r A and independent of w., then by the Markov property, we have for (s, x) C Q and r < t t h a t

E{It<~e-~'h(s+t, xt) dtlhr} = e-~"Es+r,~.It-~<~e-~'-'h(s+t, xt-r)

(a.s.),

which implies

'yAl-

ul(s,x) =Es,z

fJO

LT

e - ~ h ( s + t , xt)dt+Es,xIr>~

e - ~ ' h ( s + t , xt) dt

= E~,~ fo ~ Ae - ~ f r A ~ e -e~ h(s + t, xt) dtdr r

+

/o

Ae -x~

/?

Es,~I~>~E{lt<.e-~~

1"

= E~,~ +

Ae -x~

e-~~

+ t, x~)lT~} dtdr

+ t, xt) dt

E~,~I~>re-~~

-=-Ra(Q)h(s, x)+Es,x

//

Ae - ~ - ~

+ t, xt-r) dtdr

/o

E~+r,~Ip<~e-e'h(s+r+p, Xp) dpdr

= R~(Q)h(s, x) + AR:~(Q)R(Q)h(s, x). In this way we arrive at the Hilbert identity

R(Q) = R~(Q) + AR:~(Q)R(Q). Similarly, 7r(Q) = zr~(Q) + AR~(Q)Tc(Q), so that

A(ARx(Q)ul - ul) = -AR~(Q)h,

A(AR:~(Q)u2 - u2) = -ATr~(Q)u2.

Hence (7.1) follows from Theorem 6.3 and Lemmas 6.1 and 6.2. In the general case take an increasing sequence of bounded domains Qn c Qn c Q such that Q = u n Q n and take functions ~n E C~((O,T) x ][~d) SO t h a t ~n = 1 outside Qn+l, 0 < ~ < 1, and ~,, = 0 on Qn. Also continue h outside Q as zero and for k > 1 define

60

N. Krylov uk~ (s, x) = E~,~

f0 T-s

e-~,-k~,

h(s + t, xt) dt,

where %ont := f0 t ~n(s + r, xr) dr. By T h e o r e m 5.14 Luk,~ - k(nuk,~ + h = 0 in the sense of generalized functions on (0, T) • I~d. In particular, Luk,~ + h = 0 in Q~. Also, obviously the functions uk~ are uniformly bounded on any compact subset of Q. It follows t h a t to prove the first equality in (7.1) we only need to prove t h a t ukn --+ u at any point (s, x) E Q as first k -+ 0o and then

n -+ oo. While proving t h a t uk~ --+ u we may an will assume t h a t h > 0. In this case, obviously, ukn decrease as k increase and Ukn >_ R ( Q n ) h . In addition observe t h a t T~ := 7Q~ ~ "rQ. Hence, R ( Q ~ ) h ~ u,

lira lim uk~ >_ lim R ( Q ~ ) h = u.

r~--+ (~ k--+ oo

n ---~oo

(7.2)

On the other hand, by general form of the strong Markov property

~k~(8,x) ; Es,~{

f

Tn§

e-~*-k~"h(s + t,x~)at

JO

<_ R ( Q ~ + l ) h ( s , x) + NEs,~ exp(-k~o,~§ where N = T sup h _> sup Uk~- Here lim E~ ~ exp(-k~n,~+~) = P,,~{~on,~+~ = 0}.

k--+oO

The last probability is zero if ( s , x ) E Q,~+I, since 0 < ~-~+1 < T - s for any and for r close to T~+I we have ~(s + r, x~) > 0. This and (7.2) prove that uk~ --+ u and finish the proof of the first equality in (7.1). A standard measure-theoretic argument shows t h a t in the proof of the second equality in (7.1) we m a y confine ourselves to the case of g which are twice continuously differentiable in (t, x) and have bounded derivatives of the first and second order. In that case by ItS's formula u2 = r ( Q ) g =g + R ( Q ) ( D t g + Lg), so t h a t by the above result L(u2 - g) + Lg = 0, which coincides with the second formula in (7.1). The theorem is proved. For elliptic equations or more generally for the case T = cx~ one has the following result.

Kolmogorov's equations

61

7.2. Assume that generalized function (5.26) is a locally finite measure on (0, oc) x D, where D is a domain in ~d. Assume that we are given bounded Borel functions h(t, x) and g(t, x) defined in (0, co) x D and OD, respectively. Also assume that the function Theorem

u i ( s , x ) : = Es,~

'0 T

e-V~h(s + t, xt)dt,

where T = inf{t > 0 : xt ~ D}, is well defined and locally bounded in (0, co) z D and let u2(s,

g(s +

) :=

Then ui and u2 satisfy again (7.i) in the sense of generalized functions on (0, co) x n . To derive this t h e o r e m from T h e o r e m 7.1 it suffices to notice t h a t by the strong Markov property, for any T E (0, co), b o u n d e d d o m a i n G C G C D, and (s,x) E Q : = (0, T) x G we have

ul = R(Q)h + ~r(Q)ui,

u2 = lr(Q)u2.

In general one c a n n o t expect much more t h a n stated in T h e o r e m s 7.1 or 7.2. E x e r c i s e 7.3. Let d = 2, di = 1, D = B2 \ / ~ i , dx~ = dwt, dx 2 = O,

where T(x) is the first exit time from D of the t i m e - h o m o g e n e o u s process xt starting at x and g(x) = Ix] - 1. By T h e o r e m 7.2 U~ITI

---- 0

in (0, co) x D or in D in the sense of generalized functions. Show that, for any x 2 satisfying Ix21 < 2, the function u(x i, x 2) is indeed (piecewise) linear with respect to x i b u t is discontinuous as a function of x on the lines x 2 = • Finally, we give an extremely i m p o r t a n t extension of T h e o r e m 7.1 to a situation occurring in applications to controlled diffusion processes where we are more interested in cases when K o l m o g o r o v ' s equation becomes an inequality. One uses inequalities like (7.4) when it is known t h a t the second order derivatives of u are b o u n d e d from below in the sense of generalized functions and one wants to estimate t h e m from above. T h e idea is as follows. If for a function u of two variables (x, y) we have uzz, Uyy ~__ O, then from the inequality like uxx + uy v < 1 one gets t h a t u ~ ~ 1,Uvy < 1, SO t h a t l u ~ ] < 1, ]UyyI < 1. For more details see [KR89].

62

N. Krylov

7.4. Let u(t, x) be a bounded Borel functions defined in a bounded domain Q c (0, T) • ll~d. Assume that either (i) the generalized function (5.25) is a locally integrable function on Q,

Theorem

or

(ii) the function f(t, x) is continuous in x and there exists a constant N such that ]u(t,x) - u ( t , y ) ] <_ N I x - yl,

Ic(t,x) -c(t,y)] ~_ N [ x - y]

for all (t, x), (t, y) E Q. Finally, assume that for any (s, x) C Q u(s,x) > Es,x[

f

~'At

f ( s + r , xr)e - ~ dr+u(s+TAt, x~At)e-~^~]+o(t)

(7.3)

JO

as t $ O, where o(t) is uniform for (s, x) in any closed subset of Q. Then Lu + f <_ 0

(7.4)

in the sense of generalized functions on Q. Proof. Define g(s, x, t) = o(t)/t. It is convenient to assume t h a t r x, t) is defined for all t > 0. F u r t h e r m o r e , without loss of generality we assume t h a t e(s, x, t) = ~(s, x, T - s) for t _> T - s. This is possible since T < T - s so t h a t the first t e r m on the right in (7.3) does not change for t _> T - s and the l e f t - h a n d side does not depend on t at all. Also ]o(t)[ need not be larger t h a n sup over (s, x) 6 Q and t < T - s of the absolute value of the difference of the l e f t - h a n d side and the first t e r m on the right in (7.3). Hence, we m a y assume t h a t Io(t)[ is bounded, and since e(s,x,t) -+ 0 as t $ 0 uniformly on any closed subset of Q, we m a y also assume t h a t for (s, x) in such a set, ~(s, x, t) is b o u n d e d by a constant independent of t. T h e n by the d o m i n a t e d convergence t h e o r e m

Ix(s,x) := X2

/0

te(s,x,t)e-Xt dt =

/0

e ( s , z , t / X ) t e - t dt ~ 0

as )~ --+ oo uniformly on any closed subset of Q or in the sense of generalized functions on Q. After this p r e p a r a t i o n s m u l t i p l y (7.3) by ~e -)'t and integrate over (0, oo). T h e n by Fubini's t h e o r e m we easily get

u(s, x) >_ R~(Q)(f +/~u)(s, x) + 7r~(Q)u(s, x) + A-1I~(s, x). It follows t h a t in the sense of generalized functions on Q lim A ( A R ~ ( Q ) u - u) < -

lim A R ~ ( Q ) f -

lira ATr~(Q)u

and it only remains to use T h e o r e m 6.3 and L e m m a s 6.2 and 6.1. T h e t h e o r e m is proved.

Kolmogorov's equations

63

R e m a r k 7.5. Take a bounded Borel function g(t, x) and define u = R ( Q ) f + 7r(Q)g. Then under assumption (ii) of Theorem 7.4 we have L u + f = 0 in Q since by the strong Markov property we have equality in (7.3) with o(t) - O. The same is true under condition (i), however this is a weaker statement than T h e o r e m 7.1.

References [KO] A.N. Kolmogorov, (]ber die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., Vol. 104 (1931), 415-458. [KR89] N.V. Krylov, Smoothness of the value function for a controlled diffusion process in a domain, Izvestija Akademii Nauk SSSR, serija matematicheskaja, Vol. 53, No. 1 (1989), 66-96 in Russian; English translation: Russian Acad. Sci. Izv. Math., Vol. 34, No. 1 (1990), 65-96. [KR92] N.V. Krylov, On first quasiderivatives of solutions of It6's stochastic equations, Izvestija Akademii Nauk SSSR, serija matematicheskaja, Vol. 56, No. 2 (1992), 398-426 in Russian; English translation: Russian Acad. Sci. Izv. Math., Vol. 40, No. 2 (1992), 377-403. [KR] N.V. Krylov "Nonlinear elliptic and parabolic equations of second order", Nauka, Moscow, 1985 in Russian; English translation: Reidel, Dordrecht, 1987. [KR1] N.V. Krylov and B.L. Rozovsky, Stochastic evolution equations, "Itogy nauki i tekhniki", Vol. 14, VINITI, Moscow, 1979, 71-146 in Russian; English translation in J. Soviet Math., Vol. 16, No. 4 (1981), 1233-1277. [M] P. A. Meyer, "Probability and potentials", Blaisdell Publishing Company, A Division of Ginn and Company, Waltham, Massachusetts, Toronto, London, 1966.

L P - a n a l y s i s of finite and infinite d i m e n s i o n a l diffusion o p e r a t o r s Michael R5ckner Fakult~t fiir Mathematik Universit~t Bielefeld Postfach 10 01 31 D-33501 Bielefeld Germany

1. Introduction The purpose of these lectures is to present an approach to Kolmogorov equations in infinite dimensions which is based on an LP(#)-analysis of the corresponding diffusion operators w.r.t, suitably chosen measures #. More precisely, we solve

du d--[ = L u , u(O, .) = f

(1.1)

for functions f, u(t, .) on an infinite dimensional (vector) space E and diffusion operators L (cf. Definition 3.1 below) by a C~

u(t, .) = Ttf generated by some closed extension L of L on LP(E; #) for some appropriate measure #. Here we think of L as a-priori only given explicitly on a set D of nice (smooth) functions. In principle there are at least two types of measures that appear appropriate in applications: (1) A given reference (e.g. Gaussian) measure p so that L can be written as a sum of an operator L0, which is symmetric on L2(E; #), and a small perturbation in the sense of sectorial forms. (This case is refered to in the text below as the "sectorial case"). (2) # is an invariant measure for L (or more generally for a - L, a ~ 0) in the sense that

flu

d#

0 for all

U

ED

9

(This situation we refer to below as the "non-sectorial case"). So, in case (2) the first problem is to find such a measure #. The second problem is then to construct a closed extension L of L on LP(E;#) that generates a C~ on LP(E; #) ("existence problem"), and the third is to prove that it is the only such extension ("uniqueness problem").

AMS Subject Classification (1991) Primary: 60H30, 35K 15 Secondary: 31C25, 35 K 10, 6O J 60

Key words and phrases. Kolmogorov equations, diffusion operators, sectorial forms, Dirichlet forms, strongly continuous semi-groups, martingale problem

66

M. RSckner

In these notes we address all three problems presenting results obtained over the last few years in implementing this program. For the precise structure of the corresponding parts of the text, i.e. Sections 2. - 5. we refer to the list of contents. We would only like to point out here that we also include results on properties as "positivity preserving", "(sub-)Markovian" of the C ~ solving (1.1), in regard to their probabilistic relevance as transition functions of Markov processes. The latter is made precise in Section 6.. In Section 7. we present a case study where E is an infinite dimensional manifold rather t h a n a vector space. In particular, we discuss ergodic properties of the above C~ in this situation. The main purpose of these lectures is to give an updated and an as complete as possible presentation of what has been achieved so far and what the aims are, rather than to repeat detailed proofs. We complement instead arguments needed to derive the statements given here from those recent results, giving precise reference for the latter. The material presented in these lectures is taken from a number of joint papers with various co-authors as well as the recent work of Andreas Eberle and Wilhelm Stannat. [ would like to thank the latter two as well as all my c o authors, in particular Sergio Albeverio, Vladimir Bogachev, Yuri Kondratiev, Vitali Liskevich, Tatjana Tsikalenko and Tu-Sheng Zhang for permission to include very recent, so far unpublished joint results. Finally, it is a great pleasure to thank C.I.M.E. and, in particular, the organizer of this session, Giuseppe Da Prato, for a very stimulating summer school and a fantastic time in Cetraro. I would also like to thank my two colleague lecturers Nick Krylov and Jerzy Zabczyk for teaching me so much through their lectures and intensive discussion. I also would like to express my deep thanks to all other participants for their steady interest in the subject and for creating a really marvellous atmosphere of a type I had never experienced before.

2. Solution of Kolmogorov equations via sectorial forms 2.1

Preliminaries

In this subsection we recall some well-known classical results. We assume the reader to be familiar with the notion of a C ~ (Tt)t>_o resp. its generator (L, D(L)) on a Banach space (cfl e.g. [Pa 85]). We only recall here t h a t if u E D(L) then Ttu E D(L) for all t > 0 and d d~ Ttu = LTtu = TtLu for all t _> 0 .

(2.1)

Below let (B, II " II) be a real Banach space with dual B* and corresponding dualization ( , ). Let L with domain D be a densely defined linear operator on B.

LP-analysis of finite and infinite dimensional diffusion operators

67

D e f i n i t i o n 2.1. (L, D) is called dissipative, if for all u E D there exists u* E B* such that

(u*,u) = Ilul[~ = Ilu*l[~ and (u*,Lu) < 0

(2.2)

or, equivalently, (cf. [Pa 85, Ch. I, Thm. 4.2])

lieu- Lull >_ al[ul[

for all c~ > 0, u 9 D .

Remark 2.2. If (L, D) is dissipative, then it is closable. Below we shall denote its closure by (L, D) or shortly L. T h e o r e m 2.3. (Lumer/Fhillips) Suppose ( L , D ) is dissipative. Then-L is

the generator of a C~ of contractions (i.e., the operator norms are bounded by 1) on B if and only if (a - L)(D) is dense in B for some (or equivalently all) a > O. Let L* : D(L*) C B* --+ B* denote the adjoint of (L,D). Then as a consequence of Theorem 2.3 and the Hahn-Banach theorem one immediately obtains. C o r o l l a r y 2.4. (1 st adjoint criterion) Let ( L, D) be dissipative. Then L generates a C~ of contractions on B if and only if ker(a - L*) -- {0}

for some (or equivalently all) a > O. This in turn obviously implies: C o r o l l a r y 2.5. (2 nd adjoint criterion) Let both (L, D) and (L*, D(L*)) be

dissipative. Then -L generates a C ~

of contractions on B.

Later we shall also use the following "uniqueness result": T h e o r e m 2.6. Suppose there exists a C~ (Tt)t>o on B such that its generator extends (L, D). Then the following assertions are equivalent:

(i) D is a core for the generator of (T~)t>_o (i.e., D is dense in the domain of the generator w.r.t, the norm induced by its graph norm). (ii) The closure -L of (L, D) generates a C ~ (iii) (Tt)~>_o is the only C~ on B whose generator extends (L, D). If (Tt)t>_o consists of contractions and if c > O, then (i) - (iii) is equivalent to (iii') (Tt)~>_o is the only C~ on B such that its generator extends ( L , D ) and [[Ttl] _< e E~ for all t >_ O. Proof. See [Eb 98, Ch. I, Thm. 1.2] and JAr 86, A - II, Thm. 1.33].

[]

A sufficient condition for uniqueness (which will, however, turn out to be typically not applicable to cases of interest in these lectures) is the following:

68

M. Rhckner

P r o p o s i t i o n 2.7. Let (T~)t>o be a C~ on B whose generator extends (L, D). If there exists a dense subspace Bo C B such that Tt(Bo) C D for all t > O, then (Tt)t>_o is the only C ~ whose generator extends (L,D).

Proof. See e.g. [EthKur 86, Ch. 1, Prop. 3.3].

[]

2.2 S e c t o r i a l f o r m s . The terminology w.r.t. "sectorial forms" used in these lectures is according to [MR 92]. Below we fix a Hilbert space ( n , ( , ) ) and set I1" II := ( , ) 1 / 2 . D e f i n i t i o n 2.8. A bilinear form C : D(E) • D(C) -+ ]R with dense domain D(s C 7/is called a (closed) coercive form if it satisfies the following conditions: (i) Its symmetric part (~, D(~)) defined by ~(u, v) := 1 (E(u, v) + E(v, u)); u, v E D(~) := D(E), is positive definite and closed (i.e., D(E) is com,al/2 plete with respect to the norm ~1 coming from the symmetric part of C~(u,v) := E ( u , v ) + a ( u , v ) ; a > 0; u,v e D(E) for a := 1). (ii) (E, D(E)) satisfies the weak sector condition, i.e., for one (and hence all) a C (0, oo) there exists Ka E (0, oo) such that

fEa(u, v)l < Ka E~(u, u) ~/2 Ea(v, v) t/2 for all u, v C D(E) .

(2.3)

We shall use (2.3) below almost always with a := 1 and set K := K1.

Remark 2.9. (i) If (E, D(E)) is a coercive form on 7/, then so is (E*,D(E*)) defined by C*(u,v) := E(v,u) ; u,v 9 D(E*) := D(~) .

(2.4)

The same holds for (6, D(~)). (ii) The antisymmetric part (~, DE)) of a bilinear form (E, D(E)) on 7-I is defined by E(u,v) := ~l(E(u,v) - E ( v , u ) ) ," u,v e D ( g ) : = D(E) . it is easy to see that if (C, D(~)) is positive definite, then (E, D(E)) satisfies the weak sector condition if and only if for some (hence all) c~ 9 (0, oo) there exists Ca 9 (0, oo) such that < ca

&(v,v) */2 for all u,v 9 D ( g ) .

A coercive form is always associated with a linear operator which generates a C~ of contractions on 7/. We restate and reprove this fact here (cf. e.g. [MR 92, Proposition 2.16]), which is an easy consequence of the Lax-Milgram Theorem.

LP-analysis of finite and infinite dimensional diffusion operators Proposition

69

2.10. Let (g, D(E)) be a coercive form on 7{. Define

D(L) := {u 9 D(g)lv ~-+ g ( u , v ) is continuous w.r.t. II" II on D(E)} , (2.5) and/or u 9 D(L) let Lu denote the unique element in 7{ such that ( - L u , v) = g(u,v) for all v 9 D(E) .

(2.6)

Then ( L , D ( L ) ) is a linear operator on 7{ that generates a C~ (Tt)t>_o of contractions on 7/. Proof. Clearly (L, D(L)) is a well-defined linear operator on 7/ and applying the above definition to the coercive form (6", D(E*)) we likewise obtain an operator (L*, D(L*)) which obviously is exactly the adjoint operator of (L, D(L)) on 7/. C l a i m 1. (1 - L)(D(L)) = 7 / a n d D(L) is dense in 7-/. Indeed, applying for f 9 7{ the L a x - M i l g r a m Theorem to the linear functional v ~+ (v, f ) on D(E) (which is continuous w.r.t. ~ / 2 ) we obtain that there are G l f , G~f 9 D(g) such t h a t for v 9 D(g) (f ,

= El ( a l f , v) = E;

f,

=

(v,

f).

This in turn implies t h a t Gaf 9 D(L), G'{f 9 D(L*) and t h a t ( 1 - L ) G x f = f and (1 - L*)G~f = f. Consequently, (1 - L)(D(L)) = 7/ and if g 9 7-/such t h a t (g, u) = 0 for all u 9 D(L), then for all f 9 7/ 0 = (g, G l f )

= E

(alf, a g) = (f, a

g).

Hence G~g = 0, therefore, 9 = (1 - L*)G~g = 0. So, D(L) is dense in 7-/. C l a i m 2. L is dissipative and closed. Clearly, for all u 9 D(L), a > 0, we have c~llu[I2 < g~(u,u) = ( ( ~ - L)u,u) < I]au- Lull Hull. In particular, 1 - L is invertible with bounded inverse (1 - L) -1 which by Claim 1 is defined on all of 7/. Hence (1 - L) -1 and thus also L is closed. Claims 1 and 2 prove the assertion. [] It follows by (2.3) that the operator L in the previous proposition is weakly sectorial, i.e., for all ~ E (0, oo) there exists K s 6 (0, cx3) such that

I((c~ - L)u,v)I < ((c~ - L)u,u)l/2((~ - L)v,v) 1/~ for all u,v 9 D(L) . (2.7) In fact Proposition 2.10 establishes a o n e - t o - o n e correspondence between sectorial forms and such operators resp. C~ of a certain type. 2.11. The association of L and hence (Tt)t>_o to (E,D(E)) in Proposition 2.10 defines bijective maps between the following three sets of mathematical objects:

Theorem

70

M. RSckner

(a) Sectorial b r m s ($, D(g)) on ?-l. (b) Weakly sectorial, closed, dissipative operators (L,D(L)) on ~l for which (1 - L)(D(L)) is dense in ~ . (c) C ~ (Tt)t>_o of contractions on ~ having the property that for their natural extensions (TtC)t>_o to the complexification ~lc of ~ the semigroups (etTtC)t>o are restrictions of holomorphic contraction semigroups (eZT~)zc~: where K is a sector in C containing [0, (x~). For the proof of Theorem 2.11 we refer to [MR 92, Ch. I, Sect. 2] (in particular, cf. the diagram on p. 27). We only note here that (Tt)t>_o in Proposition 2.10 is, therefore, analytic. In particular,

Tt(7-l) C D(L) for all t > 0 (cf. e.g. [ReS 75, Corollary 2 of Theorem X.52]) and thus for all f E 7-/

d d~ Ttf = LTtf for all t > 0 To f = f . 2.3 S e c t o r i a l f o r m s o n L2(E; m ) Let (E;B;m) be a measure space and 7-I := L2(E;m) := L2(E;B;m) the corresponding (real) L2-space with its usual inner product. In particular, now has a vector lattice structure w.r.t, the usual order "<" (defined representativewise m-a.e.). As usual for f E L2(E; m) we set f T := sup(f, 0) and f - := - inf(f, 0). In this subsection we recall a few results giving conditions on a generator (rasp. the associated coercive form, if it exists) ensuring that the C ~ semigroup respects this order. These results (at least for symmetric semigroups) are known under the key-word "Beurling-Deny criteria". We first recall D e f i n i t i o n 2.12. A C~ preserving, if

(Tt)t>_o

on L2(E;m) is called positivity

f C L2(E;m) , f > O ~ Ttf > O for all t > O , and sub-Markovian if

f E L2(E;m) , O< f <_ l ~ O < Ttf < 1 for a l l t > 0 . If re(E) < ~ , a positivity preserving C~ Ttl = 1 for all t > 0.

is called Markovian if

It is easy to see that "sub-Markovian" implies "positivity preserving".

LP-analysis of finite and infinite dimensional diffusion operators T h e o r e m 2.13. Let (Tt)t>o be a C~ with generator (L, D ( L ) ) . Then

71

of contractions on L2(E; m)

(i) (Tt)~>o is positivity preserving if and only if (Lu, u +) _o is sub-Markovian if and only if ( L u , ( u - 1 ) +) < 0 f o r a l l u e D ( L )

.

Proof. For (i) see e.g. [MR 95, Theorem 1.7] and for (ii) see e.g. [MR 92, Ch. I, Proposition 4.3]. [] Unfortunately, the conditions on L in the previous theorem are often hard to check. As we shall see below, this is different if (Tt)t>>_ois associated with a sectorial form, since then there are respective easily checkable conditions on the latter. T h e o r e m 2.14. Let (s163

sponding (generator L and) C~

be a sectorial form on L2(E;m) with corre(Tt)t>_o of contractions. Then:

(i) (Tt)t>_o is positivity preserving if and only if for all u e D(s D(s and

also u + E

E ( u , u +) >_ 0 .

(ii) (Tt)t>_o is sub-Markovian if and only if for all u E D(E) also u # := inf(u +, 1) E D(C) and g(u, u - u #) >_ 0 .

Proof. (i): [MR 95, Theorem 1.5 and Proposition 1.3]. (ii): [MR 92, Ch. I, Theorem 4.4] and its proof.

[]

Remark 2.I5. (i) In case (i) of Theorem 2.14 the coercive form (g, D(g)) is also called positivity preserving while in case (ii) it is called a Semi-Dirichlet form. If also (g*, D(g*)) is a Semi-Dirichlet form, then (g, D(g)) is called a Dirichlet form. (ii) We note that if (g, D(g)) is positivity preserving, then so is (g*, D(g*)) and therefore so is (g, D(g)). (Indeed, setting u,~ := u + - 88 we obtain for all n ~ N

= E(~n, ( ~ ) +) + n C ( - ~ , (-~n) +) + ~E(~=,u) > ~E(~-,~) .) But if (g, D(E)) is a Semi-Dirichlet form, (g*, D(E*)) might be not (cf. [MOR 95, Remark 2.2 (ii)]). So, not every Semi-Dirichlet form is a Dirichlet form. Since in many applications a coercive form (g, D(E/) is explicitly given only on a core D C D(E), we recall the following useful result to check whether (g, D(C)) is positivity preserving. Analogous results hold for (Semi)Dirichlet forms (see e.g. [MR 92, Prop. 4.7 and Ex. 4.8]).

72

M. RSckner

L e m m a 2.16. Let ( g , D ( g ) ) be a coercive form on L~(E;m). Let D be a

subset of D(g), dense w.r.t. ~1 such that for all u E D the following holds: for all c > 0 there exists a function ~e : I~ -+ [-~, oo) such that ~e(t) = t for all t E [0, co), 0 < ~oE(t2)- (fl~(ti) ~ t2 - t l i/t1 ~ t2, ~o~ o u E D(g) and sup g ( ~ o u , ~ o u )

<_ const, g(u,u)

e>0

as well as limsup $(u,~oe o u ) _> 0 . ~--+0

Then (g, D(g) ) is positivity preserving.

Proof.

[MR 95, Proposition 1.2].

[5

2.4 Examples and Applications 2.4.1 E : = o p e n s u b s e t o f R~. Let E := U C [r U open, B := B(U) the Borel a-algebra on U, and let dx denote Lebesgue measure on U. For 1 < i, j < d let aij E L~oc (U; dx) such that aij

~(aij - a j i ) E L ~ ( U ; dx), bi, [~i E L ~ ( U ; dx) + Ld(U;dx), c E Llloc(U;dx) such that c- E L ~ ( U ; d x ) + Ld/2(U;dx). Define A := (aij)l<_~,j<_d, b := ( b l , . . . , b d ) , b = (D1,...,[~d) and let ( , >Rd denote the Euclidean inner product on I~d. Assume that there exists 5 > 0 such that for all ~ E I~d (A(x)~, ~}R~ _> 5 (~, ~}~ for dx-a.e, x E U . Define for u, v E C ~ (U) (: = the set of all infinitely differentiable real-valued functions on U with compact support).

E(u,v)

._-

f

f

v(x)

Then there exists an a E (0, c~) such that (g~, C ~ ( U ) ) is a closable, positive definite, symmetric bilinear form on L2(U; dx). Let (g~, D(g~)) denote its closure. ($~, C ~ ( U ) ) satisfies the weak sector condition, and hence has a unique (w.r.t. ~ / 2 ) continuous bilinear extension to the domain D(ga), and this extension, denoted by (ga, D(ga)) (with D ( g a ) : = D(ga)), is a coercive form on L2(U; dx). Furthermore, ($~, D(E~)) is positivity preserving.

LP-analysis of finite and infinite dimensional diffusion operators

73

The proofs for all these statements are contained in [MR 95, Subsect. 2.1] (and they will be essentially repeated in the infinite dimensional case in the next subsection). Let L (~) denote the generator of (s sponding positivity preserving C~ Define

L Tt

and (T(~))t>_o the correof contractions on L2(U; dx).

:= n (~) + a , D(L) := D(L (~)) , := e a r T (~) t > O

Then for all f E L2(U;dx)

d__Tj dt Tof

= L T t f for a l l t > 0 = f,

(and a l l t > 0 ,

i f f ED(L))

where L (heuristically) is given by

Lu= div(AVu+bu)-(b,

Vu)-cu.

(2.9)

(Tt)t>o is positivity preserving, since so is (T(~))t_>0. 2.4.2 E : = t o p o l o g i c a l v e c t o r space. Let E be a locally convex topological real vector space which is a (topological) Souslin space, that is, the continuous image of a separable complete metric space (e.g. E is a separable real Banach space). Let B := B(E) denote its Borel a-algebra and E' its dual space. Let (H, ( , )g) be a separable real Hilbert space, densely and continuously embedded into E. Identifying H with its dual H ' we have that E' C H C E densely and continuously

(2.10)

and the dualization E' ( , ) E between E ' and E restricted to E' x H coincides with ( , IH. Here the dual E' of E is endowed with the strong topology. H should be thought of as a tangent space to E at each point. For a dense linear subspace K C E ~ we introduce the usual test function space 9cC~ (K) on E consisting of finitely based smooth bounded functions, i.e., ~ C C := .TC~(K) := {f(g4,... ,g,~) I rn E N, g l , . . . , ~ , , G E ' , f e C ~ ( ~ ' ~ ) } ,

(2.11)

where C ~ (]~m) denotes the set of all infinitely differentiable functions on ]~m with bounded partial derivatives of all order. Note that since 13(E) = a ( E ' ) , :TC~ is dense in L2(E; m) for any finite positive measure m on (E, B(E)). Let u E i T C h . For k E E as usual we set

74

M. Rhckner

Ou ( z ) :

=

d u(z + sk)l~= 0

d-/

(2.12) m

i=1

where 0i denotes derivative w.r.t, the i-th coordinate. The latter equality follows by the chain rule if u has the representation u = f ( g l , . . . , g m ) as in (2.11). For z C E the H-gradient VHu(z) E H is defined as the unique element in H such that

Ou
(2.13)

Ou (z) is continuous (where we note that by (2.10), (2.12) the linear map h ~-~ ~-~ on (H, ( , )g)). We now fix a finite positive measure # on (E, I3(E)) such that #(U) > 0 for all n o n - e m p t y open subsets U C E. Define S.(U,V) ::

/E (VHU'VHV)H d# ; u,v E S C ~ .

(2.14)

Then (E,, 5~C~~ is a positive definite symmetric bilinear form on L2(E; #) with dense domain. We assume

(C)

(gu, .~C~) is closable on LZ(E; #).

For sufficient (and essentially necessary) conditions implying (C) we refer to JAR 90, Sect. 3]. Next we are going to "build" coercive forms on L 2 (E; #) of type (1.7) starting with (gu, 5cC~) 9 Let Z;~o(H) denote the set of all bounded linear operators on H with operator norm II II. Suppose z ~-+A(z), z E E, is a map from E to s such that z ~ (A(z)hl, h2)H is B(E)-measurable for all hi, h2 E H and 1 [141] C LI(E; #), ]]AI] c L~(E; #), where A := ~(A + A'), ~] := 89 - A ' ) and A'(z) denotes the adjoint of A(z), z E E, on H. Furthermore, assume that the following "ellipticity" condition holds: (E) There exists 5 E (0, oo) such that (~)(z)h, h)H > 6 (h, h)g for all h E H . Let c C L 1(E; #) and define :=

v.v(x)>..(dx)

+

.(dx) ;

e SC? ~

I

I

Then by (E) also (Q,hcC~) is closable on L2(E;#) (see [MR 92, Ch. II, Subsect. 3 e)]). Let (Q,D(Q)) denote its closure. Let b,/~ E L2(E ~ H ; # ) , c E L ~(E; #) such that they define a "small form perturbation" of (Q, D(Q)), i.e., they satisfy

LP-analysis of finite and infinite dimensional diffusion operators

75

(SFP) (i) there exist ~ E (0, 1) and a E (0, oo) such that

f ((b+b, VHU)H-C-u)ud#>_-~ Q ( u , u ) - a / u 2 d# for all u E J : C ~ ; (ii) there exists C E (0, c~) such that

{b - b, VHU}H V d# < C Ql(U,U)U' Ql(v,v) 1/2 for all

u,v E .TC~ (where for ~ > 0 again Qa(u,v) := Q(u,v) +

~ f u v d~). Then define 8(u,v) :=

.f (A(x)VHU(X) , VHV(X)}H #(dx) (b(x), VHU(X)}H V(X) #(dx) +/E U(X) (b(x), VHV(X)}H #(dx)

+s

u(x)v(x)c(x) ~(dx); u,v e ~ C ~ 9

(2.15)

It follows by (SFP) (i) that for all u E .~C~

~c~+I(U,U)(::E(U,U)-~-(O:'+" 1 ) / u 2 d#) >_ (1

-~) Ql (U, u)+ E Q(u, u)+ ./'( (b + [~,VHU)H + (a --c-)u) u d# Ql(u,u)

_> ( 1 - r

(2.16)

and by (SFP) (ii) that for all u, v E 9rC~

I~(~,v)L <_(M + C) QI(~,~) ~/~ Q~(v,v) ~/~ where M := # -

ess sup I[/i(x)[I. Hence by Remark 2.9 (ii) there exists

xcE

K~+~ E (0, 0c) such that for all u, v E ~ C ~

IEa+l(U,V)l <_Ka+l Qa+l(U,u) 1/2 Qa+l(v,v) 1/2 9

(2.17)

(2.16) and (2.17) imply that (S~, f C ~ ) is a closable, positive definite, symmetric bilinear form on L2(E; #), and for its closure (~a, D(S~)) we have that D(~a) = D(Q). By (2.17) S~ uniquely extends to a (w.r.t. Q1/2 a+lJ continuous bilinear form with domain D(Q), and if D(Sa) := D(Q), then (2.16), (2.17) imply that (s D(Ea)) is a coercive form on L2(E; #). Now we want to show that (S~, D(S~)) is positivity preserving. We shall use Lemma 2.16. So, for c > 0 let ~ E C~(I~) such that ~(I~) C [-c, oc), ~ ( t ) = t for all t E [0, oc), 0 < ~'~ < 1, and ~ ( t ) = - ~ for t E (-cr

76

M. Rbckner

Then the assumptions in Lemma 2.16 with D := Y C ~ are fulfilled. Indeed, obviously for all u E $-C~, e > 0, 9~ o u C $-C~ C D(g~) = D(Q) and

Q ~ + ~ ( ~ o ~, ~ o ~) _< V~+~(~, ~ ) , hence by (2.16), (2.17) sup g,~((p~ o u , ~

ou) ~ const, g~+l(u,u) .

e>0

Furthermore, since I~(~)1 ~ I~1, ~ ( ~ ) ~

e'-~o

l{~___0}u, I~'=(~)1 ~ 1, ~(u) ~'~--~o

1{~_>o}, we obtain for en := ~, n E N, by Lebesgue's dominated convergence theorem lim g~ (u, ~

= =

o u)

/I{~>_O}((AVHU, V H U ) H + ( b + [ 4 V H U ) H U + U 2 ( C + a ) ) lim n --~ oo

(/'

~

d#

(u) 2 (AVHu, VHU)H dp

n---+ o o

So, Lemma 1.17 implies that (g~, D(g~)) is positivity preserving. Let L (~) denote the generator of (g~,D($~)) and (T(~))t>o the corresponding positivity preserving C~ of contractions on L2(E;#). Define L := L (~) + a , D(L) := D ( L (a)) , Tt : = e at T (~) t > O Then for all f E L2(E; p) d d~ T t f

Tof

= L T t f for all t > O (and all t _> O, if f E D ( L ) ) :f,

where L (heuristically) is given by

Lu = div~(AVHu + bu) - (b, VHU)H -- CU .

(2.1s)

Here div~ denotes the divergence w.r.t, to #, i.e., ( - 1 ) times the adjoint V ~ ~ of VH on L2(#). (Tt)t>o is positivity preserving, since so is (T(c~))t>_o. To conclude this subsection we discuss conditions on b,/~, c - that ensure that (SFP) holds. This is e.g. in fact the case if b,/~, c - are bounded. This is easy to see and the details can be found in [MR 92, Ch. II, Subsect. 3 e)]. The unbounded case is more delicate. In finite dimensions (see Subsect. 2.4.1)

LP-analysis of finite and infinite dimensional diffusion operators

77

(SFP) follows under the "(L d + L~)-condition '' from Sobolev-embedding theorems, more precisely from the fact that

lul 2--~ dx

< const.

(/

IVul 2, dx

(cf. [MR 92, Ch. II, Subsect. 2 d)]). Such Sobolev embeddings do not hold in infinite dimensions even for very simple (e.g. Gaussian) measures #. However, as a substitute in some cases the following, so-called defective log-Sobolev inequality holds for our triple (E, H, #): (DLS)

There exists ;~1, A2 E [0, cxD) such that

~[ u21~

d# <_ )U / IVHUl2Hd# + lluH22(,)logHu'IL2(.) + X211ull~2(,) for all f e 5cC~ 9

Here HuilLp(,) := (f [ulpd#) 1/p. (DLS) e.g. holds even with A2 = 0 if (E, H, it) forms an abstract Wiener space (cf. [Gr 93]). We shall see other examples in the next subsection. L e m m a 2.17. Suppose that in addition to (C), (E) above, (DLS) holds for (E,H,#), that IlJill E L ~ ( E ; # ) , and that etb- t ,

L (E;

p:>O e Ib+gl~ E L m (E;/~) , c c- E L p2(E;/~) with Pl,P2 C [1, oo) satisfying ~ + ~1 < 5/4)u (where 5, )~1 are as in (E)

resp. (DLS)). Then (SFP) holds and all the above applies. Proof. The proof is essentially a direct consequence of the following Young inequality, i.e.,

st <_s l o g s - s + e t for a l l t E I~, s E [0, ec) .

(2.19)

Let us first check (SFP) (i): Let u C ~ C ~ . We may assume that []UblL2(~) = 1. Then by (2.19) and (DLS)

/ ( b + b , VHUlH U d # - / u 2 c - d t t ,~ 1/2 / /.

-, 1/2

>_-

f P /

1 / --> --2p--~- U2 (pzib+b[ 2 ) > (2is)

l(f

d.-

u 2 logu 2 - 1 +

2pl

f

1/ p2--

u 2 (p2c-)

e mlb+~l~ dp

) 1

> (DZS)

P2

-

d#- ~1 / [VHUl2Hd#

78

M. Rhckner

where,:=

~+~

--+7< (

1

I by assumption and

1)

1 (ob+b~ +__ P2

m

p2

--1)

_ 1).

So, (SFP) (i) holds. To show (SFP) (ii) let u, v 9 : P C ~ . We m a y again assume t h a t IlU[IL2(,) = I]V]lL~(,) = 1. T h e n by (2.19) and (DLS) for any p > 0

(b - b, VHU)H

V

d#

(Sv. <

< v~l

1

(/

.

v2 logv 2 d# - 1 +

/

eplb_~l~ d#

)'" 1

__2~ Q(v, v) + 2)~e + j]elb_~l~ ][pL~O,) _ 1

).1

Q(u, u) 1/2

- ~ Q(u, u)~/2

< K Ql(u,u) 1/2 Ql(v,v) , where K := v ~

max

1, 5 ,

-1

and we used t h a t

II~H/~(.) = Itv[Ii=(~) -- 1.

[]

In order to a p p l y all this to a given K o l m o g o r o v equation du -

dt -

z

Lu , u(O,.) = f

we have to find H and # as above so t h a t L is of t y p e (2.18). Indeed, this is possible in a n u m b e r of cases. We start investigating this in the next section by first looking at s y m m e t r i c cases (i.e., where b = b = 0).

3. 3.1

Symmetrizing

measures

The classical finite dimensional

case

Consider the following o p e r a t o r on an open set U C R d

Lu := A u -

(b, Vu)Ra , u E D ,

(3.1)

with d o m a i n D := C~(U). Here A is the usual Laplacian on I~d and b : U --+ IRd a vector field. So, we are in a special case of the situation considered in Subsection 2.4.1 (more precisely: A = Id•d, b =- 0, c = 0).

LP-analysis of finite and infinite dimensional diffusion operators

79

However, we do not assume t h a t the c o m p o n e n t s of b are in L ~ ( U ; dx) + Ld(u; dx), so c a n n o t apply the results of Subsection 2.4.1. Instead we assume that b= VW for some W : U --+ ~ say C 1. Setting p : = e - W we can rewrite (3.1) as ~Tn

Lu = A u + ( " ~, V u ) ~ P

; u c D ,

and see t h a t for # : = p dx we have

- f g L U V dp = / u ( V u , Vv)~d dp = - / u Lv dp ,

(3.2)

i.e., L is s y m m e t r i c on L 2 (U; #). (This observation goes back to Kolmogorov, c.f. [Ko 37]). If we define for a vector field V on U va divu V : = d i v V + ( ; ,V)~td ( =-V*'uV (by (3.2)),

(3.3)

(where V *'u is the adjoint of V on L2(#)) we have L = div u V on D = C ~ ( U ) ,

(3.4)

i.e, L is a so-called #-Laplacian. So, indeed L fits in our f r a m e w o r k , more precisely the one described in Subsection 2.4.2 (with A ==- IdR, b = b ==- O, c = 0, cf. (2.18)) at least if U = ~d and ~(]~d) < OO. However, also the general case above with even less restrictions on p (resp. W) can be handled using coercive forms. We refer to [MR 92, Ch. II, Subsect. 2 a)]. T h e above case resp. its "manifold version" is, of course, quite easy since we are more or less given H = ~g and A in (2.18), i.e., we basically know in advance the underlying differential g e o m e t r y and only have to find a measure Iz t h a t symmetrizes L, i.e., satisfies (3.2). So, this technique seems not suitable for infinite dimensions. It is, however, a remarkable fact, t h a t for a large class of operators, so called diffusion operators, which includes all operators (also the infinite dimensional ones) which a p p e a r in these notes, this is sufficient. More precisely, if there exists a symmetrizing measure # for such a diffusion operator, then there exists a " ( c o - ) t a n g e n t bundle" (in a generalized sense) and a gradient ~ so t h a t L is a corresponding # - L a p l a c i a n , L = div~V with div = - V *,t'. This result in its m o s t general form (valid even in n o n - s y m m e t r i c cases) is due to Andreas Eberle and we shall briefly explain this result in m o r e detail in the next subsection. Subsequently, we shall see concrete infinite dimensional examples and applications.

80

M. R5ckner

3.2 R e p r e s e n t a t i o n o f s y m m e t r i c d i f f u s i o n operators The material in this subsection is taken from [Eb 98, Ch. 1, Appendix B and Ch. 3 Appendix D, 7)]. Let (E, B, m) be a a-finite measure space and fix p E [1, oc). D e f i n i t i o n 3.1. A densely defined linear operator ( L , D ) on L P ( E ; m ) is called (abstract) diffusion operator if (i) ~ ( u l , . . . , u k ) E D for all k E N; U l , . . . , U k E D, and all ~ E C~(I~ k) such that ~(0) = 0, and, in this case k

k

L(fl(Ul,..., uk) = E Oidp(ul""' Uk) Lui+E OiOjr i=1 i,j----1

Uk) F(ui, uj) ,

where

F ( u , v ) := ~1 (L(uv) - u L v - vLu) ; u,v E D (ii) F(u, u) >_ 0 for all u E D. The bilinear operator F : D • D -+ L ( E ; m ) is called the square field operator of L, (where L(E; m) denotes all m-equivalence classes of functions on E). For the rest of this section we assume L2(E; m) to be separable and fix a diffusion operator ( L , D ) on L 2 ( E ; m ) . Note that by (i), D is automatically an algebra. We additionally assume that that D C L ~ ( E ; m), that D is dense in L2(E; m), and for simplicity that 1 E D (in particular, re(E) < cx)). Note that (i) (applied to k = 1, ul = 1, ~ _-__1) implies that L1 = 0. D e f i n i t i o n 3.2. Let T ' E = (T~zE)zEE be a measurable field of Hilbert spaces over E (cf. e.g. [ED 98, Ch. 3, Appendix D, 6)]). A map d : D ~ L2(E T ' E ; m ) (:= space of m-square integrable sections in T ' E ) is called L 2differential (w.r.t. the co-tangent bundle T ' E ) if: (i) The span of {u dv]u,v E D ) is dense in L2(E --+ T ' E ; m ) . (ii) d is linear. (iii) d(u . v) = u dv + v du for all u,v E D.

T h e o r e m 3.3. (A. Eberle) Assume that m is a symmetrizing measure for L (i.e., f L u v d m = f u Lv dm for allu, v 9 D). Then there exists a measurable field of Hilbert spaces T ' E = (T'zE)zeE, an L2-differential d : D --+ L2(E -+ T'E; m) such that Lu = - d * d u , u 9 D ,i.e. , -/Luv

d m = . / ~ ( d z u , dzV)T, E m (dz) / o r a l l u , v 9

where d* : dom (d*) C L2(E--+ T ' E ; m ) --+ L 2 ( E ; m ) is the adjoint o/ d.

LV-analysis of finite and infinite dimensional diffusion operators

Proof.

[Eb 98, Ch. 3, Theorem 3.11].

81 []

In [Eb 98] it is also proved that the above representation of L is unique "up to isomorphisms" and that there is a non-symmetric variant of Theorem 2.3 above. In applications however, it is necessary to find natural and suitable representatives of T ' E and d. This will be done in quite simple "flat" cases in the next section. There, the choice of the tangent bundle and the gradient will be more or less obvious, so the only issue will be to find the symmetrizing measure #. An important class of "non-flat" infinite dimensional examples for which the natural tangent bundle and gradient were only identified recently, will be discussed in Section 6.

3.30rnstein-Uhlenbeck

type operators.

Let (H, ( , ) H ) be a Hilbert space and ( A , D ( A ) ) a self-adjoint operator on H such that for some ~ > 0, (Ah, h)H > ~ (h, h)H for all h E H. For u E .T'C~(D(A)) (cf. (2.11) with H = E, K := D(A) C H =_ H')

Lu(x) := TraceH u"(x) - (x, A(VHU(X)))H , x E H ,

(3.5)

which makes sense, since by (2.12), •HU(X) E D(A). Here u" denotes the second Fr~chet derivative of u. Following the same idea as in Subsection 3.1 the corresponding symmetrizing measure should be given by the following heuristic formula

#(dx) = "exp [ - 89

x>] d~ ''

(3.6)

(where " d ~ x '' denotes "infinite dimensional Lebesgue measure"), i.e., # should be a mean-zero Gaussian measure with covariance operator A -1. Of course, # will only exist on H if A -1 is trace class. But it always exists on a larger Hilbert (or Banach) space E as we shall see below. So, to find a symmetrizing measure in general we have to consider L on a larger state space E. In fact, E can be chosen in such a way that, as we shall see later, the solution of the corresponding Kolmogorov equation is given explicitly by a Mehler formula. More precisely, we have the following result: T h e o r e m 3.4. There exists a separable real Hilbert space E and a C ~ semigroup (~,-tAE~ Of linear operators on E with generator ( - A s ) such that

(i) H C D(AE) with dense Hilbert-Schmidt embedding (where D(AE) is equipped with the graph norm given by AE). (ii) e -tA is the restriction of e -tAE to H for all t >_ O.

82

M. RSckner

This result was first obtained in [BRS 97, Theorem 1.6] (even for the non-symmetric case, i.e., arbitrary generators A of C~ on H). The simplest proof is given in [FR 97, Prop. 2.3]. Let A~ be the adjoint of the generator AE. Then A~ is a linear operator on E' and if K := D(A~E) (i.e., K denotes its domain), then

KCE'

cD(A) CHCE,

(3.7)

and K is dense in E ~. So, looking at test functions in the smaller space ~ C ~ (K), we see that they have unique extensions to E, and L with domain .TC~(K) becomes an operator on the state space E, i.e., for u 9 .TC~(K)

Lu(x) := Agu(x)

-

E<X,A~E(VHU(X)))E' , x E E ,

(3.8)

where for x E E AH

(z) : : T r a c e n

,

= Ei,'~:l OiOjf(~l ( x ) , . . . ,*,~(x)) <~{,(j>H ,

(3.9)

if U = f ( g l , . . . ,g,~), gl,---,g,~ E K , f E C ~ ( ~ ' > ) , i.e., /%-H i8 the GrossLaplacian. Clearly (3.8) is exactly (3.5), if x E H. u" in (3.9) is the second Fr4chet derivative of u on E and Vuu in (3.8) is defined as in (2.13). Let now # be the Gaussain probability measure on (E, B(E)) with Fourier transform given by

E eiE'
(3.10)

# exists since the embedding H C E is Hilbert-Schmidt and A -1/2 is bounded (cf. e.g. [Ya 89, Theorem 3.1]). Then it is easy to check that for all u, v E

~C~(K) fE Lu(x) v(x) #(dx) = fE u(x) Lv(x) #(dx)

= fE, ,(dz) =: E, v)

(3.11)

In particular, (8~, .TC~(K)) is closable and its closure ($,, D ( $ , ) ) is a symmetric, positive definite, densely defined bilinear form, hence in particular a coercive form on L2(E; #). (C,, D ( ~ ) ) is even a Dirichlet form (cf. [MR 92, Ch. II, Prop. 3.8]). So, the results of Subsect. 2.4.2 (with A =- IdH, b = b - 0, c - 0, a = 0) apply to give a solution to the corresponding Kolmogorov equation (2.4.2) via the sub-Markovian C~ (Tt)t>_o on L2(E; #) associated with (E~, D($,)). We note that if in this case for vector fields V : E --+ H of type

V(x) := ~ u i i:1

gi

(3.12)

LP-analysis of finite and infinite dimensional diffusion operators

83

~ E K, u~ E 5 r C ~ ( K ) , n E N, we define n

div~ V := ~

(H -- ui E' E) ,

i=I

then L = div, VH 9 By our special choice of E the semigroup (Tt)~_>0 is given explicitly by a Mehter formula. In fact, it follows by [BR 95a, Sect. 5] that for any bounded B(E)-measurable f : E --+ ]~, t ~ 0, and #-a.e. x ~ E

(3.13) =: p,s(x) Remark 3.5. (i) Obviously, (Pt)t>o from (3.13) is a Feller semigroup, i.e., pt(Cb(E)) C Cb(E) for all t >_ 0. (ii) It can be checked that for all u E 9rC~ (K) d d-t (p~ u (X))l,~o = Lu(x) for all x E AE

(3.14)

(cf. the proof of [BR 95a, Thm. 5.3]). So, restricting to H again we obtain that p~u is a solution to Kolmogorov's equations on our original Hilbert space H. So, if one is satisfied with initial conditions u E 9cC~(K) (or u E C~(E)), and "pointwise-differentiation" (i.e., for fixed x) in (3.14), one can avoid the machinery of sectorial forms in this special case by using Theorem 3.4 and the Mehler formula (3.13). (iii) The above shows that really in general H C E in this Gaussian case. Therefore, the framework presented in Subsection 2.4.2 was chosen in that way.

(iv) (T~)~_>0 above is just the first quantization of (e -~A) t>o and the generator of (E~, D(E~)) is the second qnantization of A (see e.g. [AR 91, Sect. 7 I]). Therefore, (DLS) holds for (E, H , # ) even with A2 = 0 (cf. [Re S 75, Thm. X.61] and [Gr 93, Thin. 3.12]). Therefore, the results of Subsect. 2.4.2 also apply, if we add small perturbations given by b, b, c to L in (3.5).

3.4 O p e r a t o r s w i t h n o n - l i n e a r drift. Let (E, ( , )E), (H, ( , )H) be separable Hilbert spaces such that H C E with dense Hilbert-Schmidt embedding. Again we identify H r with H so

(3.15)

84

M. RSckner E'CHCE

(of. (2.10)). Let {~ln 9 N} c E' be an orthonormal basis of H so that {ang,~In 9 N}, resp. {%~g~ln 9 N} are orthonormal bases for E' resp. E for some (~,~, % 9 (0, ce), n 9 N. Define the projectors Pnx := ~_, E'(gm,X)E gm , x 9 E .

(3.16)

m=l

The following result was obtained very recently (cf. [AKRT 98]). T h e o r e m 3.6. Let E1 be a Hilbert space containing E obtained by tenorrealizing the basis vectors gm, m 9 N, and let/3 : E --+ E1 be B ( E ) / B ( E 1 ) measurable such that the following conditions are satisfied: (i) There exists N 9 N such that for all m 9 N there exists Cm 9 (0, oo) such that IE~ (gm,/3(x))E~ I <- Cm (1 + lixllE) N for all x 9 E .

(ii) There exist continuous ~n = ( ~ n l , . . . ,~,~): ]~'~ --+ I ~ , n 6 N, such that for all n C N (a) ~n is the vector logarithmic derivative of some probability measure #~ on ~n, i.e., f

Omud#~=-f

JR n

u~,~m dpn f o r a l l l < m < n

and a l l u 6 C ~ ( I I ~ ) ;

JR n

(b) the maps ~,~ : E -+ E' defined by

fl~(x) := ~

~,,~(Pnx) g,~ , x C E ,

m----1

satisfy the following coercivity condition (uniformly in n): there exist M E [N, oo), 5 > O, and C ( M , 5) > 0 such that for all n C N and all x 6 E \ { 0 } -E > 2 M + IIPnE' c g , , H I l ~ s + 5 - C(M,~) IIP~xll~ T M ;

(c) for all m 6 N, c > 0 there exists n(E,m) E N such that for all n >

n(E,m) and all x 6 E

I.~<~m,Z(x) - Zn(x)>E~ I < ~(1 + fixity) u Then there exists a probability measure # on (E, B ( E ) ) such that # is a symmetrizing measure for the operator Lou(x) := A g u ( x ) + E~ (~(X), VHU(X))F.'~ fo~ all u e : r C F ( K ) , where K ( < E~) is the lib,ear span of { ~ l n 9 N}.

(3.17)

LP-analysis of finite and infinite dimensional diffusion operators

85

For the proof and further details, in particular, the relation and application to Gibbs measures in statistical mechanics we refer to [AKRT 98]. So as in the previous subsection all results in Subsection 2.4.2 with b =/~ = 0, c - 0 apply to L0. We now give a concrete example to which Theorem 3.6 applies.

Example 3. 7. Consider the lattice Z d and the space ~2 (zd) of square summable sequences indexed by Z d, so

e2(z d)

c

.

Define, in addition, for p C Z the weighted sequence spaces ~2p(Zd) := {x C Rz~lltxt]~ := E

( 1 + ]kl)2px2k < ~ } .

(3.18)

kEZ d

Let {ek[k ~ J } be the standard orthonormal basis in e2(Z a) and A :----(akj)k,j~Zd giving a linear operator on e2(z a) such that (i) there exists 6 > 0 such that (Ax, x),~ > 5 Ilxll~= for all x c e2(zd); (ii) akj = a(k - j ) for all k , j e Z d ("translation invariance"); (iii) there exists r > 0 such that

a(k - j )

= 0 if Ik - j l R a > r ("finite range").

Let P be a polynomial on ]~ of even degree,

P(t) = CNtN q- ... -I- clt -{- co

, CN > 0 , N

C N ,

N even.

(3.19)

Define/~ := (/3k)kcz~ : e2pl (Zd) --+ e2-p2(za), where P2 > N p l + d / 2 , pl > d/2, and ~k(x) := - E

a(k-j)xj

-P'(xk),

k e Z ~ , xeg2_p~(Z d) 9

(3.20)

jEZ d

Then Theorem 3.6 applies w i t h E : = ~ 2 - - P l (Z d) ' E1 := ~2- - P 2 \( ~ d~/ ' H : = ~2 (Z d) to give a symmetrizing probability measure # on E for the operator

nou(x) = AHu(x) -- e~ (Ax + (P'(xk))keZd , VHU(X)}e~ , --P2

x E ~2_B~(Zd), u e J : C ~ ( K ) with g := lin. span {~klk e Zd}.

Remark 3.8. The measure # in Example 3.7 is well-known. It is a Gibbs measure for the so-cMled P(~)d-lattice model (cf. [AKR 97, Subsections 5.1, 5.2]). Furthermore, we have (DLS) in the case of Example 3.7 for small cj, I<j
86

M. RSckner

T h e o r e m 3.9. Let C o , . . . , c g in (2.181 be sufficiently small. Then (DLS) holds with ~2 = 0 for (E, H, #) as in Example 3. 7. In particular, all results

from Subsect. 2.4.2 apply. Proof.

[AKR 95, Theorem 6].

[]

The corresponding semigroup (Tt)t>_o constructed as in Subsect. 2.4.2 in the special case A =_ IdH, b =- 0, then satisfies the Kolmogorov equation (2.4.2) with

Lu = AHU-- ~

--P2

(Ax + (P'(xk))keZd + b(x) , VHU(X)>t~ 2 (3.21)

In this case for vector fields V : E -+ H as in (3.12) we define n

div~V(x) := E ( V H U i ( X ) , ~i)H --Ui(X) 12_p2(Ax + (P'(xk))k~Z~ , ~)l~2 9 i=1

(3.22/ Then L0 = div~VH and L - - divt~VH- ~2 (b, VH.}t~2- c. --P2

4. Non-sectorial cases: perturbations by divergence free vector fields So, far we were only able to solve cases of Kolmogorov's equation on L p (more precisely L 2) in symmetric situations or "small perturbations" (i.e., in the presence of a sector condition) thereof. In this section we shall look at a class of infinite dimensional operators which neither have a symmetrizing measure, note are sectorial, but are perturbations of such by a divergence free vector field. This class is of special importance for applications (cf. Subsect. 4.4 and Sect. 5. below).

4.1 D i f f u s i o n o p e r a t o r s o n LP(E; m ) All the material in this subsection is taken from [Eb 98, Ch. 1, Appendix A and B]. Let (E, B, rn) be a ~r-finite measure space and p E [1, cx~). Let (L, D) be a linear operator on LP(E; m) with dense domain D.

LP-analysis of finite and infinite dimensional diffusion operators

87

L e m m a 4.1. (i) (L, D) is dissipative if and only if f

sgn(u) lul p-1 Lu dm < 0 for all u E D ,

where sgn (x) = 1(0,~) - 1(-~,0). (ii) Let )~ > O. Suppose that for every increasing ~b E C(I~) such that r and r < Islp-1 for all s E ~, we have

= 0,

/ r o u Lu dm < )~ / lulP dm b r all u E D . Then (L - A, D) is dissipative on LP(E; m). Proof. (i) is obvious. (ii) [Eb 98, Ch. 1, Lemma 1.7]. Choose smooth increasing functions ~bn : R -+ I~, n E N, such that Cn(s) = 0 if Is I <_ n -1, Cn(s) = Islp-1 9 sgn(s) if Is[ >_ 2n -1, and lr < Is[p-1 for all s. Then for any u E D, Cn o u is in L q (E; m), ~1 + 1 = 1, and

by dominated convergence.

[]

The following is important to check dissipativity. P r o p o s i t i o n 4.2. Suppose (L,D) is a diffusion operator (in the sense of Definition 3.1) and that for some a E [0, c~) m is a sub-invariant measure

for (L - a, D), i.e., U,

LuELI(E;m)

and f L u d m < _ a f u d m f o r a t l u E D

withu>_O.

,]

d

(4.1)

Then ( L - p , D ) .

is dissipative on LP(E;m). In particular, the closure -L

of (L,D) generates a C~ LP(E; m) Sot some >

on LP(E;m) if ()~ - L)(D) is dense in

Proof. [Eb 98, Ch. 1, Lemma 1.8] Let r : R -+ I~ be a smooth increasing function such that Ir < Is]p-1 for all s, and let ~ := fo r Then o < ~,(t) <_Itlp/p for all t E ~, and L ( ~ o f) = r

f Lf +r

f F(f,f) >r

f L f m-a.e.

for M1 f E D. Hence, by the sub-invariance,

f ~b o f L f dm

<

f L(~, o f) dm

<_ a f

o f drn < a--f l.flP

for all f E D. This implies the dissipativity by Lemma 4.1 (ii). The remaining part of the assertion now follows by Theorem 2.3 above. []

88

M. R5ckner

Condition (4.1) also implies that the C~ Proposition 4.2 is sub-Markovian since we have:

in the last part of

T h e o r e m 4.3. Let (L,D) be a diffusion operator. Suppose (4.1) holds and that ()~ - L ) ( D ) is dense in LP(E; m) for some )~ > a/p. Then the semigroup (Tt )t>_o generated by the closure -L of ( L, D) is sub-Markovian. Proof.

fEb 98, Ch. 1, Lemma 1.9].

[]

Below we shall mainly be working in the case p = 1. The corresponding results then have automatically consequences for LP(E; m) by the following: L e m m a 4.4. Let (T(P))t>o be a sub-Markovian C~ on LP(E; m) and let r e Lv, c~ ) . Then the restrictions o f T (p) , t > O, to L ~( E ; m )MLP ( E ; m) are bounded w.r.t. II ]ln'(m). The unique continuous extensions T (r), t > O, of these operators to L r ( E ; m ) form a sub-Markovian C~ on L~(E; m). Furthermore, if (L (~), D(L(r))) denotes the generator of (T(r))t>o on L~(E; m), then DI := {u e D ( L (p)) ] u,L(P)u E L ~ ( E ; m ) } C D ( L (r)) and L(P)u = L(r)u for all u C D1 9 If T (p), t > O, are contractions on LP(E;m), then so are T (r), t > O, on Lr(E;m). Proof.

feb 98, Ch. 1, Lemma 1.11].

[5

L e m m a 4.5. Let (Tt)t>_o be a C~ of symmetric operators on L 2 ( E ; m ) . Then each Tt, t > O, is a contraction and the assertions of Lemma 4.4 hold for all r E [1, c~). Proof.

[Dav 89, Thm. 1.4.1].

[]

4.2 S o l u t i o n of K o l m o g o r o v e q u a t i o n s o n L I ( E ; m ) .

The results of this subsection are due to W. Stannat (cf. [St 97]). We start with the finite dimensional case.

LP-analysis of finite and infinite dimensional diffusion operators

89

4.2.1 E : = ]~a. Let E := ]~d and ~o e H~o2c(I~a;dx), i.e., ~o is locally in the Sobolev space (w.r.t. Lebesgue measure dx) of order 1 in L 2 (~a ; dz). Assume ~ ~ 0 dx-a.e. and define # : =~o 2 dx , (4.2)

/3" is hence the vector logarithmic derivative of #. Define

E.(.,v) := fR a, d~ ; ~,v c C ~ ( ~ a) .

(4.3)

Then (Cu, C ~ (]~d)) is closable on L2(]~a;#) and its closure (E~, D(Eu)) is a Dirichlet form (cf. Subsect. 2.4.1 and [MR 92, Ch. I, Subsect. 2 a)]). Let L0 denote the corresponding generator (cf. Proposition 2.10). Then Lo is nothing but the Friedrichs' extension of L0 = A + (/3., V-)Rd

(4.4) = divuV ,

D(Lo) = C ~ ( ~ d)

(cf. (3.5)). Let (T~ be the sub-Markovian C~ generated by/~o on L 2(~d; #).

of contractions

Remark 4.6. In this and the next subsection the roles of (Eu, D(Cu)) (resp. Lo, L0) could be replaced by a more general coercive form (resp. corresponding operators) as in Subsecton 2.4.1. Only for simplicity L0 was chosen as in (4.4). Now fix b E L~oc(l~a --+ I~d ; #) satisfying the following "divergence condition"

(DC)

f (b, Vu)Ra d# = 0 for all u C C ~ ( ~ d) ,

or shortly written divu b = 0 .

(4.5)

Now consider the operator

LbU := LoU - (b, Vu)Rd (4.6)

= A u + (flu - b, Vu)Re , u C C~(II~d) 9

90

M. RSckner

Remark 4.7. (DC) is equivalent to f Lbu d# = 0 for all u E C ~ ( ~ d) ,

(4.7)

i.e., # is an invariant measure for Lb in the sense of Sect. 5. below. This holds since obviously, f Lou d# = 0 for all u E C~(]Rd). Hence by Proposition 4.2, (L5, C ~ (Rd)) is, in particular dissipative, hence closable (cf. Remark 2.2) on L 1(E; #). Concerning the existence of solutions to Kolmogorov's equation for at least one extension Lb of (L5, C~(IRd)), i.e. to

d__uu= Lbu u(O,.) = f (4.8) dt which are positivity preserving w.r.t, the initial condition f, we have the following result by Wilhelm Stannat. Below for W C L 1(]~d;#) we set Wb := { f ~ W l f bounded} , W0 := {f E W I supp I f l ~ compact} ,

WO,b := Wo • Wb

T h e o r e m 4.8. Suppose condition (DC) holds. Then there exists a closed extension ]Jb of (Lb, C~(E{d) ) o n Ll(I~d; #) which generates a sub-Markovian C~ (Tb)t>_o of contractions on LI(IRa;p). Lb has the following properties: (i) D(Lb)b C D(g~) and

E,(~. v) + f
for all u e D(Lb)b, V E D(E,)O,b 9 In particular, Lb is uniquely determined by the (non-coercive) bilinear form on the left hand side of (4.9) (since (D(Lb)b is dense in D(Lb) w.r.t, graph

norm). (ii) Proof.

E, (u. ~) < - f Lbu u d~ for u c D(Lb)b 9

[St 97, Prop. 1.1].

(4. 10) []

As a consequence of Theorem 4.8 we get the desired solution to (4.8). However, it is not at aI1 clear whether (Lb, D(Lb)) is unique. There could be many such extensions of the operator Lb explicitly given on C ~ (IRd) in (4.6). The same drawback we also have in the case b = 0, where by (4.9) L0 is just the Friedrichs' extension of (L0, C ~ (fEd)) considered on LI(I~d; #) (cf. Lemma 4.5). Also there, we could have many extensions that are generators. This problem (related to Theorem 2.6) will be addressed in Subsect. 4.3 below.

LV-analysis of finite and infinite dimensional diffusion operators

91

4 . 2 . 2 E :-- t o p o l o g i c a l v e c t o r space.

Let E, H, # and K be as in Subsect. 2.4.2, hence in particular #(E) < c~. Define g , ( u , v ) := /EH #(dx) ; u,v E ~ C ~ ( K ) .

(4.11)

Suppose that

(c)

(g,, :7:C~(K)) is closable on L2(E; #).

Then the closure is a Dirichlet form (cf. Subsect. 2.4.2 and [MR 92, Ch. I, Proposition 3.8]). Let L0 be the associated generator and (T~ the corresponding sub-Markovian C~ of contractions on L 2(E; #). Note that since obviously, 1 E D(L0) and Lol = 0, it follows that T ~ = 1 for all t k 0. So, (Tt~ is even Markovian. Now let b E L2(E --+ H; #) again satisfying the "divergence condition" div, b = 0 ,

(4.12)

more precisely, (De)

rE(b,

VHU)H d# = 0 for all u E JzC~(K)

@

We assume for simplicity

(4.13)

.~C~(K) c D(Lo)

(see [St 97, Sect. 4] how to avoid this condition) and consider the operator LbU

:=

L0 u -

(b, VHU)H , U E J:C~(K) .

(4.14)

Remark 4.9. We note that also in this case (DC) is equivalent to Lbu d#

0 for all

u

E J:C~(K)

@

(4.15)

Consequently, (Lb, Y C ~ ( K ) ) is dissipative, hence closable on L I(E; #) (cf. Proposition 4.2, Remark 2.2 respectively). T h e o r e m 4.10. Suppose conditions (C), (DC), and (~.13) hold. Then there exists a closed extension Lb of (LD, iPC~(K)) on L I ( E ; # ) generating a

Markovian C~ (Ttb)t>_o of contractions on LI(E;#). Lb has the following properties. (i) It is the closure of Lb, but considered with the larger domain D(Lo)b. (ii) D(Lb)b C D(g•) and for all u ~ D(Lb)b, v E D(g,)b 9

(4.16)

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M. RSckner

In particular, Lb is uniquely determined by the (non-coercive) bilinear form on the left hand side of (4.16). (iii)

$~(u,u) < - / L b U

u d# for all u E D(Lb)b 9 (4.17)

Proof. [St 97, Prop. 4.1].

[]

As in Subsect. 4.2.1 we obtain the desired solution of the corresponding Kolmogorov equation with the same drawback concerning uniqueness.

Remark 4.11. We emphasize that conditions (C) and (4.13) hold for # defined as in the cases discussed in Subsect. 3.3 and Example 3.7.

4.3 U n i q u e n e s s p r o b l e m . Let (E,/3, m) be a a-finite measure space and p E [1, oo). Let (L, D) be a linear operator on LP(E; m) with dense domain D. In the preceding subsection we presented results (for p -- 1) ensuring that (L, D) has a closed extension L which generates a C~ on LP(E; m). In this subsection we are concerned with the question whether ], is the only extension generating a C~ on LP(E; m). By Theorem 2.6 this is equivalent to the question whether the closure L of (L, D) on LV(E; m) generates a C~ on LB(E; m). Most of the results below are again taken from [St 97]. We recall that a C~ (Tt)t>_0 on L 1(E; m) is called conservative, if for its dual semigroup (T~)t>o on L~176 m) we have that Tt*l = 1 for one (hence all) t >_ 0.

4.3.1E::~. We consider the situation in Subsection 4.2.1 adopting all notations introduced there.

Remark 4.12. Obviously, (Tt~

is conservative if p(I~d) < oo.

T h e o r e m 4.13. The closure-Lo on L l ( ~ d ; # ) of (Lo, C~(l~d)) generates a

C~ if and only if (T~ (i.e., the C~ generated by the Friedrichs ' extension of (Lo, C~ (~d ) ) on L I (E, p ) ) is conservative. This, in particular holds if p(~d) < oo. Proof. [St 97, Thm. 2.1] (generalizing [Dav 85] to the case of non-smooth coefficients). []

LP-analysis of finite and infinite dimensional diffusion operators

93

As in Subsection 4.2.1 consider now Lb = A + R~ on C ~ ( I ~ ~) with b 6 L~oc(~d -+ ~ d ; # ) satisfying (DC). Define D(E~)lo c to be the set of all u : I~d -+ I~ such t h a t X " u E D ( E , ) for all X 6 C ~ ( ~ d ) . T h e n the conservativity of (Tb),_>0 can be characterized as follows: Proposition

4.14. The following are equivalent

(i) (TD~>o is conservative. (ii) There exist X~ E D ( ~ ) l o c, n E N, and a E (O, oc) such that (X~ tim Xn = 0 #-a.e. and

1 ) - E D(s

n --~ oo

g,,~(X~,v) >_ / ( b , Vx~>R~v d# for all v E D(E~)O,b, v >_ 0 .

(4.18)

(iii) The closure on L z (I~d ; #) of L5, but considered on the larger domain D( Lo)b, generates a C~ on L l (Nd; #) Proof.

[St 97, Proposition 1.9].

[]

T h e following l e m m a provides simple conditions which ensure t h a t the equivalent properties (i) - (iii) in Proposition 4.14 hold: 4.15. Proposition 4.14 (i) - (iii) hold if one of the following conditions are satisfied:

Lemma

(i) b E L z(]~d;p). (ii) /3~ - b E L 2 ( ~ d ; # ) and #(]~d) < C~. (iii) There exists a L y a p u n o v - t y p e function for Lb, i.e., a C2-function V : ]~d __+I~ such that lim

V(x)=ooand

lim

LbV(x)=-oo.

(4.19)

This is particularly the case if there exists M E [0, c~) such that ((/3" - b) (x), x>~t~ < M (~n(]x]2d + 1) + 1) for all x E l~d 9

(4.20)

Proof. For (i) see [St 97, Prop. 1.10 (a)]. (ii): B y T h e o r e m 5.8 (iii) below,/3" - b E L 2 (l~d ; #) implies/3", b E L 2 (]~d ; #). In particular, b E L 1 (Rd;#), so (i) implies the assertion. (iii): See [St 97, Proofs of Prop. 1.10 (b) and Prop. 2.8]. [] Now we have the following generalization of T h e o r e m 4.13 to the n o n s y m m e t r i c case. 4.16. The following are equivalent: (i) The closure of (Lb, C~(][~d)) on LI(I~d;#) generates a C~ LI(I~d;p). (ii) One of the equivalent conditions in Proposition 4.14 holds.

Theorem

on

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M. RSckner

Proof. By [St 97, Lemma 2.3] the proof is word by word the same as that of [St 97, Thm. 2.1]. [] Remark 4.17. The equivalent properties (i) and (ii) in Theorem 4.16 may not hold. Indeed, in [St 97, Example 1.12] it is proved that they do not hold in the following situation where # is even Gaussian : ~(x) := e -~2/2, b(x) := 6e ~2,

z~

d.

So far, we have discussed uniqueness results only in Lt(~a;/~), which hold under the weakest assumptions. Uniqueness in LP(~d;#) for p _~ 1 (in particular, for p = 2, i.e., if b - 0, the question of essential self-adjointness) has also been studied by many authors. We refer to [Eb 98] for numerous new results and also for a survey, as well as [L 98] for a more recent special result which is not discussed in [Eb 98]. To give an idea of at least one type of results proved so far, we recall T h e o r e m 4.18. Suppose # ( ~ d ) < O0 and that I/3~ - bI E LPloc(l~d;dx) for some p > d >_ 2 such that

[bl e Lq(~d ; , ) for some q 9 [1, r

.

Let r := 2 - q 2~ (where ~1 := 0). Then the closure of (Lb, C ~ ( R a ) ) on L r (~d ; p) generates a C~ -semigroup on L r (Rd ; #). Proof.

[ABR 97, Theorem 1.4 (iii)].

[]

4.3.2 E : = t o p o l o g i c a l v e c t o r s p a c e . We consider the situation of Subsection 4.2.2 adopting all notations introduced there. In particular, we assume (4.13) to hold (for simplicity).

Remark 4.19. Obviously, (Tt~

is always conservative.

Unfortunately, there is no analogue of Theorem 4.13 in this infinite dimensional case. T h e o r e m 4.20. The closure of ( L o , i T C ~ ( K ) ) does not always generate a C~ on L 1(E; p).

Proof. See the counterexamples in [Eb 98, Subsect. 5 b)].

[]

As in Subsection 4.2.2 consider now L6 :=/~0 - (b, VH ")H on J z c ~ ( g ) with b C L2(E -+ H; #) satisfying (DC). In this non-symmetric situation we have: T h e o r e m 4.21. Suppose the closure of ( L o , i T C ~ ( K ) ) on L I ( E ; # ) generates a C~ on LI(E; p), then so does the closure of (Lb, J z C ~ ( K ) ) on L 1(E; #).

LP-analysis of finite and infinite dimensional diffusion operators

Proof. [St 97, Proposition 4.3].

95 []

Theorem 4.21 is the infinite dimensional generalization of Theorem 4.16 in part. Indeed, under our present assumptions it follows that if E = H = Na , and if p = qo2dx for some ~ e H~o2c(II~d;dx) (which in fact follows from (4.13)), then Theorem 4.16 (ii) holds by Lemma 4.15 (i), since in the present subsection b is assumed to be globally #-square integrable and #(E) < co.

Remark 4,.22. Also in the present infinite dimensional situation the uniqueness problem has been investigated for p E [1, co) rather than just the case p = 1. We again refer to [Eb 98, Ch. 5], and also [LR 97]. The latter paper proves LP-uniqueness results in infinite dimensions for all p E [1, co) which are not covered by [Eb 98, Ch. 5]. They particularly apply to the operator generating the dynamics in the stochastic quantization of Euclidean field theory in finite volume (in contrast to Eberle's results also in case p > 2!). Remark 4.23. We emphasize that the closure of (Lo,~C~(K)) even on L~(E; #) generates a C~ on L2(E; #) for # as in the cases discussed in Subsect. 3.3 and Example 3.7. For the latter this follows by [AKR 97, Theorem 5.13]. For the Gaussian situation in Subsect. 3.3 this follows by the explicit form of (Tt~ given by the Mehler formula (3.13) and Proposition 2.7 applied to B0 := lin. span {cos ( E' (g, ")E) , sin ( ~, (g, "iS) Ie e K} (cf. [BRS 96, Prop. 6.1]). So, Theorem 4.21 applies to all these cases.

4.4 C o n c l u d i n g r e m a r k s . If we want to apply the results in Subsections 4.2, 4.3 above to an operator of type i = A + (B,V-)n~ on C ~ ( ~ d) for some vector field B : R d ~ II~d (or to its infinite dimensional analogue) in order to solve the corresponding Kolmogorov equation, we have to find a measure # so that the decomposition

B=~3"-b is such that all our assumptions are satisfied, By Remarks 4.7, resp. 4.9, we, therefore, have to look for invariant measures # for L. If we can prove regularity for #, i.e., that # = p dx with p regular enough, then we can take Vp b:= ---B. P This program will be pursued in the next section.

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M. RSckner

5. I n v a r i a n t uniqueness

measures:

regularity,

existence

and

Let (E, B) be a measurable space and let L(E, B) denote the linear space of all B-measurable real-valued functions on E. Let D C L(E, B) be a linear subspace and L : D --+ L(E, B) a linear operator. D e f i n i t i o n 5.1. A probability measure # on (E, B) is called an invariant measure for (L, D) (or shortly for L if D is fixed) if

Lu C L I ( E ; # ) and f L u

d# -- 0 for all u E D .

(5.1)

We abbreviate (5.1) by L*# = 0.

Remark 5.2. Suppose # is an invariant measure for (L, D) and that (L, D) is a linear operator on L 1 (E; #) such that it has a closed extension L which generates a C~ (Tt)t>_0 on L 1 (E; #). Then obviously # is (Tt)t>_oinvariant, i.e.,

if and only if (Tt)t>0 is conservative (of. Subsect. 4.3). So, by Remark 4.17, (5.1) in general does not imply (5.2). Clearly, however, (5.2)

f Lu d# = 0 for all u e D(L) .

(5.3)

So, by Theorem 4.16 for (L, D ) : = (Lb, C~(I~d)), as considered there, (5.2) holds if and only if/~ = closure of (L, D) on L 1 (E; #).

5.1 S e c t o r i a l case We consider the situation of Subsect. 2.4.2 with b - 0, c -= 0. We assume that (C) and (SFP) hold. Let L0 denote the generator of (Q, D(Q)) (defined as in Subsect. 2.4.2 with c ~ 0). As in Subsect. 4.2 for simplicity we assume that J:C~(K) C D(Lo) (5.4) and set

LbU :~- Lou - (b, VHU)H

,

U

E :7:C~(K) .

(5.5)

So, L5 is nothing but the restriction of L :-- L (~) + a to ~ C ~ ( K ) where L (~) is the generator of the sectorial form (E~,D(C~)) defined in Subsect.

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97

2.4.2 with A = IdH b = 0, c - 0. Let (T(t~))t>_o be the corresponding C ~ semigroup and set Tt := e~tT(t~), t > O. Since LbU is only defined as a class of functions being #-a.e. equal, invariant measures in the sense of Definition 5.1 are only well-defined if they are absolutely continuous w.r.t. #. In the next subsection we shall present results about existence and uniqueness of such measures.

5.1.1 R e s u l t s . 5.3. Suppose (C), (E), (SFP) and (5.4) hold. Assume furthermore: (i) (DLS) holds (cf. Subsect. 2.4.2). (ii) There exist In E E ~, n E N, forming an orthonormal basis of H such for all N E N

Theorem

~ N :---- ~ O ( ~ l , - ' ' , ~ N )

-1

= PN dx

for some B(RN)-measurable pN : ]RN -+ [0, oo) such that R(pN) :=

{

y E NN I

1

xllx-yl~N<e} pN(X-----~dx < oc for some e > 0

}

(5.6)

has full ~N-measure. Then there exists p E D ( C ) ( = D(C,)), p > O, such that u := p. # is an invariant measure for Lb. Proof.

[BRZ 97, Theorem 3.6 and R e m a r k 2.9]

[]

5.1.2 A p p l i c a t i o n t o t h e u n i q u e n e s s p r o b l e m . We still consider the situation of the previous subsection Lemma

5.4. In the situation of Theorem 5.3 we have that

~P H E

L2(E; ~). Proof. Let k E N. Then In(p + ~) E D ( $ ) ( = D(gu)) and hence by [BRZ 97, L e m m a 2.10] for all k E N

E' VHp , VHP)H ( p n u l ) d#

---- - n l i m s

VH@n(P+k)))Hinf(p,n)d,

P

< /_ IblH IVHPl. d, _<

IbiS/d~

g.(p,p)l/2 < oc.

Now the assertion follows by Fatou's Lemma.

[]

98

M. RSckner Let us assume that # has the following property:

(U) If for all p E D ( g , ) , with fVHp/plH e L2(E;p.#) and p >_ 0, we define

Lpu := Lou + (VHp P

VHU)H " U E :7:C~(K)

then (Lp, :PC~ (K)) is well-defined on L I (E; p. p) and its closure generates a C~ on L I(E; p. #).

Remark 5.5. (i)There is a lot of examples of measures # known to satisfy condition (U). For instance it follows by [Eb 98, Corollary 5.4] that (U) holds for many Gaussian measures as in Subsect. 3.3. At present we do not know, however, whether (U) holds for Example 3.7, though we expect it is the case. (ii) In condition (U) we set v,~p := 0 on {p = 0}, and to be precise we have to P consider Lp on the (p. #)-equivalence classes 9cC~ (K)~ of ~rCff (K). Since possibly #({p = 0}) > 0, there might be several different representatives in f C ~ ( K ) for the same class in J z C ~ ( K ) - . Therefore, we have to assume that Lp is well-defined representativewise. T h e o r e m 5.6. Assume # satisfies (U) and consider the situation of Theorem 5.3. Let v : : p . # be as defined there. Suppose ]b[H, Lou E L 4 ( E ; # ) for all u E JrC~. Then the closure of ( L b , ~ C ~ ( K ) ) on LI(E;L,) generates a

C~

(T[)t>_o on Ll(E; p). Furthermore, T ~ f = Ttf ~,-a.e. for all

t_>0, f E L I ( E ; # ) .

Proof. Applying Theorem 4.21 with # : : ~, L0 : : Lp, b : : v~p p _ b which is possible by Lemma 5.4, we obtain the first part of the assertion. The second part is obvious. [] Remark 5. 7. (i) Theorem 5.6 assures us that even if we might loose uniqueness on L I(E; #) by passing from (L0, ~-C~ (K)) to (LD, JrC~ (K)), we keep at least uniqueness on L I(E; ~) where z, = p 9# is an invariant measure for Lb. (ii) Again we note that Theorem 5.6 applies to all cases in Subsect. 3.3. (iii) If (E, H, p) is an abstract Wiener space and if elbt~ 6 ~ LP(E; #), it has p)2

recently been proved in [Sh. 98, Sect. 3], that the closure of (Lb, JzC~(K)) on L2(E; #) generates a C~ on L2(E; #) (hence on LI(E, #)). So, in this case one can analyze LD on L 1 also w.r.t, the initial Gaussian measure #, instead of the invariant measure.

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99

5.2 N o n - s e c t o r i a l cases In this section for simplicity we shall assume that the second order part of our operator is the Laplacian, resp. the Gross-Laplacian in the infinite dimensional case. For the general case we refer to the corresponding underlying literature [BR 95], [BKR 96], [BDPR 96], [ABR 97], [BKR 97], [BR 98]. 5.2.1 R e g u l a r i t y a n d a p p l i c a t i o n s t o K o l m o g o r o v e q u a t i o n s . a) Let us start with the finite dimensional case: E := ]~d, /3 := B(~d). Let B : I~d --+ I~d be Borel-measurable and define

LBU := Au + (B, V u ) ~ , u C C ~ ( ~ ~) 9

(5.7)

Note that if # is a probability measure on (~d, B(]~d)) such that L ~ # = 0 in the sense of Definition 5.1, then, in particular, LBu E Ll(]~d;#) for all u E C ~ (~d), hence necessarily B e L{oc(]~d; #) 9

(5.8)

We recall the following special case of the regularity result [BKR 97, Theorem

1]. T h e o r e m 5.8. Suppose p is a probability measure on (R d, B(I~d)) such that

L ~ , = 0. Let for p > d, IB[R~ 9 LPloc(I~d;dx) or [BIRd and p := ~dl~ has the following properties:

(5.9) 9

LPloc(~d;lz). Then I~ << dx

1,p d ; dx) 5.e., is locally in the Sobolev space (w.r.t. dx) of order (i) p 9 gloc(I~ 1 in LP(I~d; dx)). In particular, p is HSlder continuous of order 1 - diP. (ii) inf p > 0 for all compact K C ]~d. (iii) If IBIR~ 9 L2(]~d;#), then ~P 9 L2(]~d;~). Proof. (i): [BKR 97, Theorem 1]. (ii): This is a consequence of the Harnack inequality for LB proved in [Tr 73]. (iii): [BR 95, Theorem 3.1]. [] Consider the situation of Theorem 5.8 and decompose LB as follows L B u := A u + (/3", V u } - (b, V u ) ; u e C ~ ( ~ d )

(5.10)

where/3 ~ := W p , b := j3" - B. Then by Remark 4.7 and Theorem 4.13 we know that Theorem 4.16 applies. In particular, in case ]BIRd E L2(~d; #), we

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hence conclude by Proposition 4.14 (ii) that the closure of (LB, C ~ ( ~ d ) ) on L I(IRd ; #) generates a C~ on L 1 (~d ; #). b) Consider the situation in Subsect. 3.3 with A := IdH, hence (E, H, #) is an abstract Wiener space and K = E'. Define as in (3.8) L o u ( x ) := A n n ( x ) -

E<X, VHU(X)>E, , U e J z c • ( g )

, x 9E .

(5.11)

Let B : E --+ H be B ( E ) / B ( H ) measurable and define LBU :---- Lou + H , U 9 J : C ~ ( K )

.

(5.12)

Then we have the following regularity result for invariant measures (see also [ABR 97, Theorem 4.3] for a generalization). T h e o r e m 5.9. Suppose L, is a probability measure on ( E , B ( E ) ) such that L*BU = O. A s s u m e in addition that E,(~,'>E 9 L 2 ( E ; - ) /or all ~ 9 E ' and that ]BIH 9 L 2 ( E ; - ) . Then • << # and p := ~d, 9 D ( E , ) . Proof.

[BR 95, Theorem 3.5].

[]

Consider the situation of Theorem 5.9 and decompose L B as follows LBU := Lou + (_V_~, VHU>H -- (Vgp _ B , VHU>H , u 9 ~ C ~ ( K )

. (5.13)

since by Remark 5.5 condition (U) is satisfied we conclude by Remark 4.9 and Theorem 4.21 that the closure of (LB, ~ C ~ ( K ) ) on L I ( E ; ~) generates a C~ on L 1 (E; #). 5.2.2 Existence

and uniqueness

E :---- I~d.

Consider the situation of part a) of Subsect. 5.2.1. We recall the following special case of the main result in [BR 98]. T h e o r e m 5.10. Suppose IBI

6

LPloc(]~d ; dx)

f o r p > d and assume that there

exists a L y a p u n o v - t y p e function of LB, i.e., there exists V 6 C2(R d) such that lim Y ( x ) -- oc , lim L B Y ( x ) = - o c . Then there exists a unique probability measure # on (]~d,B(I~d)) such that L'B# = O. Proof.

[BR 98, Theorem 1.6 and Remark 1.10 (ii)].

[]

There is a close connection between uniqueness of solutions to L'B# = 0 and whether the closure of (LB, C ~ ( l ~ d ) ) on L I ( R d ; # ) generates a C ~ semigroup on L 1(Rd;#). More precisely, we have the following. T h e o r e m 5.11. Suppose IB] E LPloc(~d;dx) for p > d. Let ]C be a convex set of probability measures # on (~d;B(~d)) such t h a t / o r every # E

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101

(i) L'B# = O; (ii) the closure of (LB, C~(I~d)) on L l(Rd; #) generates a C~ on L 1 (]~d; #). Then #IC < 1. In particular, f Vp @ ~ #] L*B# = O and B e L ~(]~d;#) with p : = ~xx I P

Proof.

[ABR 97, T h e o r e m 1.2] and the discussion following it.

<1" []

Finally, we shall look at the special case where B is a gradient. More precisely, assume B is a (fixed) Borel-measurable dx-version of

(5.14)

2v-~ for some ~ C H~o2c(I~d;dx), f ~ 2 d x = 1. Here as usual we set ~ := 0 on {~ = 0}. T h e n obviously #o := ~2 dx belongs to the following set of probability measures on (]~d, B(]~d)) ~41 := {#1 L ~ # = 0 such t h a t IBIRd ~ L~oc(~d; #) and IB - v--2pI E L2(I~d;#)} 9

(5.15)

Clearly, #0 is symmetrizing for (LB, C~(]~d)), and, in fact, by the following result so are all other measures in ~41. In some cases we even have uniqueness. Theorem

5.12. Let B be as in (5.14) and #o := ~2 dx. Then:

(i) Every # C .A41 is a symmetrizing measure for (LB, C ~ ( ~ d ) ) . (ii) If B E L}oc(U;dx) for some connected open set U C ~d such that dx(l~ d \ U) = O, then M1 = { , 0 )

9

Remark 5.13. T h e condition in T h e o r e m 5.12 (ii) cannot be dropped. Indeed, let d = 1 and ~(x) := x (27r)-l/4e -x2/4, x e ]~. T h e n B(x) = - x + 2 x E ~, by (5.14), and B ~ L~oc(IR;dx ). Let ~Pl : : (f0c~ ~ d x ) - l l [ 0 , ~ ) ~ O and ~2 := ( f ~ Then a~ldx c~ E [0, 1] (cf. [BR 95, Example 6.1]).

+ (1 - a ) ~ 2 d x E .h41 for all

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M. RSckner

5.2.3 E x i s t e n c e a n d u n i q u e n e s s for E : = t o p o l o g i c a l v e c t o r s p a c e . Consider E, H , K as in Subsect. 2.4.2 and let B : E --+ E be B ( E ) / B ( E ) measurable. Define LBU := A H U -~- E(B, VHU>E ' , U 9 ~ C ~ ( K )

.

(5.16)

Assume that also E is a Hilbert space with inner product ( , )E. In this situation we have the following existence result: P r o p o s i t i o n 5.14. Assume that lim

(B(x),x)E = -~

,

(5.17)

that x ~+ E'(g,B(X))E is weakly continuous on E for all g 9 E', and that there exist C1, C2, o~ 9 (0, co) such that IB(x)IE <_ C1 + C2 IXrE for all x 9 E .

(5.18)

Then there exists a probability measure # on (E, B(E)) such that L'B# = O. Proof.

[BR 95, Theorem 5.2].

[]

The above result is not optimal. There is work in progress towards generalizations. The question, which conditions imply uniqueness for invariant measures of (LB, ~ C ~ ( K ) ) , are in this general case largely open. Concerning the question under what conditions invariant measures are symmetrizing in this infinite dimensional case, we have the following result. P r o p o s i t i o n 5.15. Let (E, H, #) be an abstract Wiener space and let (LB, J : C ~ ( K ) ) be as defined in (5.12) with K := E'. Suppose B = - I d ~ +b such that b is a #-version of 2 v ~~Oo ' ~ for some ~o 9 D ( $ , ) with /~ d # = l (where VH~O/~O := 0 on { ~ o = O } ) .

(5.19)

5) Assume L, is a probability measure on (E, B(E)) such that E'(~, ")E 9 L2(E; ~') for all e 9 E', ]b]H 9 L 2 ( E ; - ) and L*Bu = O. Then u = ~2. # for some ~ 9 D($•) and VH~:)

-

VH~90 ~o

p-a.e.

(Obviously, Yo := ~ ' t t is such a measure). In particular, ~ is symmetrizing for (LB, : P C F ( K ) ) . (ii) If l - ~ o ~ IH 9 L2(E; it), then ~'o := ~0"it is the only probability measure satisfying the assumptions in (i). Proof.

[ABR 97, Theorem 4.5].

[]

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103

6. C o r r e s p o n d i n g diffusions and relation to Martingale

problems In this section we show that the positivity preserving C~ constructed above, in most cases are really transition semigroups of Markov processes. So, to call the equation d

d-t Ttu = LTtu

(6.1)

a "Kolmogorov equation" is indeed justified. Though we do not use these processes at all to construct solutions to our Kolmogorov equations, this section is enclosed to complete the picture (however, in a slightly sketchy style). In addition, the process can be used subsequently to get more specific information about (Tt)t>_oand L. 6.1 E x i s t e n c e o f a s s o c i a t e d d i f f u s i o n s 6.1.1

Sectorial

case.

Let E be a topological space and B(E) its Borel a-algebra. Let m be a a finite positive measure on (E, B(E)). Let (E, D(E)) be a sectorial form on L2(E; m) with associated generator (L, D(L)) and C~ (T~)~>0. D e f i n i t i o n 6.1. (i) An increasing sequence is called an C-nest if {u C D(E) I u = 0 m-a.e, on

(Ek)keN of

E \ Ek

closed subsets of E

for some k E N)

is dense in D(E) w.r.t. ~ / 2 . (ii) N C E is called

E-exceptional, if N C E\ ~ Ek for some E-nest (Ek)keN. k=l

A property of points x C E is said to hold E-quasi-everwhere (abbreviated E-q.e.) if it holds outside an E-exceptional set. (iii) A function f : A --+ ]~, A C E, is called E-quasi-continuous if flEA is continuous for all k C N for some E--nest

(Ek)kCN such

that U Ek c A. k:l

D e f i n i t i o n 6.2. (E, D(E)) is said to be quasi-regularif the following conditions hold: (i) There exists an E-nest (Ek)k~H consisting of compact sets. (ii) There exists an E~/2-dense subset of D(E) whose elements have E-quasicontinuous m-versions. (iii) There exist un C D(E), n C [~, having E-quasi-continuous m-versions un, n C N, and an E-exceptional set N C E such that {unln E N]" separates the points of E \ N.

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D e f i n i t i o n 6.3. Let ($, D(S)) be a Semi-Dirichlet form. A diffusion process = (~,J:, (St), (Xt)t>o, (P~)~E,~) on E with lifetime ( (i.e., a strong Markov process with continuous sample paths on [0, ()) is called associated to (8, D(E)) if for every f C L2(E; m) n L ~ ( E ; m) and any m-version ] of f and all a > 0

is an g-quasi-continuous m-version of G ~ f := J o e-~t T t f dt. Remark 6.4. M is associated to ( g , D ( S ) ! if and only if for every f E g 2 ( E ; m ) N L ~ ( E ; m ) and any m-version f or f and all t > 0 x ~ E~ [ f ( ~ , ) ]

, x E E,

(6.3)

is an S-quasi-continuous m-version of T~f. Proof.

[MR 92, Ch. IV, Prop. 2.8].

[]

T h e o r e m 6.5. Let ($~, D(s be as in Subsections 2.4.1 or 2.4.2 with a > 0 as defined there. Assume that E in Subsect. 2.4.2 is a separable Banach space. Then: (i) (S~, D(E~)) has the local property, i.e., E~(u, v) = 0 whenever u, v E D($) such that supp (1~,1 m) n supp @ 1 m) = 0 , (ii) (s D ( S , ) ) is quasi-regular. (iii) Suppose (E~, D($~)) is a Semi-Dirichlet form (which is e.g. the case if b - 0). Then there exists a diffusion process I~ on E associated with (E~, D(S,~)). Proof. (i): [MR 92, Ch. V, Examples 1.12 (i)]. (ii): [MR 92, Ch. IV, Sect. 4]. (iii): By the main result in [MR 92, Ch. IV] more precisely its "Semi-Dirichlet form -version" in [MOR 95] we know that quasi-regularity is equivalent to the existence of a so-called m-speciM standard process ~ on E associated with (s D(Ea)). Because of (i) it follows that M is indeed a diffusion (cf. [MR 92, Ch. V, Sect. 1]). A proof that (s D(C~)) is a Semi-Dirichlet form if b - 0, can be found in [MR 95, Remark 2.6]. [] Remark 6.6. Even if Theorem 6.5 only gives a process 1~ with transition semigroup determined by (T(~))t_>0 (cf. Subsect. 2.4), "unkilling" 1~ by using the multiplicative functional e ~ , t ~ 0, we obtain a process with transition semigroup determined by (Tt)t>0.

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105

6 . 1 . 2 N o n - s e c t o r i a l cases.

Now we consider the situations of Subsections 4.2.1 and 4.2.2. Let (Tb)t>_o be the sub-Markovian C~ of contractions on LI(]~d;#) resp. LI(E; #) appearing in Theorems 4.8 and 4.10 respectively. In [St 96] a theory has been developed, called the "Theory of Generalized Dirichlet forms", which includes the theory of sectorial forms, time dependent versions of it, as well as the cases studied in Subsections 4.2.1 and 4.2.2 above. Also the notion of quasi-regularity extends to this more general framework, yielding (as in the sectorial case in [MR 92]) a complete analytic characterization of all generalized Semi-Dirichlet forms associated with (minimally regular) strong Markov processes (in the sense of Definition 6.3). So, there is a complete analogue of Theorem 6.5 for the semigroups (Ttb)t>o above (cf. [St 97, Thm. 3.5, Prop. 3.6 resp. Thm. 4.6, Prop. 4.7]). We only mention here that in the situation of Subsect. 4.2.2 we even have that ~ = (x). The only draw-back is that Remark 6.4 above does not hold in these cases. The function is still an m-version of Ttf, but in general no longer g-quasi-continuous.

6.2 S o l u t i o n o f t h e m a r t i n g a l e p r o b l e m The following is a special case of a general result in IT 97, 98]. T h e o r e m 6.7. (i) The diffusion process I~ in Theorem 6.5 solves the martingale problem for (L,D) with D := C~(]~ d) resp. D := JzC~(K) in the

situations of Subsections 2.4.1 resp. 2.~.2, i.e., for all u E D (a) f~ Lu(Xs) ds, t >>_O, is Px-a.s. independent of the #-version for Lu for #-a.e. x C E . t (b) u(Xt) - u(Xo) - fo nu(Xs) ds, t >>O, is an (J:t)-martingale under Px for #-a.e. x E E. (ii) An analogous statement holds for the cases discussed in Subsect. 6.1.2 with Lb replacing L.

6.3 U n i q u e n e s s We recall that if M = (/2, $-, (iVy)t>0, (Xt)t>o, (P~)~cEa) is a diffusion process on E, a probability measure is called subinvariant for M if for all f : E --+ ]ILF, B(E)-measurable

/ E~[f(Xt)] #(dx) <_ f f(x) #(dx) for all t _> O.

(6.4)

106

M. RSckner

The entire Sect. 4. of these notes was devoted to obtain criteria to ensure t h a t the closure of a diffusion operator (L, D) on L 1 (E; It) generates a C ~ semigroup on L l ( E ; I t ) . One application of this is to obtain uniqueness for Markov processes solving the corresponding martingale problem: 6.8. (i) Consider the situation of Theorem 6.7(i). Assume that the closure of (L,D) on L I ( E ; # ) generates a C~ on L I ( E ; # ) . Let [~/~' = (Y2',~',( t)t>_o, (Xt)t>_o. (P~)xeEA) be a diffusion (or even only a right) process on E solving the martingale problem for (L,D) such that # is subinvariant for 1V~~ (1~ from Theorem 6.5 is such a process). Then x ~-~ E~[f(Xt) ] is a #-version of T t f for all f C L l ( ~ d ' # ) n L ~ ( ~ d ; # ) and all #-versions ] of f , i.e., ~/~ = 1~ up to It-equivalence. (ii) An analogous statement holds for the cases discussed in Subsect 6.1.2 with Lb replacing L. Theorem

I

Proof.

^

I

[AR 95] and [St 97, Prop. 2.6].

[]

7. Appendix 7.1 K o l m o g o r o v e q u a t i o n s in L 2 ( E ; it) for i n f i n i t e d i m e n s i o n a l manifolds E: a case study from continuum statistical mechanics The purpose of this section is to give an important "non-fiat" example where the measurable field of Hilbert spaces, to represent an m - s y m m e t r i c diffusion operator L as in Theorem 3.3, can be constructed explicitly in a natural way giving rise to numerous applications. Let us first describe our framework. 7.1.1 F r a m e w o r k

a n d r e l e v a n t o p e r a t o r s L.

Let X be a connected, oriented C ~ Riemannian manifold such that re(X) = oo where m is the volume element. Let ( , ) T X denote the Riemannian metric and V X, A X the corresponding gradient resp. Laplacian. There is a natural and simple infinite dimensional structure associated with X, the so-called configuration space Fx over X defined as follows:

Fx := {V C X I V n K is a finite set for each compact K C X } .

(7.1)

7 C l"z is identified with the Z+ U {+c~}-valued Radon measure

:: Z

(7.2)

xC~f

where Ex denotes Dirac measure at x. Fx can therefore be endowed with the vague topology. Let B ( F x ) denote the corresponding Borel a-algebra.

LP-analysis of finite and infinite dimensional diffusion operators Functions

107

f C C~(X) on X are lifted to Fx as follows: f(x) = I x f(x) 7(dx) .

(f, 7) : : E

(7.3)

xC')"

This gives rise to the following algebra of ) r C ~ :-- {g((fa, " ) , . . . ,

test functions on Fx:

(fN, ")) I N e N, f l , - . . , fN e C ~ ( X ) , g E C~x~(I~N)} 9

Typical differential operators of interest on and in particular the references there in):

Fx are (cf. [AKR 98 a, b], [R 98]

LF('y) N

E

OiOjg((fl,'Y),..., ( f N , ~ / ) ) / : ( V : % ( x ) , v X f j ( x ) ) T o X ~/(dx)

i,j-:l N

+ Z Oig((fl, ~ ) , . . . , (fN, ~/)) f . AX fi(x) "y(dx) g 2~

i=l N

- E gig((fl'~/)''"' (fN'~/)) E (VXC(x'Y)' vXfi(x))T~X(7"4) i=1 {x,y}c~ where F = g((fl,'),...,(fg,')) E 2cC~ and r : X x X --+ 1~ is a given function satisfying suitable conditions. Below for simplicity we shall only handle the case r - 0 and refer for the case with non-trivial interaction potential r to [AKR 98 b] and [R 98]. One immediately checks that L in (7.4) is a diffusion operator in the sense of Definition 3.1. There are various ways to find a corresponding natural "differential geometric" structure on Fx giving a concrete model for the representation in Theorem 3.3. We shall obtain such a structure by lifting the geometry on X to I x . 7.1.2 A R i e m a n n i a n - t y p e

s t r u c t u r e on configuration space.

a) Lifting of flows Let v C Vo(X) (:= all smooth vector fields on X with compact support). Let r t C I~, be the corresponding flow on X, i.e., the unique solution to d d---t r

= v(r

We define a corresponding r

, r

= x CX .

(7.5)

flow on Fx by

:= { e l ( x ) [ x e 7} = E ~ r xC7

We immediately get from this the following:

9

(7.6)

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M. RSckner

b) Lifting of directional derivatives On X for v E Vo(X), f E C ~ ( X ) we have

v x f ( x ) :=

d

=

9

(7.7)

The corresponding directional derivative on Fx is hence for F = g((f,, .),..., (fN, ")) E YC~ ~ and T E r x

V(F(T) : : =

ad F (r v (T))l,=o Y~.N1 & g ( ( f l , T ) , . . . , ( f N , 7 ) ) f x ( V X I ( x ) , v ( x ) ) T ~ X 7(dx) 9 (7.s)

Comparing (7.7) and (7.8) we obtain: c) L i f t i n g o f gradients and t a n g e n t b u n d l e Define for F = 9((fl, "),..., (fN, "}) E ~-C~ , T E F x , N

Oig((I1,T},..., (fN,T>)vXfi

VFF(T) := E i=l

(E

Vo(X)).

(7.9)

Then by (7.7), (7.8) for v E Vo(X) = f ( V r F ( T ) ( x ) , v(x)}T.X T(dx) = L2(X-+TX;~) ,

VrF(7)

(7.10)

where L 2(X --+ TX; T) denotes the space of (T-classes of) T-square integrable sections in the tangent bundle T X = (T~X),cx. Hence the appropriate "tangent bundle" on F x is given by

T~rx := L2(X --~ T X ; T ) , T E F x

,

(7.11)

and the metric ( , )T~rx is just given by the usual inner product in this vector-valued L2-space. -It-

1

~

7.1.3 L as a /~-~,aplaclan. Let V J r C ~ denote the set of all vector fields V on F x (i.e., all sections in T F x := (T~Fx)~crx) of the form N

T ~ V(T) : ~

Fi(T) v~ (E T ~ r x ) , Fi E :vC~ ,

(7.12)

i:1

v~ E Vo(X), N E N . For such V define N

TF-~ d i v r v ( T ) : = E ( V v r F i ( T ) + F i ( T ) ( d i v X v i , T ) ) i=1

.

(7.13)

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109

Note t h a t this is the exact lifting of the divergence on X , since by (7.3) we have necessarily d i v r v ( 7 ) := ( d i v X v , 7 ) , 7 E F x , hence requiring the usual p r o d u c t rule for div r we arrive at (7.13) for vector fields V as in (7.12). Obviously, for the o p e r a t o r (L, i T C h ) defined in (7.4) with r - 0 we have L = d i v r V r =: A r on : T C ~ .

(7.14)

B u t so far we have no m e a s u r e # so t h a t L is really a / ~ - L a p l a c i a n , i.e., - div r is the adjoint of (V r , J : C ~ ) on L2(Fx; #), or shortly so t h a t - ( V r ) *'u = div r on V J : C ~ .

(7.15)

In fact, one can characterize all such measures. Recall there exists a unique m e a s u r e 7r,~ on ( F x , I 3 ( F x ) ) with Laplace transform

rx e 7r,~(dT) = e fx(~f-1)d'~ for all f e C ~ ( X )

(7.16)

called (pure) Poisson measure with intensity (or mean) m e a s u r e m (cf. e..g. [R 98, Subsect. 2.3]). 7.1. Let # be a probability measure on ( I x , B ( F x ) ) such that

Theorem

rx [
(7.17)

(i.e., the m e a n m e a s u r e of # is Radon). Then the following are equivalent: (i)

( v r ) *,"

= -

div r.

(ii) p is a m i x e d Poisson measure, i.e., there exists a probability measure on ([0, oo), B([O, oc))) such that P= f

7rz.m )t(dz) .

(7.18)

J[o ,~)

In particular, in this case for all F, G E JzC~

Jrx

j/'

x

i.e., L = d i v r V r is a #-Laplacian. Proof.

[R 98, T h e o r e m 2.2] and [AKR 98 b, T h e o r e m 4.1].

[]

110

M. RSckner

Remark 7.2. (i) Note that since we a-priori assume (7.17), A in (7.18) must satisfy f[0

,oo)

z A(dz) < co

(7.20)

D

(ii) Tracing through the proof of [AKR 98 b, Theorem 4.1, (i) ~ (ii)] one sees that the set of all measures, which have Radon mean and symmetrize L, is indeed exactly the set of all mixed Poisson measures with A satisfying (7.20). There is a well-known theorem in differential geometry characterizing the volume element m on a Riemannian manifold X as the up to a constant unique Radon measure a on X so that div z is the adjoint of V r on L2(X; a). In this sense Theorem 7.1 can be interpreted as follows: it characterizes the mixed Poisson measures with A satisfying (7.20) as exactly the volume elements on our "manifold" F x (with tangent bundle (T~Fz).rer). Theorem 7.1 has its analogue for non-trivial interaction potentials r (cf. [AKR 98 b]) with so-called Ruelle measures as symmetrizing measures. It also has a number of applications. One concerns ergodicity and will be presented in the next subsection.

7.2 E r g o d i c i t y The results of this subsection are of a very general nature and hold for arbitrary diffusion operators. We shall explain them here only in the ("non-flat"!) situation of the previous subsection. We refer to [Eb 98, Ch. 3, Theorem 3.8] for the general case and to [AKR 97 a, Sect. 5], [AKR 98, Sect. 4], [AKR 98 b, Sect. 6] for cases of various types of Gibbs measures on the lattice or in continuum including infinite volume Euclidean quantum fields and also including the situation of Example 3.7 above. We introduce the following notation # #~ := ]

(7.21)

rrz.m A(dz)

r ,oo) where ~rz.m is as defined in (7.16) with z. m replacing m for z E [0, co) and A is a probability measure on ([0, co), B([0, co)) satisfying (7.20). Let M denote the set of all such #x. By Theorem 7.1, A4 is exactly the set of all volume elements on F x . 7.2.1 S o b o l e v s p a c e s o n F x . Fix #~ E A/[ and consider

g.~ (F,C) :=

f T~rx#~(d.y), Jrx

F , G 9 J::C~~

g

(7.22)

LP-analysis of finite and infinite dimensional diffusion operators By T h e o r e m 7.1 (ii) ~ (i), (~,r~,gcC~) is closable on S r v ( ua' D (8ua)) denote its closure. T h e n we set

H~'2 (Fx;/Z:~) : :

Lz(Fx;#a).

D (8u~) ,

Fx,

equipped with the

(F,G)H].2(Fx;~) := fFx(VFF, VFG}TFx d/zx + f FG d/zx : sF F 1 2(rx;/zx) ; F, O e H o' HH~.2(Vx;u~).We

Let

(7.23)

i.e., we consider D(Ef~) as the (1, 2)-Sobolev space on inner p r o d u c t

and corresponding norm II

111

(7.24)

are now going to construct

an extension of H~'2(Fx;/Z~) which is the exact analogue on Fx of a weak Sobolev space on a finite dimensional manifold. This is more or less a special case of a construction by A. Eberle in [Eb 98, Ch. 3 b)]. We first note that, as is easy to check, the/z~-equivalence classes determined by V.TC~ (cf. (6.13)) are dense in L2(Fx -+ TFx;/Z)~), i.e., the Hilbert space of (/z-classes of) /z-square integrable sections in TFx. Let ((divV) *''x, n ( ( d i v r ) * ' ~ x ) ) be the adjoint of ( d i v V , P $ - C ~ ) as an operator from L2(Fx -+ TFx;#~) to L2(Fx;#~).

Remark 7.3. As an adjoint the operator ((divV) * , ~ , D((divV)*'u~)) is automatically closed, and by definition it is an operator from L2(Fx;/Z~) to L2(Fx --+ TFx; /Z~). Furthermore, again by definition, G E L2(Fx; /Z~) belongs to D((divV) *'~x) if and only if there exists 170 E L2(Fx -+ TFx;/Z~) such t h a t f G divVV

f ( v a , V)TFx d/z

d/z~ = -

for all V C );$'Cff ) .

(7.25)

d ux := (divV) T M

(7.26)

Va.

In this case (divV)*'U~G =

Because of R e m a r k 7.3 we set

WI'2(Fx;/Z~) and think of product

IF,

:= D((divV)

WI'2(FX;/Z)~)

aS a

d/' x

weak

TM)

,

(1, 2)-Sobolev space on

d
Fx

+ [ _ pc 41'x

with inner

(7.27)

F, G c and corresponding norm [[. that

[Iwl.2(Vx;~x). R e m a r k

.TC~ C WI'2(Fx;#x) hence

7.3 immediately implies

and d ~ = V r on : T C ~ ,

(7.28)

112

M. Rbckner

H~'2(Fx;#~) C W l a ( F x ; # ~ ) and d ~ = V v on H~'2(Fx;#~) .

(7.29)

T h e a b o v e construction can be carried out for a large class of probability m e a s u r e s # on ( F x , B ( F x ) ) (cf. [R 98, Subsect. 4.1]) and in general we do not have equality in (7.29) with such # replacing #x. It is a r e m a r k a b l e fact t h a t we do have equality for mixed Poisson measures #~ in m a n y cases. 7.4. Suppose X is complete, and A = gz for some z 9 [0, oo) or Brownian motion on X is conservative. Then

Proposition

H~'2(Fx; #,x) = W I ' 2 ( F x ; #;~) 9 Proof.

[R 98, P r o p o s i t i o n 4.6].

(7.30) []

7.2.2 Ergodicity and extremality. Obviously, the set 2~4 introduced at the beginning of Subsect. 7.2 is convex. Let A 4 ~ denote the set of its e x t r e m e points.

Remark 7. 5. It is not hard to check t h a t

2ae

= { zm I z 9 [0, c o ) } .

(7.31)

Now we can f o r m u l a t e the m a i n result of Subsect. 7.2. T h e o r e m 7.6. Suppose the conditions in Proposition 7.4 hold and let I-t C .All. Then the following are equivalent:

(i) ~ 9 :~e~ (i.e., by Remark 7.5, , = ~ . ~ f o r some z 9 [0, oo)). (ii) {~, 9 Ad I " = P ' # for some p : Fx -+ [0, oo), bounded, B ( F x ) measurable} = {#} .

(iii)

is

irreducible (ie., F 9 D(Ef), E f ( F , F ) = 0 implies

F = const.). (iv) If (Ll, , D(Lt,) ) denotes the generator of the Dirichlet form (Ef , D ( E f ) ) (i.e., (L,,, D(L~)) is the Friedrichs' extension of (A v, ,~C~), cf. (7.14)), then: F 9 D(Lt,), Lt~F = 0 implies F = const. ("uniqueness of ground state "). (v) (Tt~)t>_o := (etL,)t>_O is irreducible, (i.e., if F 9 L 2 ( F x ; # ) such that T ~ ( F G ) = FTt~G for all G 9 L ~ ( F x ; # ) , t >_ O, then F = const.). (vi) F 9 L 2 ( F x ; # ) , Tt~F = F, t >_ O, implies F = eonst. (vii) (Tt")t>_o is L2-ergodic (i.e., for all F 9 L2(Fx; p) f

Tt~F -

Fd#

d# --+ O as t --+ oo) .

Remark 7. 7. It can be shown t h a t (8~, D ( C , ) ) as in T h e o r e m 7.6 has a diffusion process associated to it in the sense of Definition 6.3 and t h a t (i) - (vii) above is equivalent to the t i m e - e r g o d i c i t y of t h a t diffusion if it is s t a r t e d with distribution # (cf. [R 98, Prop. 3.17]).

LP-analysis of finite and infinite dimensional diffusion operators

113

Proof. (Proof of Theorem 7.6 cf. [R 98 , Theorem 9.4]). The equivalence of (iii) - (vii) is standard and is proved e.g. entirely analogously to [AKR 97a, Proposition 2.3]. So, we only prove (i) r (ii) ~=~ (iii). (i) ~ (ii): Assume (i) holds. Let p : Fx ~ [0, cx~) be bounded and B(Fx)measurable such that u := p. # E A4, and let M := sup p('y). Define ~fEFx

M-p m

.-

M ~

~ "

Then #1 E 34 and # = - ~ # 1 + -~u. By assumption (i) it follows that p = 1, and (ii) is proved. (ii) ~ (i): Assume (ii) holds. Let #1, #2 E 3.t and t E (0,1) such that # = t#l + (1 - t)#2. Then they are both absolutely continuous w.r.t. # with bounded densities. By assumption (ii) it follows that #1 = # = #2. Consequently # E f14. 1,2 (ii) ~ (iii): Assume (ii) holds. Let G E Ho ( F x ; p ) ( = D(E~)) such that 8 , ( G , G) = 0. Let G~ := sup(inf(G, n ) , - n ) . Then lim JIG - G,~HH~,2(rx;,) = 0

n--~ oo

(cf. e.g. [MR 92, Ch. I, Prop. 4.17]) and

E . ( a ~ , a ~ ) <_ E . ( a , c ) = o (el. e.g. [MR 92, Ch. I, Theorem 4.12]). Since G = const, if all G,~ = const., we may assume that G is bounded and, since 1 E D(g,) and g,(1, 1) = 0, also that G >_ 0 (otherwise we add a large constant) and that f G d# = 1. Define u := G . #. Then, since V r G = 0 and # E 34, we have

frx (v r F ,

=

d.

= -fF G d i v r v d# = - f F d i v r v du for all F E f C ~ , V E Y f C ~ . Hence , E 34 and thus by assumption (ii), G=I. (iii) ~ (ii): Assume (iii) holds. Let p : Fx ~ [0, cx~), B( Fx )-measurable and bounded so that u := p 9# E 34. Then there exist F~ E f C ~ , n E N, such that ft --4OO

Hence for all V E VScC~, since u E 34,

divrv P d # = O It follows by Remark 7.3, (7.25) and (7.26) that

p E WI'2(Fx;#) and d~'p = 0 . Proposition 7.4 and (7.29) now imply that p E H~'2(Fx; #) and that V r p = d"p = 0, i.e., g,(p, p) = 0. By assumption (iii) we conclude that p = 1. []

114

M. RSckner 12

R e m a r k 7.8. If we are dealing with measures # where H o' (F x ; #) ~ W 1'2 ( F x ; #), there is an analogue of Theorem 6.6 where H o1,2 ( F x ; # ) is replaced by W I ' 2 ( F x ; # ) . We refer to [R 98, Sect. 9] (or [AKR 98 b, Sect. 6]) for de-

tails and examples.

References [ABR 97] S. Albeverio, V.I. Bogachev, M. RSckner: On uniqueness of invariant measures for finite and infinite dimensional diffusions. SFB-343-Preprint 1997. To appear in: Commun. Pure and Appl Math., 46 Seiten. [AKR 96] S. Albeverio, Y.G. Kondratiev, M. RSckner: Dirichlet operators via stochachstic analysis. J. Funct. Anal. 128, 102-138 (1995). [AKR 97a] S. Albeverio, Y.G. Kondratiev, M. RSckner: Ergodicity of L2-semi groups and extremality of Gibbs states. J. Funct. Anal. 144, 394-423 (1997). [AKR 98] S. Albeverio, Y.G. Kondratiev, M. RSckner: Ergodicity for the stochastic dynamics of quasi-invariant measures with applications to Gibbs states. J. Funct. Anal. 149, 415-469 (1997). [AKR 98a] S. Albeverio, Y.G. Kondratiev, M. RSckner: Analysis and Geometry on conllguration spaces. J. Funct. Anal. 154, 444-500 (1998). [AKR 98b] S. Albeverio, Y.G. Kondratiev, M. RSckner: Analysis and Geometry on configuration spaces. The Gibbsian case. J. F~nct. Anal. 157, 242-291 (1998). [AKRT 98] S. Albeverio, Y.G. Kondratiev, M. RSckner, T. Tsikalenko: Existence and exponential moment bounds for symmetrizing measures and applications to Gibbs states. Preprint (1998). Publication in preparation. JAR 90] S. Albeverio, M. RSckner: Dirichlet forms on topological vector spaces closability and a Cameron-Martin formula. J. Funct. Anal. 88,395-436 (1990). JAR 91] S. Albeverio, M. RSckner: Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms. Probab. Th. Rel. Fields 89, 347-386

(1991). [AR 95] S. Albeverio, M. R5ckner: Dirichlet form methods for uniqueness of martingale problems and applications. In: Stochastic Analysis. Proceedings of Symposia in Pure Mathematics Vol. 57, 513-528. Editors: M.C. Cranston, M.A. Pinsky. Am. Math. Soc.: Providence, Rhode Island 1995. JAr 86] W. Arendt: The abstract Cauchy problem, special semigroups and perturbation In: One-parameter semigroups of positive operators, Edited by R. Nagel. Berlin: Springer 1986. [BDPR 96] V.I. Bogachev, G. Da Prato, M. RSckner: Regularity ofinyariant measures for a class of perturbed Ornstein-Uhlenbeck operators. NoDEA 3,261-268 (1996). [BKR 96] V.I. Bogachev, N. Krylov, M. R5ckner: Regularity ofinvariant measures: the case of non-constant diffusion part. J. Funct. Anal. 138, 223-242 (1996). [BKR 97] V.I. Bogachev, N. Krylov, M. RSckner: Elliptic regularity and essential self-adjointness of Dirichlet operators on ]~d. Ann. Scuola Norm. Sup. Pisa. C1. Sci., Serie IV, Vol. XXIV. Fasc. 3, 451-461 (1997). [BR 95] V.I. Bogachev, M. RSckner: Regularity of invariant measures on finite and infinite dimensional spaces and applications. J. Funct. Anal. 133, 168-223 (1995).

LP-analysis of finite and infinite dimensional diffusion operators

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[BR 95a] V.I. Bogachev, M. Rbckner: Mehter formula and capacities for infinite dimensional Ornstein-Uhlenbeck processes with general linear drift. Osaka J. Math. 32, 237-274 (1995). [BR 98] V.I. Bogachev, M. Rbckner: A generalization of Hasminski's theorem on existence of invariant measures for locally integrable drifts. SFB-343-Preprint 1998. To appear in: Theory Prob. Appl., 18 Seiten. [BRS 96] V.I. Bogachev, M. Rbckner, B. Schmuland: Generalized Mehler semigroups and applications. Probab. Th. Rel. Fields 105, 193-225 (1996). [BRZ 97] V.I. Bogachev, M. Rbckner, T.S. Zhang: Existence and uniqueness of invariant measures: an approach via sectorial forms. SFB-343-Preprint 1997. To appear in: Appl. Math. Optim. 28 Seiten. [Dav 85] E.B. Davies: L1-Properties of second order elliptic operators. Bull London Math. Soc. 17, 417-436 (1985). [Dav 89] E.B. Davies: Heat kernels and spectral theory. Cambridge University Press 1989. fEb 97] A. Eberle: Uniqueness and non-uniqueness of singular diffusion operators. Doctor-Thesis, Bielefeld 1997, SFB-343-Preprint (1998), 291 pages, publication in preparation. [EthKur 86] S.N. Ethier, T.G. Kurtz: Markov processes. Characterization and convergence. New York: Wiley 1986. [FR 97] M. Fklhrman, M. Rbckner: Generalized Mehler semigroups: The nonGaussian case. FSP-Universit/it Bielefeld-Preprint 1997, 37 Seiten. To appear in: Potential Analysis. [Gr 93] L. Gross: Logarithmic Sobolev inequalities and contractive properties of semigroups. Lect. Notes Math. 1563, 54-82. Berlin: Springer 1993. [Ko 37] A.N. Kolmogorov: Zur Umkehrbarkeit der statistischen Naturgesetze. Math. Ann. 113, 766-772 (1937). [L 981 V. Liskevich: On the uniqueness problem for Dirichlet operators. Preprint 1998. [LR 97] V. Liskevich, M. RSckner: Strong uniqueness for a class of infinite dimensionM Dirichlet operators and applications to stochastic quantization. SFB343-Preprint 1997. To appear in: Ann. Scuola Norm. di Pisa, 25 Seiten. [MOR 95] L. Overbeck, Z.M. Ma, M. Rbckner: Markov processes associated with Semi-Dirichlet forms. Osaka J. Math. 32, 97-119 (1995). [MR 92] Z.M. Ma, M. Rbckner: An introduction to the theory of (non-symmetric) Dirichlet forms. Berlin: Springer 1992. [MR 95] Z.M. Ma, M. Rbckner: Markov processes associated with positivity preserving forms. Can. J. Math. 47, 817-840 (1995). [Pa 85] A. Pazy: Semigroups of linear operators and applications to partial differential equations. Berlin: Springer 1985. [ReS 75] M. Reed, B. Simon: Methods of modern mathematical physics II. Fourier Analysis. New York - San Francisco - London: Academic Press 1975. [R 98] M. Rbckner: Stochastic analysis on configuration spaces: basic ideas and recent results. In: New directions in Dirichlet forms, 157-231. Editors: J. Jost et al. Studies in Advanced Mathematics, International Press, 1998. [Sh 98] I. Shigekawa: A non-symmetric diffusion process on the Wiener space. Preprint 1998. [St 96] W. Stannat: The theory of generalized Dirichlet forms and its applications in analysis and stochastics. Doctor-Thesis, Bielefeld 1996, SFB-343-Preprint (1996), 100 pages. To appear in: Memoirs of the AMS. [St 97] W. Stannat: (Nonsymmetric) Dirichlet operators on LI : existence, uniqueness and associated Markov processes. SFB-343-Preprint (1997), 38 pages. Publication in preparation.

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[Tr 73] N.S. Trudinger: Linear elliptic operators with measurable coe~cients. Ann. Scuola Normale Sup. Pisa 27, 265-308 (1973). IT 97] G. Trutnau: Stochastic calculus of generalized Diriehlet forms and applications to stochastic differential equations in infinite dimensions. SFB-343Preprint (1998), 29 pages. Publication in preparation. IT 98] G. Trutnau: Doctor-Thesis, Bielefeld, in preparation. [Ya 89] J.A. Yah: Generalizations of Gross' and Minlos' theorems. In: Azema, J., Meyer, P.A., Yor, M. (eds.) S~minaire de Probabilit~s. XXII (Lect. Notes Math., vol. 1372, pp. 395-404) Berlin - Heidelberg -New York: Springer 1989.

Parabolic equations on Hilbert spaces J. Zabczyk

1. P r e f a c e The notes are an expanded version of 8 lectures given to a School on Kolmogorov Equations organized by Prof. G. Da Prato in Cetraro, Italy, August 2 0 - September 2, 1998. The lectures were intended as an introduction to second order parabolic equations on a separable Hilbert space H . Let L be a second order elliptic operator on Rd: 1

d

L~(x) = -~ E

d

~r

qij(x) 0 2 ~ ( x ) + E fi(x) ~x~ (x),

i,j=l

x e R d,

(].1)

i=1

where, for each x E R d , Q(x) = (qij(x)) is a nonnegative definite matrix and F(x) = ( f l ( x ) , . . . , fd(x)) a vector from R d. Under appropriate regularity conditions imposed on Q, F and a function ~, there exists, see [Fr] and [Kr], a unique solution u(t, x), t > O, x E R d of the following parabolic equation:

-~(t,x)OU = u(o,x)

Lu(t,z),

=

t > O,

(1.2)

e Rd

The equation 1.2 is one of the most studied equations in mathematics. The theory of its infinite dimensional version, with R d replaced by a separable Banach space H , was initiated by L. Gross[Gr] and Yu. Daleckij [Dal]. The papers [Gr], [Dal] used, in an essential way, connections between parabolic PDEs and probability theory. Let us recall that the solution u of 1.2 is of the form:

u(t,x) =/RdPt(x,y)~(y)dy,

t > O, x e R d.

(1.3)

where the positive function pt(x, y), t > O, x, y C R d, satisfies the relations,

/R pt(x,y) dy = 1,

pt+s(x,y) = JRf P'(X'z)Ps(z'Y)dz,

x E R ~, t,s > O, x,y e R d.

(1.4) (1.5)

The research supported by the KBN grant No. 2 PO3A 037-16 "R6wnania paraboliczne w przestrzeniach Hilberta"

118

J. Zabczyk

Thus the function p, called also the f u n d a m e n t a l solution of 1.2, is a density of a probability measure, for any t > 0 and x E R d. It was proved by Kolmogorov [Ko] t h a t the fundamental solutions are in fact transition densities of a Markov process and this is why the equation 1.2 is often called the KoImogorov equation. The relation 1.5 is known as the C h a p m a n - K o l m o g o r o v equation. Although the formula 1.3 has a probabilistic meaning it does not allow to construct a solution to 1.2. A constructive way, based on the theory of stochastic processes, was proposed by K.It6 [It]. Assume t h a t the diffusion matrix Q ( x ) can be factorized: Q(x) =

x c R

where G(x), x E R d is a d • k matrix valued function. Stochastic characteristics of the equation 1.2 are solutions XX(t), t > 0 of the following It6 stochastic equation: dX = F(X)dt + G(X)dW(t),

X(O) = x,

(1.6)

where W is an R k valued Wiener process on a probability space (f2, ~-, 17). Under rather general conditions imposed on the coefficients the equation 1.6 has a unique solution, which can be constructed by successive approximations. If in addition, the function ~ is bounded and smooth then solution to 1.1 is given by the stochastic formula: u(t, x) = E ( ~ ( X ~ (t)),

t >_ 0, x e R d

(1.7)

in which E stands for the integral with respect to the measure P. For each t > 0 and x C R d, pt(x, .) is the density of the distribution of the random variable X ~ ( t ) . The C h a p m a n - K o l m o g o r o v equation is a consequence of the Markovian character of the stochastic solutions to 1.6. The regular dependence of the function u on t and x follows from a regular dependence of the process X on t and on the initial data x. This way several results on the regularity and existence of solutions to 1.2 can be obtained by probabilistic methods. Stochastic constructions can be generalized to the infinite dimensional situation and several results in L. Gross[Gr] and in Yu. Daleckij [Dal], as well as in a number of recent papers, were obtained this way. The same approach will be followed in the present notes. Although the infinite dimensional theory is, in m a n y respect, similar to the classical one, there exist also m a n y new infinite dimensional phenomena, which make the subject specially interesting, see e.g. w167 and w A different approach to P D E s on Hilbert spaces was proposed by [Le]. The lectures constitute only an introduction to the theory and analytical techniques like the continuation method and Schauder estimates in infinite dimensions, see [DaPr2], are not discussed here. We have not covered also i m p o r t a n t results by A. Piech [Pie] and H.H. Kuo [Ku], as well as variational inequalities on Hilbert spaces see [Za3]. Familiarity with elementary concepts of analysis and functional analysis and in particular with differential calculus

Parabolic equations on Hilbert spaces

119

on Hilbert and on Banach spaces is taken for granted. Special results on linear operators and on stochastic processes are gathered in the Preliminaries. For more details we often refer to [DaPrZa2] and [DaPrZa4]. The lectures are divided into several chapters. Chapter 2 contains preliminary results on linear operators, probability measures, Wiener process and stochastic integration. Chapter 3 deals with the infinite dimensional heat equation. Majority of the results presented here are due to L. Gross. Chapter 4 is devoted to the semigroup t r e a t m e n t of the parabolic equations with an emphasis on the heat semigroup. The heat equation perturbed by first order terms is the subject of Chapter 5. The existence of solutions is the main subject here. Chapter 6 is on parabolic equations with the state dependent coefficients. It starts from a construction of stochastic characteristics and examines the regularity of the generalized solutions given by 1.7. The uniqueness question is addressed in Chapter 7. Parabolic equations with Dirichlet boundary conditions are discussed in Chapter 8. Applications to nonlinear parabolic equations on Hilbert spaces are treated in Chapter 9. In the same chapter a financial model leading to an infinite dimensional parabolic equation is described. The Appendix is devoted to theorems on implicit functions. They allow to show existence and regular dependence on initial data of solutions to stochastic equations in a simple, hmctional analytic, way. A c k n o w l e d g e m e n t s All topics discussed in the notes were an object of numerous discussions with Prof. G. Da P r a t o and I thank him for his help. Thanks go also to A. Chojnowska-Michalik, B. Goldys, E. Priola and A. Talarczyk, who read some parts of the manuscript and made valuable comments.

2. P r e l i m i n a r i e s Results on linear operators, measures and on stochastic processes, which will be used throughout the lecture notes, are gathered here. 2.1 Linear o p e r a t o r s The space of all linear bounded operators from a Banach space H into a Banach space U, equipped with the operator norm, will be denoted by L(H, U). 2.1.1 O p e r a t o r s o f l p c l a s s e s . A linear bounded operator R : H --+ H , defined on a separable Hilbert space H , is said to be trace class if it can be represented in a form: +oc

Ry = ~ ( y , ak)bk, for some sequences (ak), (bk) such that

y EH

120

J. Zabczyk +co

Ilakil Jtbkll < + ~ k=l +co

and the trace norm LiRGJl is the infimum of ~-~k=l ]]akJ] Jibki] over all possible representations. If an operator R is trace class then its trace, Tr R , is defined by the formula: +oo

TrR : E

< Rej ,ej >,

j:l

where (ej) is an orthonormal and complete basis on H. The definition is independent on the choice of the basis and

IW~Ri < lIRII1 A linear operator S : H --+ H is Hilbert-Schmidt if for arbitrary orthonormal and complete basis (ek) +oo

E

I(Sek'ei)12 < +oo

(2.1)

k,j:l

If 2.1 holds for some, then also for all, complete orthonormal basis, and the Hilbert-Schmidt norm IJSJJHS is given by, +oo

IIsll~/~ :

+co

] ~ I<se~,eJ>l~= Z k,j=l +oo

=

Z

+c~

trse~ti 2 = Z

k=l

I<e~'S*eJ>t~

k,j=l

liS*eJtl 2 = ilS*ti~s.

j=l

P r o p o s i t i o n 2.1. Assume that S and T are Hilbert-Schmidt operators on a separable Hilbert space H. Then the operator R = S T is trace class and its trace class norm ]lRiJl satisfies the inequality

liRJ]I <_ IISIBHsilTIIHs Proof. Let (ek) be a fixed, complete and orthonormal basis, then +oo

+oo

Ty = Zek:Ze~ k:l +oo

STy

=

k:l

E (Y,T* ek>Sek. R:I

Consequently IISTI]I

<

E

JiT*eki[ IISekji <-

k=l

<

IlTJIHsJlSIIHs.[]

JiT*ekJi2

]iScrim2

Parabolic equations on Hilbert spaces

121

More generally let S be a compact operator on a separable Hilbert space H. Denote by ()~k) the sequence of all positive eigenvalues of the operator ( S ' S ) 1/2, repeated according to their multiplicity. Denote by lP(H), p > 0, the set of all operator S such that

IiSll,

--

(+s

(2.2)

<

k:l

Operators belonging to 11(H) and 12(H) are precisely the trace class and the Hilbert-Schmidt operators. The trace class and the Hilbert-Schmidt norms coincide with ll(H) a n d / 2 ( H ) norms. The space/2(H) is denoted also by LHs(H). The following result holds, see [DS]: P r o p o s i t i o n 2.2. If S E IP(H), T E lq(H) and

1 r

1 p

-=-+-,

1 q

p,q,r>0,

(2.3)

I[STIIr ~ 21/rllSllplITllq.

(2.4)

then S T E l r(H) and

2.1.2 R a n g e i n c l u s i o n s a n d p s e u d o i n v e r s e s . The following result is due to R. Douglas [Do] and is a consequence of the closed graph theorem, see [za21.

T h e o r e m 2.3. Assume that H1, H2, H are Hilbert-spaces and $1 E L(H1, H), S2 C L(H2,H). Then Range Sl C Range S2

(2.5)

if and only if there exists a constant c > 0 such that

IlS~hlIH1 ~ cllSF~hl[H~, for

all h E H.

(2.6)

If U, H are Hilbert spaces and S E L(U, H ) , then for arbitrary y E RangeS, there exists a unique element ~ E U such that, S(~) = y and if S(x) = y then II~ll ~ Ilxll. The element ~ is denoted by S - l y and the transformation S -1, from Ranges into U is linear, called the pseudoinverse of S. The graph of S -1 is a linear, closed subset of H • H.

122

J. Zabczyk

2.1.3 S e m i g r o u p s o f l i n e a r o p e r a t o r s . Any family Pt, t >_ 0 of bounded linear operators on a Banach space E such that P0 = I and

Pt+, = Pt(P,),

t, s > 0.

(2.7)

is called a semigroup of operators. The semigroup (Pt) is said to be strongly continuous or a C0-semigroup if for arbitrary W E E, the E-valued function

t --+ Pt~,

t >0

(2.8)

is continuous at t = 0 ,see [Yo]. It turns out that then the functions given by 2.8 are continuous at all t > 0 and there exist constants M > 0, w E R l, such that, IIPtll ~ Me ~t, t >_O. (2.9) The fundamental characteristic of any strongly continuous semigroup (Pt) is its infinitesimal generator A. It is a linear, usually unbounded operator, defined on the domain D(A) consisting of all W C E such that there exists the limit,

Pt(fl-W

as t $ 0 .

t For W E D(A) one sets, ]

AW = lim --(Ptw - W). t$o t The set D(A) is a linear and dense subspace of E moreover the graph of the operator A : {(W, A~); W C D(A)} is a closed subspace of E • E. If 2.9 holds and A > w then the operators A~ = AA(AI - A) -1 , are bounded and called Yosida approximations of A. Moreover if one denotes:

+~ (tA~)n P? = etAx = E n! ' rt=0

then for arbitrary W E E,

Pffw -~ Pew,

as ~ -~ +c~,

uniformly in t from bounded subsets of R~_. There exists a one-to-one correspondence between C0-semigroups and their generators however only very seldom one can find an explicit description of the domain D(A) and an explicit formula for A. A linear subspace K C D(A) is said to be a core of the generator A if K is dense in the space D(A) equipped with the graph norm:

II~II~(A) = IIwII2 + IIA~II 2. Note that the generator A is completely determined by its values on a core and for many applications the knowledge of the values of A on a core is sufficient. If a linear subspace K C D(A) is dense in E and invariant with respect to (Pt) then K is a core for (Pt), see IDa].

Parabolic equations on Hilbert spaces

123

2.2 M e a s u r e s a n d r a n d o m v a r i a b l e s We start from recalling basic concepts of the probability theory. 2.2.1 P r o b a b i l i t y s p a c e s . A measurable space ( E , g ) consists of a set E and of a a - f i e l d g . If # is a nonnegative measure on a measurable space (E, g) such that #(E) = 1 then # is called a probability measure, shortly probability, and the triplet (E, g, #) is called a probability space, see [Bi]. If (~2, jr) and (E, g) are two measurable spaces then any measurable transformation X : Y2 --+ E is called a random variable. Assume that (~2, jr, ?) is a probability space and X a random variable with values in E. The image # of the measure P by the transformation X, ~(r) =P(~:X(~)eF),

FeS,

is called the law or the distribution of X and denoted by s If a probability measure # on (E, 8) is given then (E, g, #) is a probability space and the mapping X(x) = x, x E E is called the canonical random variable. Its law is identical with the measure #. Random variables X 1 , . . . , Xn with values in E 1 , . . . , E ~ are said to be independent if s on (El x ... x E~, gz | ... @ 8~) is identical with the product s • ... • s of the laws: ]~(X1 E F 1 , . . . ,

Xn ~ Fn) : I I ]~(Xk ~

Fk),

F k C gk, k = 1 , . . . , n .

k=l

An arbitrary family of random variables is said to be independent if any finite sequence of its elements is independent. Assume that H is a separable Hilbert space equipped with the Borel afield B(H). Probability measures on H will be always regarded as defined on B(H). If # is a probability measure on H then its characteristic function ~t is a complex valued function on H of the form:

"fi()~) = fH ei<:~'x>#(dx),

)~ E H .

If characteristic functions of two probability measures #, v on H , are identical : ~()~) = ~(A), )~ E H, then also the measures are identical, # = v. Therefore there exists one to one correspondence between probability measures and their characteristic functions. 2.2.2 G a u s s i a n m e a s u r e s . Gaussian probability measure on R 1 is either i (m--m)2

concentrated at one point, say m E R 1 or has a density ~ e

q

,

x E R 1 where q > 0. More generally a measure # on a Hilbert space H is Gaussian, [Ku], if all linear mappings y --+ defined on H, considered as random variables on (H, B(H), #) with values in (R 1, B(RZ)) , have Gaussian laws. R a n d o m variables are Gaussian if their laws are Gaussian. Gaussian measures are completely described by the following theorem.

124

J. Zabczyk

Theorem

2.4. A measure # is Gaussian if and only if

)~ C H,

~(,~) = e i()~'m)- 89

where m C H and Q is a self-adjoint, nonnegative, operator with finite trace. The measure p from the theorem will be denoted by N ( m , Q). If m = 0 the m e a s u r e / ~ is called symmetmc or centered. Instead of N(0, Q) we will often write NQ. Assume t h a t X is a Gaussian random variable s = N ( m , Q) and (ek) is the complete system of eigenvectors of Q corresponding to the eigenvalues

Qek = 7kek,k = 1 , . . . then, E(X, A) = (m,)~), E(X

-

;~ C H ,

m, a ) ( X - m, b) = (Qa, b),

a, b C H ,

+c~

EIx

- ml

: k:l

Vector m is called the mean vector and operator Q the covariance of N ( m , Q) and of the r a n d o m variable X. If s = N ( m , Q) then x

:

+ k

where ~1, ~2,... are independent random variables with real values, s N(0, 1). In fact one can define : ~n = ( ~ / n ) - 8 9

=

en) n = 1 , . . . .

The following proposition explains why the operator Q should be of trace class or equivalently why ~ k "~k < +c~. 2.5. Let (ek) be an orthonormal sequence in a Hilbert space H, (~k) a sequence of independent Gaussian random variables with s = N ( 0 , 1 ) , k = 1 , . . . and let 7k, k = 1,... be nonnegative numbers. The series

Proposition

V kek k

~-00

converges in H, F-a.s. if and only if ~ k = l ~/k < +C~.

(2.10)

Parabolic equations on Hilbert spaces

125

Proof. The series 2.10 converges if and only if r = Z ~kr < + c o k

Since

-~oo

E(e-~) = 1-I E ( e - ~ ) k=l

and

E(e .~ ) _ ~

,1

if E(~) = N(0, 1),

one has that

~:(e-~) = (~Ik ( 1 + 27k))1/2 However I I ( 1 + 2"~k) = +co if and only if E k

7k = +co.

k=l

Thus if ~ T k = +co then E(e - i ) = 0 and ]?(r = +co) = 1. If ~ k T k < +co then E(~) = ~ k+or = l "Yk < +co so ]?(~ < +co) = 1. [] As in the finite dimensional spaces, Gaussian measures have exponential moments also in general Hilbert spaces. 1 T h e o r e m 2.6. Assume that # = N(O, Q). Then for arbitrary s E (0, 2--Y~rQ)'

SHe'll~ll'#(dx)

< v/1 - 21 s T r Q

(2.11)

Proof. Let (ek) be a complete orthonormal basis which diagonalizes Q and (Ak) the corresponding sequence of the eigenvalues of Q. For s E (0, ~--~--0) 1 and n E N define

In(s) = [ eS~:l<~'~k)2#(dx). JH

(2.12)

Then

I,~(s)

=

f l /H eSi~'~k):#(dx)

(2.13)

k=l

fl

k=l ~ =

However if [y[ < 1 then

1 I+2

...

e sy -2~k dy

1 - e-~ ~ = ~ log(l-2~x~) k=t V~ -= 2sAk

126

J. Zabczyk +ca y m

(2.14)

-~-.

- ln(1 - y) = ~ rn----1

Therefore and consequently, from 2.12 and 2.13,

H

=

~=~ +oo

e89E m = *

~-

(2")~ Tr(Qm)

s6

Since Tr(Q '~) < (TrQ) "~, 7=

< -

1

,/1-

[]

2sTrQ"

As a corollary we have the following result. 2.7. If ~ is a Gaussian random variable with values in a separable Hilbert space H and Y_~ = 0 then, for all s 6 (0, 1

Theorem

1

E(e *lI~II~) <

v/1 - 2sEll,iT

Proof. It is enough to remark that if s

= N(0, Q) then s

= Tr Q.

[]

We finally recall the so called Cameron- Martin formula and Feldman- Hajek theorem, see [DaPrZa2] . Denote by Ql12 and Q-1/2 the positive square root of Q and its pseudo inverse. If Qx = 0 only for x = 0 then the pseudo inverse of Q1/2 is identical with the usual inverse defined on the domain equal to the image of Q 1 / 2 . For arbitrary v E H , the function y --+ (v, y) is a centered Gaussian random variable on the probability space (H, ~(H), N O). Let (ej, ~,j) be the eigensequence corresponding to Q. We set for all y E H , +ca

(g, O-l/2y) = ~ ( g , ej> (y, ej} j----1

(2.15)

V/~

The sum in 2.15 converges in L 2 (H, ]~(H), NQ) and defines a centered, Gaussian r a n d o m variable on H such that for all g, h E H ,

H(g Q-1/2y)ih, Q-1/2y)NQ(dy) = (g, h).

(2.16)

If g E ImQ 1/2 then (g,Q-1/2y} = (Q-1/2g, y), y E H. We will need the following well known result.

Parabolic equations on Hilbert spaces

127

T h e o r e m 2.8. For arbitrary ml,rn2 9 H, measures N ( m l , Q), N(m~.,Q) are either equivalent or singular. They are equivalent if and only if m l - m 2 9 I m Q1/2 and then Radon-Nikodym derivative is given by the Cameron-Martin formula:

d N ( m l , Q) (y) = e(Q-~/2(m~-m2)'Q-~/2(Y-m2))-89[Q-~/2(m~-m2)]2. d N ( m 2 , Q)

(2.17)

The next theorem is due to Feldman and Hajek. T h e o r e m 2.9. For arbitrary vectors rnl,m2 E H , and arbitrary trace class selfadjoint operators Q1, Q2 measures N ( m l , Q 1 ) , N(m2, Q2) are either _

,.~1/2

_

,.~1/2

equivalent or singular. They are equivalent if and only if Im ~1 = I m Lr2 , /-~1/2 (g3--1/2y)l/2.~{[3--1/2[31/2"l, ml -m2 9 I,-wl , and the operator ~ 1 ~r J~'~l "~2 / - I is Hilbert-Schmidt. Thus T h e o r e m 2.9 is a generalization of Theorem 2.8. 2.3 W i e n e r p r o c e s s a n d s t o c h a s t i c e q u a t i o n s 2.3.1 I n f i n i t e d i m e n s i o n a l W i e n e r p r o c e s s e s . Let U be a separable Hilbert spaces and (~, iT, ~) a probability space with a given increasing family of (r-fields J:t C iT, t >_ 0. A family W(t), t > 0 of H-valued r a n d o m variables is called a Wiener process, see [DaPrZa2], if and only if 1) W(0) = 0, 2) For almost all w E ~, W ( t , w ) , t > 0 is a continuous function, 3) W ( t l ) , W ( t 2 ) - W ( t l ) , . . . , W ( t ~ ) - W(t,~_l) are independent r a n d o m variables, 0 < tl < t2 < ... < tn, n 9 N and 4) s - W(s)) = z . ( w ( t - s)), t >_ s. It turns out t h a t if W is a Wiener process then s measure on U with the mean vector 0 :

is a Gaussian

~ ( w ( t ) ) = N(O, tQ) where Q is a nonnegative operator with finite trace. Moreover,

E ( W ( t ) , a } u ( W ( s ) , b } u = t A s(Qa, b)u,

a,b E U .

Let (ek) be the sequence of all eigenvectors of Q corresponding to the sequence of eigenvalues (Tk). If ?k > 0 then

Zk(t) = ( ' y k ) - ~ ( W ( t ) , e k ) v ,

t > O,

is a 1-dimensional, standard Wiener process (Q = 1) and the Wiener processes/3k, k = 1 , . . . are independent. It is clear that

w(t) = ~k(t)ek, k=l

t > 0.

(2.18)

128

J. Zabczyk

Conversely, if (ek) is an orthonormal basis in U, (Tk) is a summable sequence of nonnegative numbers and (~k) are independent, standard Wiener processes then the formula 2.18 defines an U - valued Wiener process. We have also the following, instructive result. P r o p o s i t i o n 2.10. Define

N WN(t,w) : E v ~ k ( t , w ) e k ,

N=I,...

t>0.

k=l Then for arbitrary T > 0 there exists a sequence N,~ --+ +oo such that W Nm (.) is uniformly convergent on [0, T], for almost all w 9 12. Proof. Let N > M then E

sup

E

~t
~-k~k(t)ek

=

E

sup

"~kt32(t)

~

\ t<_T k~-M+ 1 N

-~

N 2

E

k=M+l

7kE( sup ~ k ( t ) ) < C

tC[0,T]

E

k=M+l

7k,

where C = E(suPtc[0,T] f112(t)). Therefore one can find an increasing sequence (Nm) such that,

(suPiWNm+x(t) _ wN~(t) [ > ~___~) < 1 \t<_T _ _ 2----~,

m = 1,....

The result now easily follows.

[3

Example 2.11. Here is the simplest example of the infinite dimensional Wiener process. Denote by Z the set of all integers and by ~d the ddimensional lattice. Let (/3~)~ezd be independent real Wiener processes and W(t) = (~(t))~ez~. Then W is U-valued Wiener process when U = 12(Z d) with the weight @(~/) > 0, ~/ E Z d, ~ z d 6(7) < +co. The covariance Q is the multiplication operator (Qx)(7) = ~(7)x(7),

? E Z d, x E U.

Example 2.12. Let B(s,t), (s,t) C [0,1] • [0,1] be a Gaussian family such that

EB(s,t)B(u,v)

=

(sAu)(tAv),

(s,t),(u,v) 9 [0,1] x [0,1].

The random field B has a continuous version and is called Brownian sheet. The U = L2(0, 1) valued process

W(t)

=

B(t, .),

t 9 (0, 1]

is a Wiener process with the covariance Q:

Qx(~) = ~01 (~ A •)x(•) d~.

Parabolic equations on Hilbert spaces

129

2.3.2 S t o c h a s t i c i n t e g r a t i o n . There is a natural class of operator valued processes which can be stochastically integrated with respect to a U-valued Wiener process W. Denote by Uw the image of U by the operator Q1/2 equipped with the following scalar product:

(a, b)uw = (Q-1/2a, Q-1/2b)u (2.19) where Q-1/2 denotes the pseudoinverse of Q1/2. The space Uw is a separable Hilbert space. Denote by LHs(Uw, H) the Hilbert space of all HilbertSchmidt operators from Uw into H equipped with the Hilbert-Schmidt norm. The space LHs(Uw, H) is again a separable Hilbert space which, from now on, will be denoted by 7-/. Note that operators belonging to LHs(Uw, H) are not, in general, defined on the whole space U and that different Wiener processes may lead to the same space 7/. An N-valued process ~P is called measurable on [0, t] if it is measurable when treated as a transformation from the set (2 x [0, t], equipped with the ~r- field Yt • B([0, t]), into (7/,/3(7-/)). If the process ~P, is measurable on all intervals [0, t], t > 0, then it is called progressively measurable. A progressively measurable process ~ such that

(fo stochastically integrable. Ile(s)ll

is called

as <

t > 0

)

= 1,

(2.20)

For stochastically integrable processes the

stochastic ItO integral fo ~~(s) aW(s),

(2.21)

t >_O,

is well defined, see [DaPrZa2]. Moreover:

E (,, fo~(s) dW(s),,2H) < E ( fot ,,~P(s),,~ds) .

(2.22)

If the right-hand side of 2.22 is finite then 2.22 becomes an identity. In addition the following Burkholder-Davis-Gundy inequalities hold. For arbitrary p > 0 there exists a constant Cp > 0 such that IE (sups<~,, f0 s ~P(u)dW(u),,PH)<_cpE

(fo t ,,~(s),,2 ds) p/2 ,

t>_ O. (2.23)

Assume that kv is an 7/-valued process stochastically integrable on [0, T], r is an H-valued process, Bochner integrable on [0, T], ~-a.s. and Z(0) is an 5r0-measurable H valued random variable. Then the following process

Z(t)

= Z(0) +

f

r

ds +

f kV(s)dW(s),

t e (0, T)

(2.24)

is continuous and well defined. Let F : [0, T] • H --+ R 1 be a function uniformly continuous on bounded subsets of^J0, T] • H together with its partial derivatives Ft, Fx, Fx~. The following [tO's formula holds.

130

J. Zabczyk

T h e o r e m 2.13. Under the above assumptions on the function F and the process Z , F-a.s., for all t 9 [0, T],

F(t,Z(t))

=

F(0, Z ( 0 ) ) +

+

/o

f

{Fx(s,Z(s)),~(s)dW(s))H

(2.25)

Z(s)) +

1

+-~ Tr[(~(s)Q1/Z)*F~ (s, Z(s) )OP(s)Q1/e)]} ds. For more details see [DaPrZa2] . 2.3.3 S t o c h a s t i c e q u a t i o n s . In the lecture notes we will often deal with solutions X ( t , x) t > O, x 9 H, of the following stochastic equation,

dX

=

(AX + F(X))dt +G(X(t))dW(t),

(2.26)

X(0)

--

x 9 H,

(2.27)

on H. In the equation 2.26 - 2.27, F and G are mappings from H into H and from H into L(U, H) respectively and A denotes a linear operator, usually unbounded. We wilt require that: The operator A generates a C0-semigroup S(t), t >_ O, of linear operators on H. A solution to 2.26-2.27 is a progressively measurable H - v a l u e d process X ( t ) = X ( t , x ) , t >_ O, such that for each t > O, P--a.s.:

X ( t ) = S(t)x +

fo

S(t - s ) F ( X ( s ) ) ds +

/o

S(t - s)G(X(s)) dW(s). (2.28)

In particular if U = R m, H = R ~ and A = 0 the equation 2.26 is the classical ItO equation. Note also that if F = 0 and G = 0 and H is a general Hilbert space then our assumption on A is equivalent to the requirement that the Cauchy problem

dX -- = AX, dt

X(O) = x

(2.29)

is well posed. The equation 2.26 is called linear if F(x) = 0, x E H and G(x) = B , x 9 H for some B 9 L ( U , H ) . We write also XX(t) instead of X ( t , x ) . For more details see [DaPrZa2] .

3. H e a t E q u a t i o n The chapter is devoted to the existence of solutions to the heat equation on a separable Hilbert space. Regularity of the solutions is studied as well.

Parabolic equations on Hilbert spaces

131

3.1 I n t r o d u c t i o n We recall first basic formulae for the solutions of the heat equations in R d and then we pass to the heat equation on a separable Hilbert space H . Let Q = (qij) be a symmetric, nonnegative definite d • d-matrix. One of the most studied partial differential equations is the heat equation:

Ou

1 d 02 u . (t,x) = ~ j 1Eq i J ~ ( t ' x ) '

u(0, x) =

x

.

t > O, x = ( x l , . . . , x g ) E R d (3.1) (3.2)

R d.

It is well known, see e.g. [Fr] , and [Kr] , t h a t if the matrix Q is invertible and ~ is a bounded and continuous function on R d then there exists a unique, continuous and bounded function u on [0, +c~) • R d such t h a t partial derivatives

O---~U(x,t) D2u (t,x)i,j = 1,...,d, Ot ' OxiOxj are well defined and continuous on (0, +c~) • R d and 3.1 and 3.2 hold. Moreover the solution is given by the following explicit formula:

u(t,x)

=

=

1

v/(27rt)ldetQI

f

/R

e-1/2t(Q-I(Y-X)'Y-Z)(fl(Y)

~(x +y)nt(y)dy,

dy

(3.3)

t > 0, x C R d

JR d where :

nt(y) =

1

e -1/2t(Q-ly'y), y E R d.

(3.4)

V/(27rt)dl det QI One can check directly t h a t the functions pt(x, y) = rtt(x - y ) , t > O, x, y, C R ~, satisfy the C h a p m a n - Kolmogorov equation. It follows from 3.3 that the solution u is a C ~ function, for each t > 0 , and depends linearly on the initial function ~. In fact denote by Cb(R d) and UCb(R d) the Banach spaces of all bounded continuous and all bounded and uniformly continuous functions on R d equipped with the supremum norm. Then for each t > 0, the following formula u(t, x) = P ~ ( x ) = f ~ ~(x + y ) n t ( y ) @ ,

x C n d,

(3.5)

defines a linear bounded operator Pt transforming functions ~ from Cb(R d) onto functions in Cb(R d) and from UCb(R d) onto functions in UCb(Rd). From the C h a p m a n - Kolmogorov equation or from the fact that the formula 3.3 defines a unique solution to 3.1-3.2, it follows t h a t the family Pt, t > 0 has the semigroup property:

132

J. Zabczyk

Pt+s = Pt(Ps),

t, s > 0.

(3.6)

If one sets P0 --/, then 3.6 is valid for all t, s _> 0. Basic properties of the solution u can be expressed in terms of the semigroup 3.5 as follows.

T h e o r e m 3.1. i)Assume that the matrix Q is positive definite. Then the formula 3.5 defines a semigroup of bounded operators on Cb(R d) and on UCb(Rd). The semigroup is strongly continuous on UCb(R d) but not on

Cb(Rd). ii) For arbitrary ~ E Cb(R d) the function Pt~a(x), t > O, x E R d is of class C ~ with respect to variables t > 0 and x E R d. We omit the proof as more general results will be proved in the next chapter. We assume now t h a t Q is only nonnegative definite, symmetric matrix. Let el,- 9 9 ed be its eigenvectors and 71,- 9-, 7,~, m < d, its positive eigenvalues. If m = d then Q is invertible and we therefore assume t h a t m < d. Note t h a t

E qij C92U OxiOx~-~(x) = Tr QD2u(x), i,j

x E R d,

where D2u(x) denotes the second Fr~chet derivative of u at x and the trace of a quadratic matrix is the sum of its diagonal elements. Therefore, in the coordinates determined by the eigenvectors of Q, m

Tr QD2u(x) = E

2 Ou

3'i~x~ (x) "

(3.7)

i=1

The solution u(t,x), t >_ O, x E R d of

Out x 02u" -g-i(, ) = ~,1 ~ ~x~(t,x), u(0,x)

=

t>0,

xe

Rd

,

(3.8)

~(x).

is given, for x = ( x l , . . . ,Xm,X,~+l,... ,Xd), by the formula: ~(t, x)

=

v / ( 2 t ~ ) "~11-I~%-~ ~

In

e - ~ ~ ' ~ ' ~(~,_u,)2

(3.9)

. . . dym.

•

(3.10) If ~ is in Cb(R d) then, in general, u(t, x), t > 0, x E R d is not a C~-function. This is clear if, for instance, ~ depends only on the variables Xm+l,... ,Xa, say ~ ( x l , . . . , Xd) = ~ao(Xm+l,... ,Xd), as then

u(t,x)

= ~0(~d~+~,...,xd),

x C Rd

Parabolic equations on Hilbert spaces

133

and therefore the solution u has the same regularity as the initial function ~0. We will see that the lack of the regularizing power for degenerate heat equation in R d will be shared by all heat equations on infinite dimensional Hilbert spaces. Let finally H be an infinite dimensional Hilbert space and Q a bounded selfadjoint, nonnegative operator on H. As we already know if H = R d then equation 3.1-3.2 can be written in the following way,

-Ou ~ ( "t, x)

=

x)

=

1Tr2 QD2u(t' x),

t > O, x 9 H.

(3.11)

(3.12)

Note that the formulae 3.3-3.4 for the solutions loose their meaning if dim H = +co. They depend on the dimension d of the space and are written in terms of the Lebesgue measure for which an exact counterpart in infinite dimensions does not exist. One way to construct a solution to 3.11-3.12 in the infinite dimensional situation would be to consider a sequence of equations

Ou,~ .

1

(t, x)

=

2--TrQnD2u,~(t,x),

u,~(O,x)

=

~(x)

t > O, x E H

(3.13) (3.14)

with finite rank nonnegative operators Q~ converging to Q. For each natural n the equation has a unique solution u~ given by formula (3.9). If the sequence of solutions us were convergent then its limit could be taken as a candidate for the solution to 3.11-3.12. The following theorem shows that to define the solution this way some restrictions on the operator Q have to be imposed. Let us recall that if Q is a nonnegative linear operator on a Hilbert space then its trace is given by the formula: ~-00

Tr Q = ~

< Qek, ek >

k=l

for some (and then for all) complete, orthonormal basis (e~). T h e o r e m 3.2. Assume that ~ E Cb(H) and limlyl_++oo ~(y) = 0. If TrQ =

+c~ and (Qn) is a sequence of finite rank, nonnegative operators converging strongly to Q, then lim u,~(t,x) = 0,

n-+q-~

for all t > 0 and x E H.

Proof. It easily follows that lima Tr Qn = + ~ . Without any loss of generality one can assume that x -- 0. Let us consider first a special function ~: = e -1.12, x E H .

134

J. Zabczyk

Let e ~ , . . . , enm~ and V~,. 99 ,V,~ be the eigenvectors corresponding to all positive eigenvalues of Qn. Then, by (3.9), for the corresponding solutions Un, mn

~,~(t,0) = ( H (1 + 2t7~))- 89 k=l One can assume t h a t for some o~ > 0 2t7~ _< a,

k = 1,2,...,rn,~, n = 1,2 . . . .

Then, for a constant ~ > 0,

_< e- (E :I Since Tr Qn = ~ km~ = l 7kn --+ +C~, the result is true for the special function. If now ~ is any function satisfying the conditions of the theorem then, for any > 0 one can find a 5 > 0 and a decomposition ~a = (flo + ~ 1

,

with ~o , ~1 C Cb(H) such that, I p o ( x ) l _ < S e -]~]2,

I~l(x)l <_e,

xeH.

Consequently, for the solutions u~ , lu~(t, 0)l < 5 ~ ( t , 0 ) + ~ , and lira sup lun(t, 0)1 _< e . Since e is an arbitrary positive number the proof of the theorem is complete. [] T h e o r e m 3.2 indicates that if Tr Q = + ~ then, for a majority of initial functions ~, the equation 3.11 does not have a continuous solution on [0, + ~ ) • H . This is why we will assume that the symmetric nonnegative operator Q is of trace class: T r Q < +co. (3.15) The same condition has already appeared in the Preliminaries in connection with Gaussian measures. We go back to the heat equation. It is easy to see that if 3.15 holds and (p E Cb(H) then approximate solutions u , from Theorem 3.2 are given by the formula:

un(t, x) = /H ~(x + y)N(tQ~ ) (dy),

x C H, t >_ O.

(3.16)

They converge, as n --+ +oo, to the function u:

u(t,x) = f ~(x + y)N,Q(dy), JH

x C H, t > O.

(3.17)

Parabolic equations on Hilbert spaces

135

Moreover the operators Pt:

Pt~(x)

~(x + y)NtQ(dy),

x 9 H, ~ 9 Cb(H)

(3.18)

form a semigroup of operators. The function

u(t, x) = Pt~(x),

t > O, x 9 H

(3.19)

where (Pt) is given by 3.18 will be called the generalized solution to 3.11. This is in agreement with the finite dimensional case. In fact we have the following counterpart of Theorem 3.1. 3.3. The formula 3.18 defines a semigroup of bounded operators on Cb(H). The semigroup (Pt) is strongly continuous on UCb(H) but not on Cb(H).

Theorem

We omit the proof of the theorem as more general results will be shown in the next chapter. We will call (P~) the heat semigroup.

3.2 Regular initial functions It is instructive to realize that the heat equation 3.11 is the Kolmogorov equation corresponding to the simplest I t O ' s equation:

dX = dW

, X ( 0 ) = x,

(3.20)

on a Hilbert space H, where W is a Wiener process on H, with the covariance operator Q, see the Preliminaries. However this fact will be not used in the present chapter. It will be shown in 3.6 that for some initial functions ~ E UCD(H) the generalized solution u(t, x), t > 0, x E H of 3.11 is not Lipschitz in the state variable, for t > 0 , so it can not satisfy 3.11 in the classical sense. In this subsection we show t h a t if the function ~ is more regular then 3.17 defines a classical solution to 3.11. A function u(t, x), t > O, x C H is said to be a strict solution to 3.11 if its first time derivative and its second Fr6chet derivative, in the space variable, exist for all t > 0, x C H , are continuous on [0, + ~ ) • H and satisfy 3.11-3.12. We denote by UC~(H) the space of all real functions on H which are n-times Fr~chet differentiable with all derivatives of order k = 0, 1 , . . . , n bounded and uniformly continuous. We have the following lemma. Lemma

3.4. If ~ C UC~(H), then for arbitrary t > O, u(t,.) E UC~(H)

and Du(t,x)

=

. / , D~(x + v~y)NQ(dy)

D2u(t,x)

=

f D2~(x + v~ty)gQ(dy), JH

(3.21)

t > O, x 9 H.

(3.22)

136

J. Zabczyk

Proof. The integral in 3.21 is of Bochner type and in 3.22 is strong Bochner, see [DaPrZa2]. Let g, h E H then Du(t,x;g) D2u(t,x;g,h)

= ./ (D~(x + v~y),g}NQ(dy) = /H(D2~(x + x/~y)g,h}NQ(dy),

by the rules of classical analysis and therefore 3.21-3.22 follow. The uniform continuity of the functions u(t,-), Du(t, .), D2u(t,-) is a consequence of the formulae 3.21-3.22. [] The following theorem is taken from [DaPrZa2].

[f ~ E UC~(H), then the function u given by 3.17 is a strict solution to 3.11-3.12.

T h e o r e m 3.5.

Proof.

By the mean value theorem

u(t,x)

= ~(x) + v q.f (D~(x),y)Nq(dy) +~t i / H (D2~(x + (7(y)v/ty)y, y}NQ(dy),

where cr is a Borel function from H into [0, 1]. Note that

H(D~(x),y)NQ(dy) H(D2~(x)y,y}NQ(dy)

=

0,

= TrQD2~(x),

x EH

and consequently u(t, x) - u(0, x) t

1

2

:

Tr QD2u(O, x)

~ fH ([D2~(x + a(y)v/ty)- D2~(x)]y,y)NQ(dy).

Therefore

u(t, x) -t u(0, x)

21Tr QD2u(O, x)

<_

and x) -

for all x E H.

x)

89Tr QD2u(O, x),

as t -+ O,

Parabolic equations on Hilbert spaces

137

This way we have shown that c~+

--~u(O,x) = ~ TrQD2u(O,x), Z

x 9 H.

Now fix s > O. Since Ps+t(p = Pt(Ps~), applying the previous argument with replaced by Ps~o, we obtain for all x 9 H , a+ 1 - ~ u ( s , x) = ~ Tr QD2u(s, x),

s >_ 0, x 9 H.

(3.23)

The right hand side of 3.23 is continuous on [0, +oc) • H . In particular, for every x, the right derivative in time of u is a bounded and continuous function in s 9 [0, -t-oo). From the elementary calculus, u(., x) is continuously differentiable and the result follows. []

3.3 G r o s s L a p l a c i a n We will see in w t h a t for ~ E UCb(H) the generalized solution u might be even not Lipschitz in x for positive t > 0. However if ~ is twice Fr6chet differentiable then,

TrQD2(p(x) = ~j-~ Ti-A-j~2 (x), x E H , 'J"' i

i:l

where ~'i are eigenvalues of Q and ~ ( x )

stand for second derivatives of

in the directions of eigenvectors ei. Therefore to give a meaning to the right hand side of the heat equation on H one does not need t h a t u is twice Fr~chet differentiable in space but only that some of its directional derivatives exist and are properly summable. This way we are lead to the concept of Gross Laplacian, see [Gr]. Let (B, II ' liB) be a separable Banach space and G a linear dense subspace of B equipped with a scalar product (-, "/c and the Hilbertian norm: Itgllc = ~ , g e G. It is assumed (G, I1" IIc) is a separable Hilbert space and for some c > 0 IIgllB _< cllglIG, g 9 G. (3.24) Identifying G with its dual G* and taking into account t h a t the embedding G C B is continuous one can identify B* with a subset of G. Thus B* C G* = G C B.

(3.25)

Let E be another Banach space and u a transformation from B into E. If there exists T 9 L(G, E) such that lim

11911 - o

Ilu(x + g) - u(x) - TglIE = 0

HgiIG

138

J. Zabczyk

then T is called the G-derivative of u at x and is denoted by Dcu(x). Replacing the space E with L(G, E) one can define the same way D~u(x) as an element of L(G, L(G, E)). Identifying L(G, L(G, E)), in the usual way, with the B a n a c h space L2(G, E) of all bilinear transformations from G into E , one gets t h a t D~u(x) C L2(G, E). In a similar way D~u(x) can be defined and if D~u(x) exists then D~u(x) E Ln(G, E). Note t h a t in particular if f l , . . . , f ~ C G and

V(tl,..., tn) = U(X -t- t l f l + . . . q- tnfn),

t l , . . . , tu E R 1,

then

On(O,...,O) _ D ~ u ( x ; f l , . . f,~). Otn . . . Ot l "' If E = R 1 then D~u(x) is said to be of trace class if,

(3.26)

+oo

ID~u(x;g.~,g.d[ <

+o0

(3.27)

?n~l

for some complete o r t h o n o r m a l system (gin). T h e n 3.26 holds for all ort h o n o r m a l bases and the Gross Laplacian Aau(x) is defined by the formula +oo

Acu(x) = E

D~u(x; g.~, gin).

(3.28)

m=l

Let now B be a separable Hilbert space H and Q : H --+ H a self-adjoint nonnegative, trace class operator such t h a t Qh = 0 only if h = 0. Define G = Q1/2(H) and,

(gl,g2)a = (Q-1/2gl,Q-1/2g2>H,

gl,g2 E G.

T h e n H* C G* = G C H and H* = Q(H) = Q1/2(G) with the induced norms. Assume t h a t u : H -+ R is twice Fr~chet differentiable at x E H with Du(x), D2u(x) its first and second Fr~chet derivatives. T h e n for a r b i t r a r y

g, gl~g2 C G: (Du(x),g>H (D2u(x)gl,g2}H

= =

(Dou(x),g)a, (D2G~t(Z)gl,g2>G,

(3.29) (3.30)

where bilinear forms D2u(x), D~u(x) were identified with linear operators on H and G respectively. Since

(Dcu(x),g)c (D2cu(x)gl,g2)G

= =

(Q-1/2Dau(x),Q-1/2g>H, (Q-1/2D2cu(z)gl,Q-1/2g2>H,

(3.31) (3.32)

one arrives at the following relations:

Dcu(x) = QDu(x),

D~u(x) = QD2u(x).

(3.33)

Parabolic equations on Hilbert spaces Moreover if (hn) is an orthonormal, complete basis in H then m = 1 , . . . , is an orthonormal and complete basis in G and

Aau(x)

=

139

gm= QU2hm,

E (D~u(x)g'~'g'~)G

(3.34)

m----1

-= E (Q-1/2(QD2u(x))Q1/2hm, Q-1/2Q1/2hm)H m=l

=

~(QU2D2u(x)Q1/2hm, hm)H. m:l

Therefore Aau(x ) =

TraceQ~/2D2u(x)Q ~/2,

(3.35)

and for regular functions u the Gross Laplacian is identical, up to the constant 2, with the operator in the right hand side of the heat equation 3.11. If G = Q1/2(H) we define the Gross derivatives DQu(x) 9 H, D~u(x) 9 L(H, H) by the formulae

DQu(x) = Q-I/2DGu(x), D2Qu(x) = Q-U2 D2u(x(Q1/2. Note t h a t Trace

D~u(x) = Trace D2QU(X),

where the traces are calculated with respect to the spaces G and H respectively. 3.4 H e a t e q u a t i o n w i t h g e n e r a l i n i t i a l f u n c t i o n s In the lecture notes we always assume that the space B , from the previous section, is a separable Hilbert space H . This way our presentation is less general t h a n t h a t of L. Gross [Gr], nevertheless it covers interesting applications and allows essential simplifications. 3.6. Let u be the generalized solution corresponding to ~ E Bb(H). Then Gross derivatives DQu(t, x), D~u(t, x) exist for all t > O, x E H and ]or g, h E H, Theorem

(DQu(t, x), g) = - ~ (D~u(t,x)h,g} 1

(g, (tQ)-l/2y)9~(x + y)NtQ (dy),

(3.36) (3.37)

= ~/H[(g , (tQ) 1/2y)(h, (tQ)-l/2y) - (g, h)]~(x + y)NtQ(dy). Moreover

140

J. Zabczyk 1 ~ll~[Io

IlDQ~z(t,z)ll

~

[ID~u(t,x)ll

<_ ~ll~llo

(3.38)

2

(3.39)

Proof. We will only derive the formula 3.36-3.37. In the following calculations we will use easy to check result that if (~, q) is a Gaussian, two dimensional random vector, ( is a bounded random variable on (~, F, ~) and r

: E(e"~+~'),

~,3 e R 1

then r is a C ~162 and 0~_r (a, 3) = E ( ~ e ~ + z v )

Oa

0r

'

030a

(a, Z) = E(O?@ "~+~v).

(3.40)

Fix t > 0, x E H, vectors g, h E H and define, v(a, fl)

u(t,x+ aQ1/2g+3Q1/2h)

=

(3.41)

= ./, ~(x + aQ1/2g +/3Q1/2h + y)NtQ(dy) = /H ~(x + y)N(aQ~/2g + 3Q~/2h, tQ)(dy),

o~,3 E R i 9

It is clear that

(DQu(t, x), g) = Ov (0, 0), Ov (D~u(t, x)h, g) - 030a (0, 0).

(3.42) (3.43)

By the Cameron-Martin formula

,(~,3)

= ./.(p(x + y)e~(~9+3h'(tQ)-l/2Y)-~]ag+3h]2NtQ(dy) (3.44) =

e ~ [~

• =

~(x + y)e-~[~(9'(tQ)-I/2Y)+/3(h'(tQ)-i/2Y)]NtQ(dy)

vl (~, 3 ) ~ (~, Z), ~, 3 c R ~.

By 3.40 ~Vl

Oa (a, 3) Ov2

-

~[~lgl 2 +fl(g,h)]vl(c~,3),

1I. ~(x + y)(g, (tQ)-l/2y)

v~

• e ~ [a(g'(tQ)- 1/2y)-b~(h,(tQ)- 1/2y)] NtQ (dy), O2V2

030~ (a, 3)

= -{

~(x + y)(g, (tQ)-l/2y)(h, (tQ)-l/2y)

• e ~[a(g,(tQ)- l/2y)+~(h,(tQ)-l/2y}]NtQ (dy).

Parabolic equations on Hilbert spaces

141

Since

Ov (o, o) 0~

=

0% (~ 0) = aZa~

~(o, o) Ov2

~--j(0, 0),

vl (0, 0) 0~-0~,u, O~v2 (~ O)

02vl (0, 0).~(0,o), + o-fO--~"

the required formula follows.

[]

For arbitrary ~o C BD(H), t > 0 and x E H, the operator D~u(t, x) is of Hilbert-Schmidt type and

T h e o r e m 3.7.

IID2QU(t, x)IIHs <_-~11~11o

(3.45)

Proof. Let (en) be an orthonormal and complete basis in H. For n, m E N define

fn,m(y) = (en, (tQ)-~/2y)(em, (tQ)-~/2y)- (e~, era),

y e H.

Then

]__ f,~,,~(y)~(x + y)NtQ(dy) Jll

n,mEN

t2

2

f

1

I]D~ u(t'x)ll2Hs = -~ E

fn,n(y)~(x + y)NtQ(dy)

+ ~_,

/~,m(y)~(x + y)g~Q(dy)

.

n:/:m

Taking into account that (en, (tQ)-l/2(.)}, n =: 1, 2 are independent, Gaussian random variables on (H, B(H), NtQ) with mean 0 and variances 1 one calculates that 0

fHf~m~(y)f~,,,n, (y)NtQ(dy) =

1

if (n,m) r ( n ' , m ' ) , , if (n, m) = (n', m'), n

2

if

Therefore the system of functions fn,m

(,~,.~)

= (w, m'),

n,m E N,

= x/~ , n E N , f n , , ~ = f n , m , n , rnCN, n # m , is orthonormal in the Hilbert space 7-/= 2

L2(H, B(H), NtQ). Since

I]D~Qu(t' x)][2Hs <- -~ E I(s

V(x + "))hi 2,

n,m

therefore

IID2Qu(t,x)ll2HS < ~

~ m,

~ = ~n

~2(x + y)N~'Q(dy) <_ Ika[l~.[]

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J. Zabczyk

For the theory of the heat equation a slightly stronger result, that the trace norm of D~u(t, x) is finite, would be needed. We will show in the final section of the present chapter that even if the initial functions is in UCb(H) one can not expect that this is true. However if the initial function is more regular, the trace of D~u(t, x) is finite, see [Gr].

If a bounded function qo has a continuous and bounded Frdchet derivative Dqo, then the operator D~u(t,x) is trace class for any t >0 a n d x C H and

T h e o r e m 3.8.

Tr D~u(t, x) = -/

(y, D~(z

+ y))NtQ (dy).

(3.46)

Moreover [Tr D2Qu(t,x)[ <_ -~(Tr Q)l/e[IDqo[[o and the heat equation holds: 0 1 ~-~u(t, x) = -~ Tr D~u(t, x),

t > O, x 9 H.

(3.47)

(3.48)

Proof. By differentiating expression 3.36 in direction Q1/2h one gets (D~u(t,x)h,g } = --~ 1 / H {9,(tQ)-l/2y)ID~(x +y),Q1/2h)NtQ(dy). By 3.37 the operator

D2Qu(t, x)g

(3.49)

D~u(t, x) is self-adjoint. From 3.49,

=

-~01/21v~~ D~o(x + y)(g, (tO)-l/2y)NtQ(dy)

= Q1/2Gg(x),

g 9 H.

Since QI/2 is a Hilbert-Schmidt operator to show that D~u(t,x) is trace class it is enough to prove that also G is a Hilbert-Schmidt operator. Note that

IICI[2HS = -[1 ~

/. (ej, D~o(x + y))(ei, (tQ)_l/2y)NtQ(dy) 2

i,j=l

(ek) is an orthonormal and complete basis in H. However the sequence ((ek, (tQ)-l/2('))) is orthonormal in (H,B(H),NtQ) and therefore by the

where

Parseval identity,

[IGII2Hs <

t

f (ej, D~o(x + y))2NtQ(dy). JH

Consequently

[IGI]~s <_ -~

[D~(x q- Y)I2NtQ(dY),

Parabolic equations on Hilbert spaces

143

and

IID2QU(t,x)H1 <_ HQ1/2]IHS . II'GNHS _< ( T r a c e q ) 1/2. --~t~tllD~N0. This way inequality 3.47 has been established and in particular D~u(t, x) is a trace class operator. To show that 3.46 holds we use 3.49. If (ek, Ak) is an orthonormal sequence determined by Q, then

E(D2ou(t, x)ek, ek} = - ~ k

((tQ)-l/2ek, y}(D~(x + y), Q1/2ek}]NtQ(dy) k=l

=~ t

~(ek,y)(D~(x+y),V~kek}

N~Q(dy)

y, D~(x + y)}NtQ(dy)

and 3.46 holds. Since

~(t, x) = ]H v(x + eTy)Nq(dy), Ou

=

f.(

_

2t1 f. (D~(x + vqY), x/ty} NQ(dy)

-

2t

D~(x + V~y), ~-~y

NQ(dy)

(D~(x+y),y)UtQ(dy), []

the equation 3.48 holds by 3.46.

Remark 3.9. It is known, see [Phi, that the set of all those points were a Lipschitz continuous functions on Hilbert is not Gateaux differentiable is of measure zero with respect to any nondegenerate Gaussian measure. Using this result one can extend the theorem to Lipschitz ~o , see[Gr] for a different proof. 3.5 G e n e r a t o r s

of the heat semigroups

As we have already said the heat semigroup:

Pt(p(x) = /H (p(x + y)NtQ(dy),

~ E UCb(H),

x E H.

is Co continuous on UCb(H) In this section we construct cores for this semigroup and show t h a t its infinitesimal generator is equal to the Gross Laplaclan. For this purpose we have to introduce several function spaces.

144

J. Zabczyk

The space UC~(H) consists of all ~ C UCb(H) such that the first Gross derivative Dv~(x) is well defined for each x E H and the mapping x DQ~(X) belongs to UCD(H, H). In a similar way C~(H) consists of all those e UGh(H) such t h a t the second Gross derivative D ~ ( x ) is well defined for all x C H and the mapping x --+ D ~ ( x ) belongs to UCb(H,L(H)). The spaces UGh(H), UC~(H) are Banach with respect to the norms:

II~II1,Q = II~llo + IIDQ~ILo,

11~l12,q = II~II~,Q + IID~llo.

By L I ( H ) we denote the space of all trace class operators on H equipped with the trace norm H" [[1. The following operator Ao:

Ao~(x) = 1 Tr D ~ ( x ) ,

x EH

(3.50)

defined on the set D(A0):

D(Ao) = {qo e UC~(H); D~qo e UCb(H, LI(H))} will play an i m p o r t a n t role. If (ek, s then

(3.51)

is the eigensequence determined by Q,

-I-~)

Aoqo(x )

=

1 Z(Dh~(x)ek -~

' ek)

(3.52)

k=l

1 +oc

=

~ Es

X C H,

k=l

where D2~(x) denotes the second directional derivative of ~ at x in the directions ek 9 The following theorem is taken from [Pr]. 3.10. Let (Pt) be the heat semigroup on a separable Hilbert space H and A its infinitesimal generator. Then

Theorem

i) A is an extension of Ao, ii) D(Ao) N UC~(H) is an invariant subspace for (Pt), dense in UCb(H), iii) sets D(Ao) and D(Ao) N UC~(H) are cores for A.

Proof. i) We will show that for ~ E D(A0), t > 0 and x C H , P ~ ( x ) = ~(x) +

Ps(Ao~)(x)ds.

(3.53)

Since Ps(Ao~p) --+ Ao~ as s $ 0 in UCb(tI), the identity implies that ~p E D(A) and Ao~o = A(p. Define,

Parabolic equations on Hilbert spaces

145

?%

R.x

= ~'~<x, ek}ek, k~-I

Q~x

= ~ Ak(x, ek)ek,

x 9 H.

n~-I

Let (Ptn) be the heat semigroup corresponding to Q?%and An its infinitesimal generator. By the finite dimensional theory, D(Ao) C D(A~) and

P~(p(x) = (p(x) +

P2(An(p)(x)ds.

(3.54)

Moreover, for ~o E UCb(H),

P2~(x) = / g ~o(x + Rny)NtQ(dy) and consequently P~qo(x) --~ Pt~o(x) as n --~ +oo. Note also, that

k n

E
1 +~

-+ ~ E(D~p(x)ek,ek> = Ao~(X),

as n -+ +oo.

k=l

In addition, q-oo

IAn(p(x)l <_ -~ ~~l[ 5

IID~p(x)lll,

(3.55)

k=l

as for self-adjoint operators S and arbitrary orthonormal basis (fk), +oo

[<sA, A>I _< 118111 k=l

with the identity taking place if (fk) is determined by S. Thus the sequence (A,~) is uniformly bounded. To establish 3.53 it is enough to prove that lira

n-+-l-oo

P~(Anqo)(x)ds =

Ps(Aoqo)(x)ds.

Since the sequence of functions P2(A~)(x), s 6 [0, t] is bounded it is sufficient to verify that for s E [0, t], lim P~ ( A ~ ) ( z ) = Ps (A0~)(x).

146

J. Zabczyk

However

IPf(A~)(x)

-

P~(Ao~)(x)] <_I P ~ ( A ~ ) ( x ) - P~(An~)(x)[ +IP~(A~)(x) - P~(Ao~)(x)]

<_ ] [An~(x + Rny) - An(fl(x A- y)]NsQ(dy) + /H[Anqo(X + y) -- Ao~(x + y)]Nsq(dy) <_11 + 12. The t e r m / 2 converges to zero by the Dominated Convergence Theorem. By 3.55, for arbitrary x, z E H and n E N, [A~(x)

-

A~V(x + z)I

< 89

-

D ~ ( x + z)lll.

Since D ~ E UCb(H, L1 (H)), also 11 ~ 0, by the same Dominated Convergence Theorem. This way the proof of i) is complete. The proof of ii) starts from the lemma []

If ~ E UC~(H), t > O, ;~ > 0 then Pt~ E D(Ao)NUC~(H) and R:~ E D(Ao) N UC~(H) where

L e m m a 3.11.

R~(x)

=

e-XSPs~o(z)d.

Proof. It follows from Theorem 3.8 and from its proof that

ItD~Pt~(x)lll

<

+oo, x E H and

So

Pt~ E D(Ao). Since DPt(p(x) = ~ D~v(x + y)NtQ(dy),

x E H,

Pt~ E UC~(H) as well. In exactly the same way one shows that R ~ E D(Ao) N UC~(g). L e m m a 3.11 implies that the subspace D(Ao) MUC~ (H) is invariant. To show that it is dense in UCD(H) we use the fact that D(A) is dense in UCb(H) and take f E D(A) and set g = f - Af. Since UC~(H) is dense in UCb(H), there exists a sequence (gn) of elements from UC~(H) such that gn -+ g in UCb(H). By lemma 3.11, f~ = Rlg,~ E D(Ao) n UC~(H),

n = 1,2,...

Since f = Rig = lim~ Rlg~ = lim~ f~, the density of D(Ao)M UC~(H) in D(A) and therefore also in UCD(H) follows. The proof of ii) is complete. Since D(Ao) M UC~ (H) is invariant with respect to (Pt) and dense therefore D(Ao) M UC](H) is a core for the generator A. Since UC~(H) n D(Ao) C D(Ao) C D(A) the set D(Ao) is also a core. []

Parabolic equations on Hilbert spaces

147

3.6 N o n p a r a b o l i c i t y We present here new results showing that the theory of the heat equation on infinite dimensional spaces differs, in m a n y respects, from t h a t on the finite dimensional one. If ~ E Cb(H) then the function u given by 3.19 is continuous on [0, +oc) • H . If H = R d and Q is nondegenerate then u is even a smooth function on (0, +r • R d. But if the nonnegative matrix Q is degenerate then, as we have seen in w the solution u might be non differentiable at some points x E R d for some t > 0 . The following theorem shows t hat if dim H = +r then for all heat equations the generalized solution is not smooth on (0, +oo) • H , for some ~ 9 UCb(H). If r 9 UCb(H) then 0 r e ( x ) will denote the directional derivative of r at x in the direction v. 3.12. Assume that d i m H = +0% T r a c e Q < +co. Then for arbitrary t > 0 and sequence (xm) of elements in H there exists (p E UCb(H) such that the function Pt~ is not Lipschitz in any neigbourhood of any point xm , m E N .

Theorem

Proof. It follows from the previous considerations t h a t for arbitrary v and arbitrary Borel bounded function ~ and arbitrary x 9 H ,

OvPt~(x) = - ~

~(x + y)(Q-1/2 (tQ)-l/2y}NtQ(dy) .

(3.56)

Let (v~) be a sequence of vectors such that: v~ 9 I m Q 1/2, lv~l = 1 and lim~ IQ-1/2v,~] = +c~. For each n, m 9 N define the following linear functionals Fn,,~ on UCb ,

Fn,.~(~) = - ~

~(x + y)NtQ(dy) 9

(3.57)

Then for the norm I[F~,.~I[ of F~,.~ ,

[IFn,ml[ = - ~

[(Q-1/2v~, (tQ)-l/2y}[NtQ(dy) .

(3.58)

Since, for each h 9 H , y -+ (h, Q-UUy) is a Gaussian random variable on (H, B ( H ) , NQ) with mean 0 and the second moment Ih[ 2, one easily gets that

lbF~,~ll = ~

lQ-1/2~I 9

(3.59)

Therefore ,for arbitrary m, limn I IF~,~ I I = +~ 9 Consequently by the Banach -Steinhaus principle of the condensation of singularities, there exists ~ 9 UCb such t h a t for each m,

supnlO~ pt~(Xm) [ = SUPnlFn,m~ [ = + ~ .

(3.60)

148

J. Zabczyk

Assume t h a t for some m the function Pt~ is Lipschitz in a neighbourhood of Xm- Then there exists 5 > 0 such that for r = Pt~

Ir

+y) - r

_< Clyl

if

lyl -< 5.

(3.61)

Let an > 0 be such that ~n < 5 and [r

+ cr,~vn) - r IOA

> ll0,~r 2

(3.62)

Then

1

CIGrnl)n[

< -

-

< c,

= 1,2,...,

(7 n

a contradiction with 3.60.

[]

There are m a n y reasons to conjecture that for arbitrary t > 0, x E H there exist initial functions ~o E UCb(H) such that the generalized solution u of the heat equation in the form 3.48 is not satisfied at (t, x) . The following theorem is a partial confirmation of the conjecture. Denote by C ~ ( H ) the set of all cylindrical functions go of the form: ~o(y) = r

e l ) , . . . , (y,e~>),

y E H,

(3.63)

where r is any bounded C ~ function of n variables, n = 1, 2 , . . . and (ek) is the basis determined by Q. It is clear that C ~ ( H ) C UCb(H). By direct calculations, or by the finite dimensional theory , D~u(t, x), t > O, x E H is a trace class operator. Let us fix t > 0 and x E H and define on functions 3.63 a linear functional F:

F(~) = TrD~Pt~(x), ~ E

C~(H).

(3.64)

3.13. The functional F, given by 3.64, can not be extended from C ~ ( H ) to a continuous functional on UCb(U).

Theorem

Proof. By Proposition 3.8, setting, for simplicity of the notation, x = 0, t = 1, one gets t h a t

= T DS (1,o) n

el>,...,iy, en>)NQ(dy). j=l Integrating by parts in Rn: F(~)

=

T r D ~ u ( 1 , O)

=

y

Parabolic equations on Hilbert spaces

149

It is therefore clear that the supremum of F ( ~ ) , with respect to all functions E C ~ (H) having the supremum not greater than 1 is equal to,

Let ~j(y) - - (Y'e~)~ - 1, j = 1,2, " .. ' y E H. Then the sequence (Vj) consists Aj of independent random variables or (H, B(H), NQ), identically distributed, with mean value 0 and the finite second moment. It therefore easily follows, for instance by the central limit theorem that, lim fH 1~1 + . . . + U,~INQ(dY) = +co.

n --+ fi- o o

This proves the result.

[]

It is well known that if d i m H < +co then the heat semigroups is continuous on (0, +co) in the operator norm and is even analytic. It follows from the following theorem that neither of those properties hold if dim H = +co. The following theorem is taken from [Gu] and the proof from [NeZa]. T h e o r e m 3.14. If d i m H = +co and kerQ = {0} then the heat semigroup

is not continuous on (0, +co) in the operator norm. Proof. If t ~ s the operator (t_~l)I is not Hilbert-Schmidt and by the Feldman-Hajek Theorem, the measures N(x, tQ) and N(x, sQ) are singul a r . Consequently llP~ - / ' s l l

=

sup ~oEUCb(H)

I1~11o<_1 sup ~EUOb(H) II~llo~l var(N(x, tQ) - N(x, sQ)) = 2.[:]

Remark 3.15. It follows from the semigroup theory that analytic semigroups are operator norm continuous on (0, +co) consequently the heat semigroup on an infinite dimensional Hilbert space is not analytic, see [DaPrZa2].

4. T r a n s i t i o n

semigroups

Transition semigroups, in spaces of continuous functions and in a space of square integrable functions are introduced. Their strong continuity is discussed.

150

J. Zabczyk

4.1 Transition semigroups in the space of continuous functions As we have already seen semigroups of operators appear naturally in connection with the heat equation. It turns out that the same is true with more general Kolmogorov equations. The corresponding semigroups are called transition semigroups. In the first section we investigate their time continuity in the spaces of continuous functions. Although we are mainly interested in the properties of the heat semigroups, several results will be proved in a much more general situation, without an additional effort. In particular some results will be valid for an important class of the translation invariant transition semigroups, solving the most general Kolmogorov equation with constant coef-ficients. Let (M, d) be a metric, separable and complete space. Any function p(t, x, F), t ~ 0, x E M F C B ( M ) , with the following properties: i) for arbitrary F C B ( M ) , p(.,., 1") is a Borel function from [0, +c~) • M into [0, 1], ii) for arbitrary (t, x) C [0, + ~ ] • M, p(t, x, .1") is a probability measure on B(M), iii) for arbitrary x C M, p(0, x, .) = 5{~}, iv) p(t + s, x, F) = fM p(t, x, dy)p(s, y, F), t, s > 0, x e M, F C B ( M ) , v) p(t, x, .) --+ 5{~} weakly as t --+ 0, is called a transition function on M. If p is a transition function then the formula:

Pt(fl(x) = / M p(t, x, dy)~(y)dy ,x E M ,

(4.1)

defines a semigroup of operators on the space E = Bb(M) of bounded Borel functions on M. Semigroups of this form are called transition semigroups. For example if M is a Hilbert space H then

p(t,x,1") = N ( x , tQ)(F),

t >_ 0, x E H, 1" C B ( H )

is a transition function on H determining the heat semigroup. It is a special case of translation invariant transition functions for which, by definition,

p(t, x, F) = p(t, x + y, 1" + y) forall x, y C H, 1" e B ( n ) .

(4.2)

Transition semigroups defined by translation invariant transition functions are called translation invariant transition semigroups on the Hilbert space H. We will recall their description here as several results valid for the heat semigroups can be easily extended to them. A family of probability measures #t, t >_ 0 , on a separable Hilbert space H , is called infinite divisible if #o

=

5{o} and

#t+s

=

#t*#s,

#~--+#oweaklyast$0 for allt, s_>0,

(4.3) (4.4)

Parabolic equations on Hilbert spaces

151

where * denotes the convolution operation. T h e n the f o r m u l a

p(t, x, F) = # t { F - x},

(t, x, F) e R+ x H x B ( H ) ,

(4.5)

defines a t r a n s l a t i o n invariant transition function on H . It is clear t h a t also conversely, if p is a translation invariant transition function on H , then the family (#t) given by ,

#t (P) = p(t,O,F) t > O F E B ( 7 0 ,

(4.6)

is infinite divisible. Infinite divisible families are completely described by the so called Levy-Khinchin f o r m u l a for the characteristic functions fit()~)~ )~ E H of #t, t >_ 0. N a m e l y

fit(A) = f ei(:~'~)#t (dx) = e t*(~), JH

t >_ O, )~ C H,

(4.7)

where / ,

1 ~5

+ f

I>1

[e i(~'x) - 1 -- i()~.x)]~,(dx), I_<1

)~ e H.

In the f o r m u l a a E H , Q is a nonnegative, trace class o p e r a t o r on H and is a m e a s u r e c o n c e n t r a t e d on H \ {0} such t h a t v{x: Ix I > 1} < + o c , f

Ixi2~(dx) < +oo,

(4.8)

I_<1 see [GiSk] and [Pa]. T h e following t h e o r e m , as the other results from the present section, are t a k e n from [TeZa]. Theorem

4.1. Any translation invariant transition semigroup is strongly

continuous on UCb(H). Proof. Assume t h a t ~ E UCb(H), t > 0. T h e n IPtg(x) - Ptf(z)l

= _<

f[~(x

f[

+ y) - ~(z + y)]#t(dy)

(x + y) -

+ y)],t(dy).

For ~ > 0 there exists 5 > 0 such t h a t if Ix - x' I < 5 then If(x) - f(x')] < ~. T h u s if [ 5 - z I < 5 t h e n also [ ( x + y ) - ( z + y ) [ < 5 and [ ~ ( x + y ) - ~ ( z + y ) [ < a. This gives IPtf(z) - Ptf(z)] < E. In a similar way, IPt~(x) - ~(x)l = I fH[~(x + y) - ~(z + y)]#t(dy)]. For ~ > 0 there exists 5 > 0 such t h a t I~(x + y) - ~(x)l < ~ if lYl < 5 and x arbitrary. Therefore

152

J. Zabczyk

IP~(z(x) - ~(x) I < ~ f #~(dy) + Jly I_<(5

211~ollf

~(dy).

Jly[>~

Since #~ -+ 6{0} weakly as t $ 0, Pt~o -+ ~o uniformly.

[]

Only in special cases transition semigroups are strongly continuous on the larger space of all b o u n d e d and continuous functions on M. In particular heat s e m i g r o u p s are not strongly continuous on CD(H), b o t h in finite and infinite dimensional Hilbert spaces H , with the exception of the trivial case when Q : 0. This is why the heat semigroup is studied, in the notes, on UCb(H) only. 4.2. Let (Pt) be a transition semigroup on a metric space M. If there exists a sequence (Xn), of elements in M ,which is not totally bounded and such that for each n E N and all sufficiently small t > O,

Theorem

p(t, xn, E \ (x~}) = 1

(4.9)

then for arbitrary (~ E (0, 1), there exists qo E Cb(M) and a sequence (tn) $ 0 such that sup IP~ ~(x) - ~(x)l > ~, for all n E N xEE

Proof. One can assume, without any loss of generality, t h a t there exists R > 0 such t h a t d(xn,x,~) > R, for all n , m E N. Fix ~ > 0. Since p is a transition function we can choose, by p r o p e r t y v), a decreasing sequence (t,~) $ 0 such that

( {z:d(z, xn)<_~

p t,x~,

>l-e,

tE(0, tn].

Moreover by 4.9, for all n E N, there exists r,~ E (0, -~) such t h a t

p(t,~, xn, {x: d(x, x,~) <_r~}) < e. Let now ~p~ be a continuous function on M with values in [0, 1] such t h a t ~o~(xn) = 1 and ~on(x) = 0 for all x E M such t h a t d(xn,x) > rn. Define

~(~) =

Z ~n(x),

x e M.

n----1

It is clear t h a t ~o E Cb(M), qO(Xn) = 1 and ~o(x) E [0, 1] for all x E M . Moreover

P~o~(x~)

=

P~(~)

+ Z

p~ov~(~n)

rn~kn

<

p(tn,x,~,{x:d(x,x,~)<_r,~})+P

<

2~,

( { t,~,x,~,

x:d(x,x,~) > -~

n EN

and this completes the proof.

[]

Parabolic equations on Hilbert spaces

153

Remark 4.3. We conjecture t h a t a translation invariant semigroup is strongly continuous on Cb(H) if an only if in the Levy-Khinchin representation : a = 0, Q = 0 and the measure v is finite. T h a t those conditions are sufficient for strong continuity directly follows from an explicit expression for the corresponding infinite divisible family ( # t ) . Also the space UCb(H) is too large for the strong continuity of several transition semigroups which will be studied in the notes. In the next chapter we will investigate the so called Ornstein-Uhlenbeck semigroups for which the transition function p is of the term:

p ( t , x , F ) = N ( S ( t ) x , Qt)(F),

( t , x , l ' ) E R+ • H • B ( H ) .

(4.10)

In 4.10 , S(t), t >_ 0, is a Co-semigroup on H generated by an operator A and Qt is a nonnegative operator of the form

Qt =

S(a)QS*(a)da,

t >_ 0

(4.11)

where Q is a symmetric, bounded nonnegative operator. We assume that

Tr Qt < +c~

for all t > 0.

(4.12)

Only if 4.12 holds the definition 4.10 is meaningful. If A = 0 then the Ornstein- Uhlenbeck semigroup is identical with the heat semigroup. Theorem

4.4. Assume that (Pt) is an Ornstein- Uhlenbeck semigroup. Then,

i) For arbitrary 99 E UCb(H),

Pt~(x) --+ ~(x) uniformly with respect to ii) For arbitrary ~ E and only if the generator iii) For arbitrary ~ E A=0.

(4.13)

x in compact sets, UCb(H), 4.13 holds uniformly on bounded sets if A is bounded UCb(H), 4.13 holds uniformly on H if and only if

Proof. One can easily show that, IPt~o(x) - ~(S(t)x)l -+ 0 uniformly with respect to x in the whole H . Therefore one can always reduce the proof to the case of Q = O. i) If K is compact then sup~cK IS(t)x - x I -+ 0 at s $ O. Thus if ~ C

UCb(H) then sup xEK

as required

IPt~(x) - ~(x)[ =

sup [v(S(t)x) - 7~(x)[ --+ O, :~EK

154

J. Zabczyk

ii) It is well known that the generator A is bounded if and only if the convergence S(t) --+ I as t $ 0 is uniform on bounded sets. Since the convergence is always uniform on compact sets, one can find, if A is unbounded, a number c~ > 0, a sequence tn $ 0 and a sequence {xn} such that, Ix~l = 1, ]x,~ - xm] > c~, and IS(t~)x~ - x~] > a, n , m E N, n y~ m. Replacing t,, by properly chosen smaller, positive number, we can assume t h a t O~

IS(t=)x~ - x ~ I = -~

and

C~

IS(t~)x= - x , , I > ~ ,

n, m E N, n # m, and (tn) $ 0. Let ~5 E UCb(H) be such that ~5(0) = 1 and ~(x) = 0 i f Ix I >_ ~. ~ Define~v~(x) = ~ 5 ( x - x ~ ) , n E N , x E H and conclude as in the proof of the previous theorem iii) By ii) we can assume t h a t A is bounded and A # 0. We can find again (tn) $ 0 such t h a t S(t~) # I. Since operators S(t,~) are invertible there exists a sequence (x~) such that Ix~-x,~t > 1, [S(tn)xn-x,~[ > 1 for all n , m E N, n # m. The proof can be now completed as in ii). [] 4.2 T r a n s i t i o n s e m i g r o u p s in s p a c e s o f s q u a r e s u m m a b l e f u n c t i o n s It is well known, see [Yo], t h a t the heat semigroup Pt on R d can be uniquely extended from the set of all bounded, Borel functions with compact supports, to the Hilbert space L 2 (R d) of all square summable functions, equipped with the norm:

The extended semigroup (Pt) is strongly continuous and consists of contraction operators:

IIP wll2 ___ II~ll~,

t >_ o, ~ E L2(Rd).

The space L 2 (R d) does not have a natural counterpart if R d is replaced by an infinite dimensional Hilbert space H , as there are no natural generalizations of the Lebesgue measure to H . In this section, see [Za3], we introduce a set of measures p such that transition semigroups, and the heat semigroup in particular, have strongly continuous extensions to the space 7t -- L2(H, #) of Borel functions ~ such that

I1~1I~ ---

fH

I~(x)r2~(dz)

< +~'

with the Hilbertian norm ]1' 114, Let M be a separable metric space equipped with the Borel a-field B(M). A m e a s u r e / ~ on B ( M ) is said to be locally finite if there exists an increasing sequence of open sets (Un) such that,

Parabolic equations on Hilbert spaces #(U,~) < +oo,

155

+oo n 9 5t and U u,~ = M. n=l

Let w > 0 be a nonnegative number. A locally finite measure # on M is said to w- excessive, with respect to a transition semigroup (Pt), if and only if

MP(t X, Y)]A(dx)<_ ewt]s

forall

t > O, F E B(M).

Here p stands for the transition function determining (Pt). The 0-excessive measures are called shortly excessive.

If a measure # is w-excessive for a transition semigroup (Pt) then the semigroup (Pt) can be uniquely extended from Bb(M)M L2(M, #) to N a strongly continuous semigroups (P~) on L2(M,#). Moreover

T h e o r e m 4.5.

II?tIIL2(M,,) ~_ e 89

t k 0

(4.14)

Proof. Denote 7 / = L 2 ( M , # ) and assume that ~ 9 Bb(M) M 7/. Then IIPt~il~t

= /M ~ P(t'x'dy)~(Y) :#(dx).

By the Schwartz inequality, 2

Consequently

IiPt~l[~

_<

/M I/MP(t,x,dy)l~(y)i2]#(dx)

(4.15)

<- e~t J'M I~(Y)]2#(dY)" The second estimate obviously holds if the function ]~1~ is replaced by an indicator function of any Borel subset. Thus, by a standard limiting procedure, that estimate holds for an arbitrary nonnegative measurable function and in conclusion for I~] 2. It follows easily from 4.15 that the extension (/5) exists and that 4.14 holds. It remains to show that, lira 1I~St~ - 91in = 0,

t--~0

for all ~ E 7/.

(4.16)

Assume first that ~ C 7-/is a bounded continuous function such that for some nEN, ~=0onU~.Then

156

J. Zabczyk (4.17)

= - 2 / M Ptqo(x)~(x)#(dx) + fM IPt~(x)12#(dx) + I(pI~" However

and by the Lebesgue dominated convergence theorem lim f

t-~O J M

Pt(p(x)r

=

IN 199(x)12/t(dx) ,~

Moreover by 4.14,

lim IIP~ll~ < I1~11~,

t-+O

and consequently t~+O

Since the functions ~ with the imposed properties form a dense set in 7-/the proof of the theorem is complete. [] A locally finite measure # is said to be invariant for (Pt) if

MP(t,x,F)p(dx) = # ( F ) ,

t _> 0, F E B(M).

It is clear that invariant measures are excessive. They exist for m a n y transition semigroups and the properties of the extended transition semigroups were an object of numerous studies, see e.g. [MaRa], [Chow], [DaPrZa3], [ChojGo2] and [AhFuZa]. The Lebesgue measure on R d is invariant for the classical heat semigroup. There are no however invariant measures for nondegenerate heat semigroups on infinite dimensional Hilbert spaces. It will easily follow from the next theorem t h a t there exist many excessive measures for them. Theorem

4.6. For arbitrary locally finite measure ~ and a number w > 0,

the measure #: #(F) =

e-~~

x, F)dtu(dx),

F E B(M)

(4.18)

is w-excessive for (Pt). If the measure #, defined by g.18 with w = O, is locally finite, then it is excessive.

Parabolic equations on Hilbert spaces

157

Proof. Let Pt*~ be a measure defined as follows: P~,(F) =

fMP(t,x,F)v(dx),

t >_ O, F 9 B ( M ) .

Then Pi*+su = Pt*(Psu), t , s >_ 0 and the measure # given by 4.18 can be written more compactly:

=

_> 0).

(4.19)

From 4.19

p* #

=

f0+~ e - ~ t P ; ( p ; ~ ) d t

=

e ~s

=

f0+~ e-~tp*+t~dt

e-~Orp~udr <_ e~~ D

This proves the result. Theorem

4.7. If

~0 coN(O, tQ)dt,

# =

then the heat semigroup has a strongly continuous extension to L2(H, #). The extended semigroup consists o/contraction operators. Proof. It is easy to check, see e.g. [Gr], that measure # is finite on bounded sets. By T h e o r e m 4.6, with u = 5{0}, the measure # is excessive and by T h e o r e m 4.5 the conclusion follows. [] Remark 4.8. In the case, important for the Malliavin calculus, when H = L2(0, 1) and the covariance operator Q has the kernel q(s, t) = s A t, t, s C (0, 1), a different weight, # =

N(O, t Q ) d t ,

was proposed in paper [NuUs]. The proof that the transition semigroup has an extension to a C o - semigroup on L2(H, #), was in [NuUs] more involved and the contraction property was missing.

5. H e a t

equation

with

a first order

term

Heat equation with a first order perturbation having linear but unbounded coefficients, is studied. Regular and irregular initial functions are considered. The perturbed equation m a y have much stronger regularizing properties than the u n p e r t u r b e d one.

158

J. Zabczyk

5.1 I n t r o d u c t i o n Let A be the infinitesimal generator of a C0-semigroups S(t), t > 0 on a separable Hilbert space H. In the present chapter we will study the following parabolic equation on H.

~-[(t,x)OU u(0, x)

=

-21Tr[D~u(t,x)] + (Ax, D~u(t,x)} ,

(5.1)

=

~(x),

(5.2)

t >0, x 9

It is a Kolmogorov equation corresponding to the solution of the following linear stochastic equation, see the Preliminaries,

d X = A X d t + dW(t),

X(O) = x,

(5.3)

where W is a Wiener process on H with the covariance operator Q. Let us recall t h a t solutions to this equation are given by the formula:

X(t, x) = S(t)x +

~0t S(t

- s)dW(s),

t >_ O.

(5.4)

The corresponding transition semigroup, called Ornstein- Uhlenbeck, is of the form:

Pt~(x) = f ~(S(t)x + y)NQ, (dy), JH

x E H, t ~ 0 ,

(5.5)

where

t

Qt =

fO

S(cr)QS*(a)da,

In particular Tr Qt < §

TrQt < +c~,

t >_ O .

(5.6)

if Q is a trace class operator.

We will find conditions on the initial function ~ and on the operators A, Q , under which the function u:

u(t, x) = Pt~a(x) = /H ~a(S(t)x + y)NQt(dy),

x E H, t >_ 0

(5.7)

is a solution to 5.1 - 5.2. We will not use any result on the stochastic equation 5.3 or on the stochastic formula 5.3. Note t h a t even if u(t,.) is a smooth function, the right hand side of 5.1-5.2 has a meaning only if x E D(A). Thus if the operator A is unbounded, the Kolmogorov equation 5.1 has a coefficient which is not everywhere defined. This fact introduces additional difficulties in the study of 5.1 - 5.2.

Parabolic equations on Hilbert spaces

159

5.2 Regular initial functions We start with rather special initial functions. Note that if the first derivative Dxu(t, x) belongs to D(A*),then equation 5.1-5.2 can be rewritten as follows:

O--U-U(t,x) Ot u(0, x)

=

]~ Tr[D~u(t,x)] + (x,A*D~u(t,x)), 2 --- ~(x), t >0, xeg.

(5.8) (5.9)

We have also written the second order term in the more convenient Gross form. A continuous function u(t, x), t >_ 0, x C H , is called a classical solution to 5.1-5.2 if u(0, x) = ~ ( x ) , x E H , and 5.8 holds for all t > 0 , x E H . In particular the first derivative of the initial function ~ h a s values in D ( A * ) . Let us recall that the generalized solution to 5.8-5.9 is of the form:

u(t, x) = /H ~o(S(t)x + y)NQ~ (dy),

t >_ O, x e H.

(5.10)

T h e o r e m 5.1. Assume that T r Q < + ~ , ~o 9 UC2(H), Dx~(x) 9 D(A*) for all x 9 H and the function A*Dx~ is continuous and bounded. Then the generalized solution to 5.8-5.9 is the classical one.

Proof. By the mean value theorem (s(t)x + y)

=

(5.11)

+ +

y)

where at(.) is a Borel function from H into [0, 1]. Therefore **(t,

+ 1/s(D~(S(t)x

=

+ at(y)y)y,y)NQ,(dy),

(5.12)

and

u(t, x) - u(0, x) t

1 Tr QD~u(O, x) - ~ ( S ( t ) x ) - ~(x) 2 t

(5.13)

We show first that

~o(S(t)x) - ~(x) --+ (x,A*D~o(x)), t

as t $ O.

Again, by the mean value theorem:

~o(S(t)x) - qo(x) = (D,~o(x + {t(S(t)x - x)), S(t)x - x) where {t 9 [0, 1]. However, for arbitrary x 9 H

(5.14)

160

J. Zabczyk S(t)x - x = A

(/0'

S(s)xds

)

,

and ~(s(t)x)

-

~(~)

t

_

1 t

(

=

(

A*Dx~(x

1

/o' I'

D=~(x+~t(S(t)x-x)),A

S(s)xds

1 + ~ t ( S ( t ) x - x ) ) , -~

} }

S ( s ) = ds

t

Since S ( t ) x - x -~ O, T fo S ( s ) x ds -~ x as t -+ 0, the relation 5.14 holds. To prove t h a t the right-hand side of 5.13 tends to zero as t $ 0 we need the following lemma. Lemma

5.2. If Tr Q < +oo t h e n ~ Q t --+ Q in the trace n o r m , as t --+ O.

Proof. Note that

lfo'

~

Qt - Q = -[

t>O,

[ S ( u ) Q S * ( u ) - Q] du,

and it is enough to show t h a t IIS(u)QS*(u)

- QII: -> 0

as u --+ O.

However tlS(u)QS*(u)

- QII:

=

<

li(S(u)Q:/=)(Q:/=S*(u))

- Q:/~Q1/211:

II(S(u)Q:/2-Q:/~)Q1/2S*(,)II:

+[IQ1/2( Q:/2 S* (u) -

Q1/2)111

II ( S ( ~ ) - Z ) Q : / 2 I I H s I I Q 1 / 2 S * ( ~ ) I I H s +llQ:/=llHsllQ:/=(s*(u)

- I)llHs.

We will prove t h a t I](S(u) - I ) Q i / 2 ] l g s

--+ O,

[ I Q : / 2 ( S * ( u ) - I)I[HS ~ O, as u $ 0.

(5.15)

Denote by (Ai, ei) the eigensequence corresponding to Q, then

II(S(~) - Z)Q:/=ll~s

+oo -- ~ ~,~llS(u)e~ - e~l[ =. i:1

+oo

Since E i = I I)ItHS

A i < -t-oo, S ( u ) e i ---} e i

as u J~ 0 for all i E N, and

IlQ:/2(S*(u) -

= II(S(u) - I)Q1/211HS, 5.15 holds and the proof of the lernma is

complete.

[]

Parabolic equations on Hilbert spaces

161

Let It denote the right-hand side of 5.13 then,

2It

--

1t /H [(D2~(S(t)x + at(y)y)y, y} -- {D2qo(S(t)x)y, y)}]NQ, (dy) 1

+ ~ Tr Q t D ~ ( S ( t ) x )

- TrQD~(x)

However IIt~l

<

f ./u ID2~(S(t)x + at(y)y) - D~(S(t)x)[ {yl2NQ~ (dy)

1

-i

<

By elementary properties of Gaussian measures on Hilbert spaces, see ([DaPrZa2], p. 57), for a constant cl > 0, ly]4NO, (dy)

1_

t

< cl Tr Qt,

and Lemma 5.2 implies that for a constant c2 > 0, 1 Wr Qt <_ c2, t

t c (0, 1).

(5.16)

Moreover NQ, ~ 5{0} weakly as t $ 0, and D2~(z), z C H is bounded and uniformly continuous, therefore, taking into account 5.16, [ItII --+ 0 as t $ 0 uniformly with respect to x E H. In addition,

II~l

< _

( ~ Q t ) D:(p(S(t)x) - Q D : ~ ( x ) I1

<_

yQt -

IID~(S(t)x) - D~(x){I

+[IQ{{llID~(S(t)x) - D ~ ( x ) l [ -+ 0

as t $0,

uniformly on compact sets of x E H. This way it was shown that, 0+u(0, x) _ 1 Tr QD~u(O, x) + (x, A*nxu(O, x)},

Ot

2

for arbitrary x E H. It is easy to see that if a function to satisfies assumptions of Theorem 5.1 then also u(t, .) satisfies the same assumptions for all t _> 0 and consequently,

162

J. Zabczyk

O+u(t, x) 1 - Tr QD~u(t, x) + (x, A*D~u(t, x)) Ot 2 for all t _> 0, x E H and the proof of t h e o r e m can be c o m p l e t e d in the s a m e way as t h a t of P r o p o s i t i o n 3.5 in w []

Remark 5.3. If A*D~o is a b o u n d e d function, uniformly continuous on b o u n d e d sets t h e n it follows, from the proof of the t h e o r e m , t h a t the time derivative ~ x) is uniformly continuous with respect to (t, x) from c o m p a c t subsets of (0, + c o ) • H . A l t h o u g h classical solutions are very special we will show now t h a t a r b i t r a r y generalized solution is a limit of the classical ones. 5.4. For arbitrary ~ E UCb(H) there exists a bounded sequence of initial functions (~,~) from UC~(H), converging pointwise to ~, for which the Kolmogorov equation 5.8-5.9 has classical solutions un converging pointwise to the generalized solution u.

Theorem

Proof. A function ~ : H --+ R is called cylindrical if there exist n E N and vectors a l , 99 9 an E H such t h a t cfl(x) = ~

x, ai)ai

,

x E H,

T h e n and only then there exists r : R n --+ R such t h a t ~(x) = r

al),...,

(x, an)),x E H .

(5.17)

A cylindrical function ~ is called special if in the representation 5.17, ai E D(A*), i = 1 , . . . , n and r E UC~(Rn). Then, by direct calculations,

D~(x) D2~p(x)

~

O~i ((x, a l ) , . . . , (x, an))ai,

0r

O~j

=

((x, al ) , . . . , (X, an})ai | aj,

i,j=l

A*D~(x)

=

~--,,0r

an))A*ai, x E H .

i=10~i

Therefore ~ satisfies a s s u m p t i o n s of T h e o r e m 5.1 and the corresponding generalized solution is the classical one. To finish the proof it is enough to show t h a t for a r b i t r a r y ~ E UCb(H) there exists a sequence (~n) of special cylindrical functions such that, i) ii)

II~,~IIo _< II~llo, n E N lim,~(x) = ~(x) = ~(x),

x E H.

Parabolic equations on Hilbert spaces

163

Since the set D(A*) is linear and dense in H , there exists an orthonormal and complete sequence (ei) composed of elements from D(A*). Define

~n(~) =~(~lelW...+~nen),

~=

( ~ l , - . . , ~ n ) e Rn.

For each n e N there exists t~ > 0 such t h a t supceR~ Ir where 1

r

~b~(~)e- 2,. &?,

- r162

< n1 ~

--

~ e R '~,

and Cn E UC~(Rn), n C N. Moreover, for x E H ,

I~(~)

-

r

r

+1(r

-

9 (x, e~))l

r

~),..-, (~, ~,,))l ...

- --+ 0 as n --+ +c~.[3

--

n

5.3 G e n e r a l i n i t i a l f u n c t i o n s This section is concerned with a class of Kolmogorov equations of the type 5.1 - 5.2 for which generalized solutions are smooth functions in the space variable for each t > 0 and for each bounded Borel initial function ~. We follow basically [DaPrZal]. We will need the following hypothesis, called range condition, see [Zal],

S(t)(H) C Q~/2(H),

for all t > O.

(5.18)

If 5.18 holds we define 1

At = Q ~ S ( t ) ,

t > 0

(5.19)

where Qt-89 denotes the pseudoinverse of r)1/2 ~ t . It follows from the closed graph theorem t h a t At , t > 0, is a bounded operator. The range condition is closely related to the regularity of the generalized solution as the following theorem shows, see [DaPrZal] . 5.5. The hypothesis 5.18 holds if and only if for arbitrary ~ E Bb(H) and arbitrary t > O, Pt~ ~ C~(H).

Theorem

Proof. Assume t h a t 5.18 holds, t > 0, ~ E Bb(H) and x E H. Since S(t)x e Qt1/2 (H) the measures N(S(t)x, Qt), N(O, Qt) are equivalent and the corresponding density ~t (x, .) is given by the C a m e r o n - M a r t i n formula:

dN(S(t)x, Qt) (y) = Qt(x, y), dN (0, Q~)

y c H,

164

J. Zabczyk

where

y E H.

Qt(x,y) : e (Atx'Q[~/~y)- 89 Therefore

Pt~(x) = / H ~(Y)e(A~x'QV1/2Y)-89

(dy).

In exactly the same way as in the proof of Theorem 3.6, one shows t h a t Pt~(x) is differentiable an arbitrary number of times. In particular, for g, h E H ,

(DPt~(x),g}

= . / (Atg, Q[l/ey}~(S(t)x + y)NQ~(dy),

(D2pt~(x)h, g} =

(5.20)

/i~i

[(Atg, Qt-1/2 y)(A~h, Qt-1/2 y} - (Atg, Ath}t5.21 )

9(fl(S(t)x + y)NQ, (dy). Assume now t h a t for arbitrary ~ E Bb(H), Pt~ is a continuous function but nevertheless for some xo E H, S(t)xo r Q~/2(H). Then, for all n E N, 0 measures N(S(t) x,~ , Qt), N(O, Qt) are singular. Consequently for arbitrary n C N there exists a Borel set K~ C H such that

g(s(t)X~

N(O, Qt)(K,~)= 1,

and i [ K = N~=I K~, then

N(S(t)x--~ If ~ = X~, then P t ~ ( ~ ) = O, n G N and Pt~ is not continuous at 0.

N(O, Qt(K))= 1. Pt~(O) = 1. Therefore the function []

It is our aim to find conditions, in addition to 5.6 and 5.18, which imply that generalized solutions satisfy the parabolic equation for arbitrary ~ E

UGh(H). We start from rewriting formulae 5.20 and 5.21 in a more compact way. The precise meaning of the integrals in the following proposition are given in the proof. Proposition

5.6.

DPt(p(x) = A~

I/ ~ E Bb(H) and conditions 5.6, 5.18 hold then

[/H Q-[1/2Y(S(t)x + y)NQ(dY)] '

(5.22)

D2Pt~(x) = A~ where the integral in 5.22 is an element in H and the integral in 5.23 is a Hilbert-Sehmidt operator on H.

Parabolic equations on Hilbert spaces

Proof. For (ek) in H ,

165

arbitrary g 9 H , and an arbitrary orthonormal, complete basis

~-,/2 Y),, g, Q-,12 t Y), = ~(g, eki(e~,"r _

y9

k~-i

where ~k(') = ( e k , Q t l / 2 ( ' ) ) , k = 1,2 . . . . is an orthonormal sequence on Let r = ~(S(t)x + y ) , y 9 H. Then

L2(H,B(H),NQ,).

/H(g,Q;

1/2Y)~(y)NQt(dY)

+~

= /HIg, Eek~k(y

))r

y)NQ~(dy)

k=l

"

k=l

where ((., .)) denotes the scalar product in L2(H, B(H), noting by II1' ]11 the norm on L2(H,B(H),NQ~),

NQ,).

However, de-

I((~, 4))12 ~ 11141112= /H ~2(S(t)~ + y)NQ~ (dy) ~ I1~11o~, k

so the series ~ k = l ek((~k,

~)) converges

in H and it is natural to define,

/H Qtl/2y~(S(t)x + y)NQ~ (dy) =- E

ek((~k, ~)) 9

k=l

Moreover,

/ QV~/2yv(s(t)x+ y)NQ~ (d~) <_llvL1o.

(5.24)

It is well known that operators ek | el, k,j C N, form an orthonormal and complete basis in the space of Hilbert-Schmidt operators on H. Let g, h E H , then H

[(g, cCt Y~i ,Q~l/Vy) ~--1/2 ~ h

L\

=

k=l

g,

(ek | e j ) ( ( ~ j , r

LkCj

(g,h)]~(S(t)x + y)NQ,(dy)

j=l

+ ~

k=l

ek | ek(((~ - 1,r

9

166

J. Zabczyk

It is therefore natural to set,

f. (Q2~y 1 | Q2~y 1 - I)~o(s(t)~ + y)No, (@) -t-~

= ~(~

(5.25)

-t-~

| ej)((~,

k•j

r

+ ~e~

| e~((~ - 1,r

k--1

However, compare the proof of Theorem 3.7, +oo

+o~

i((~k~j, r kCj

2+ ~

i((r - 1, r

2, < 2ilr 2 < 211~il~

k=l

and the integral 5.25 defines a Hilbert-Schmidt operator and <_ x/-211~l10.[:] (5.26)

H (Q[1/2y | Qt- 1 / 2y - I ) ~ ( S ( t ) x + y)NQ, (dy) HS

As a corollary we have the following result with a new estimate 5.28. T h e o r e m 5.7. Assume that ~ C Bb(H) and conditions 5.6, 5.18 are satisfied. i) If the operator AtA has a continuous extension to H then DPt~(x) E D(A*) for all x C H and IIA*DP~IIo <_ IIAtAII II~llo

(5.27)

ii) If the operator At belongs to 14(H) ,then D2P~qo(x) is trace class operator and el Tr D2pt~i{o < 4ikviloliAti{] (5.28) Proof. The estimate 5.27 follows from 5.24 and 5.22. Similarly, from 5.23 and 5.26 one has that P2P,~(x) = A;GAt (5.29)

where G is a Hilbert-Schmidt operator with the estimate 5.26. Consequently, by the lemma which follows,

I TrA;GAtl < IIA;GAtlI~ <

2~llalJ21JAtJl

2 .

Taking into account the estimate 5.26, the proof is complete.

[]

We formulate and prove the lemma which has just been used. L e m m a 5.8. Assume that G is a self-adjoint Hilbert-Schmidt operator and R E ll(H). Then R*GR C 14(H) (5.30) and [JR*GRIt1 < 23/211GI]2- [IR[I42

(5.31)

Parabolic equations on Hilbert spaces

167

Proof. Assume first that G is also nonnegative definite. Then HR*GRII1 = IIG1/2R]I 2, and by 2.4,

IIG1/2Rl]2 < 211211Gll2114" IIRII4. Since HGI/2[14 = IIG[]~/2,

IIR*GRIIt

<

211G1/21I~I[RII24

<

211GII211RII249

(5.32)

In the general case the operator G has a representation: G = G1 - G2, where G1 and G2 are respectively the positive and negative parts of G. Moreover, [IR*GR[]I <_ IIR*G1R[I1 + [[R*G2R[[1, and by 5.32

[IR*GRIlt < 2[1lGl112 + [IG2IlIIIR[I2. But IIGll2 = (IIGlll 2 + IIG2112) 1/2 and therefore IIG~II2 + IIG~ll2 ___ v@lCII2. Finally

IIR*CRIll <_ 23/211Cll~lIRtl~T h e o r e m 5.9. Assume in addition to conditions 5.6, 5.18, that the operators AtA, t > 0 have continuous extensions to H, form a strongly continuous function of t > O, and the operators AtQ89 t > 0 , are in 14(H). Then for arbitrary ~ E UCb(H), the generalized solution is uniformly continuous on [0, +oo[ • H and satisfies 5.8 for all t > 0 and x 9 H.

Proof. It follows from the demonstrations of Proposition 5.6 and of Theorem 5.7 that A*DPt~(x) = (AtA)* [./~ Q t l / 2 y ~ ( S ( t ) x + y)NQ,(dy) 1 ,

(5.33)

Tr Q 89D2Pt~(x)Q 89=

(5.34)

~Q~ A~

(Q[1/~y|

y - I)~(S(t)x + y)Ne~(dy) A~Q89

Therefore, under the assumptions of the theorem, functions

A*DPt~(x), TrQ-1/2D2pt~p(x)Q-1/2, t > 0, x 9 H

168

J. Zabczyk

are continuous in both variables. Let e(A) denote the linear subspace of

UCbc(H) = UCb(H)G iUCb(H) spanned by functions ~)h, h ~ D(A*), where Ch(x) = e i(h'~>.

By direct calculations,

uh(t, x) 0 x) Nun(t, (x, A'Dun(t, x))

=

PtCh(x) = ei(~'s'(t)h)-l/2(Q~h'h)

: =

l
1 1_Tr QD~u~(t, ~) = --~u~(t,x)(QS*(t)h,

2

S* (t)h),

t > 0,

and therefore the theorem holds for ~ C e(A). It is easy to show that for arbitrary ~ C UCb(H) there exists a sequence (~n) uniformly bounded such that F~ E UCbc(H), n E N and ~ -~ ~ uniformly on compact sets. Let u~ and u be the corresponding generalized solutions to 5.8 and x C H a fixed element. It follows from 5.33, 5.34 that

A*Dun(.,x) --+ A*Du(.,x), Tr D~u,~(.,x) --+ Tr D~u(.,x), uniformly on compact subsets of ]0, +cc[, as n --+ +c~. Since functions un satisfy 5.8 on ]0, +cw[ • H therefore the sequence ~ u ~ ( . , x) converges uniformly on compact subsets of ]0, +oc[ • H. Consequently the time derivative ~ x) exists and is continuous on ]0, +oc[. By the obvious limit argument the function u satisfies 5.8. []

Remark 5.10. Instead of using exponential functions one could take in the proof cylindrical functions ~ of the form: ~(x) = r

hll,...,(x,h~)),

9 e H,

where h i , . . . , h,~ C D(A*), r C UCb(Rn), n E N. Direct computations imply that also for them the problem 5.8-5.9 is solvable. We finish this section by a result due to G.Da Prato, see [DaPr2], obtained inedependently in [ChojGol], which complements Theorem 5.7. Its starting point is a remark that for all t > 0, and r E Bb(H),

Ptr

= Pt/2(Pt/2(r

x E H.

If the condition 5.18 holds then, by Theorem 5.5, the function Pt/2(r is continuously differentiable with bounded first derivative and therefore, with a slight abouse of notation, DPtr = Pt/2(S*(t/2)DPt/2(r Let us recall that if S(t)(H) C D(A) t > 0 then the semigroup S is called differentiable. The following proof was communicated to us by B. Goldys. T h e o r e m 5.11. Assume that conditions 5.6, 5.18 hold. Then,

Parabolic equations on Hilbert spaces

169

(i) DPtr C dom(A*) for all t > O, x E H and all r C BD(H) if and only if the semigroup S(t) is differentiable. (ii) The operator DzPtr is of trace class for all t > 0 , x E H and r e BD(H). Proof. Let BD (H, H) and Bb (H, LHs(H)) , denote the spaces of all measurable, bounded mappings from H into H and from H into LHs(H) respectively. Define, for x E H , t >_ 0, Nl(t)r

= fgS*(t)r

+ y)~t(dy),

~pEBb ( H , H ) ,

and N2(t)

=

fH S*(t)r (S(t)x -4-y) S(t)#t(dy),

r G Bb (H, LHs(H)).

Starting from appropriate cylindrical functions it is easy to prove the following lemma L e m m a 5.12. The families (Nl(t)) and (N2(t)) form semigroups of linear operators on BD (H, H) and on Bb (H, LHs(H)) 9Moreover,

DPtr

= Nl(t)Dr

r C C~(H),

(5.35)

and D2Ptr

= N2(t)D2r

r C C~(H).

(5.36)

To prove part (i) of the theorem assume that the semigroup S(t) is differentiable. Then A*NI(t) is well defined for every x E H. It follows from 5.18 , Theorem 5.5 and 5.35 that

A*DP~r

= A*NI ( 2 ) DP~/2r

and sufficiency follows. Assume that DP~r r

=f({x,h)),

E Dora (A*) and take function xEg,

where h E H and f E C~ (R). Then

DPtr

= ( / H f ' (S(t)x + Y) ,t(dY)) S*(t)h.

If we choose ] in such a way that the integral does not vanish then the necessity follows. It remains to prove (ii). If 5.18 holds then, by Theorem 2.3 from the Preliminaries, for a constant C > 0,

IIS(t)*hll 2 < [IQ~/2 hi[ 1/2 h 9 H .

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J. Zabczyk

Since Qt is a trace class operator, the operator S(t) is Hilbert-Schmidt for all t > 0 and hence, by the semigroup property, it is also of trace class . It follows that N2(t)r r 9 Cb (H, LHs(H)) is of trace class for every t > 0. By 5.36

D2ptr

= N2 ( 2 ) D2pt/2r

and (ii) follows.

[]

5.4 R a n g e c o n d i t i o n a n d e x a m p l e s In this section we investigate the range condition 5.18 in more detail and present some examples. 5.4.1 R a n g e c o n d i t i o n . The range condition has a useful control theoretic interpretation, see [Za2]. Consider a linear control system:

~-t (t) = Ay +

Q1/:u,

y(O) = x.

(5.37)

For arbitrary function u(.) e L2([O,T];H), x 9 H, denote by y~'~(')(t), t 9 [0,T] the solution to 5.37. System 5.37 is said to be null controllable in time T > 0 if for arbitrary x, there exists u 9 L2([O,T];H) such that =

0.

T h e o r e m 5.13. The range condition 5.18 holds/or a given T > 0 if and

only if system 5.37 is null controllable in time T > O. Proof. Define on L2([0, T], H) a linear operator LT with values in H by the formula:

LTU =

//

S ( T - s)Ol/2u(s) ds,

u 9 L 2 ([0, T], H).

Then its adjoint L~ acts from H into L2([0, T], H) and is of the form:

L~h(s) = Q1/2S*(T - s)h,

s 9 [0, T].

Since y(T) = S ( T ) x + LTU, the system 5.37 is null controllable in time T i f and only if Range S(T) C Range LT. (5.38) r)~/2 However, by It is therefore enough to show that Range LT ---- Range "~r the Theorem 2.3 from the Preliminaries, the identity of images is equivalent to the existence of positive constants c, C , such that "

clQ1T/2hl < In,hi <_ CIQT/2hl,

for all h 9 g .

(5.39)

It follows from the formulae for L~v and QT that:

IL~hl e =

//

[Q1/:S* (s)hl 2 ds = (QTh, h) = IQ1T/2hl2

so 5.39 holds with c = C = 1.

[2

Parabolic equations on Hilbert spaces

171

The operator At defined by 5.19 has also control theoretic interpretation, see [Za2], and [DaPrZa2]. P r o p o s i t i o n 5.14. For arbitrary T > 0 and x E H ,

IATx] e = inf

lu(s)l 2 ds; s ( r ) x + LTU = 0

.

(5.40)

Therefore the number IATxl 2 can be interpreted as the minimal energy which the system 5.37 requires to drive the state x to the origin 0. We deduce now some consequences of the general characterizations. P r o p o s i t i o n 5.15. If for all x E H and some T > O,

oT IQ-1/2S(s)xl 2 ds < +oo then 5.37 is null controllable in time T and IlQ-1/eS(s)l]2 ds

[]ATII _< ~

(5.41)

Proof. Define u(s) = - ~ Q - 1 / 2 S ( s ) x , s e [0,T]. Then y~'U(')(T)

=

S(T)x +

=

s(r)x

S ( T - s)Q 1/2 -

-

S(T

Thus 5.37 holds and T 1

'ATxl2 ~-- fo

- s)S(s)x

2

TQ-1/2S(s)x

Q-1/2S(s)x

ds

a s = O.

T

ds <_ [~2 fo

I'Q-1/2S(s)I'2 ds'

so 5.41 holds as well. C o r o l l a r y 5.16. If Q = I then for arbitrary T > 0 there exists M > 0 such

that

M IIA~I[ _< ~ ,

t e [0,T]

C o r o l l a r y 5.17. If for some 13 ~ [0, 1), M > and all t > O, M

I[O-1/2S(t)l[ <_ - y ,

t > 0,

(5.42)

t~

then

M [IAtll _< ~

1 t~x/~5_~,

t > 0.

(5.43)

Assumptions of Corollary 5.17 are satisfied if Q = I , and t h e n / 3 = 0, or if the operator A generates an analytic semigroup, see e.g. [DaPrZa2], such that I m Q 1/2 D D ( - A ) ~ .

172

J. Zabczyk

5.4.2 E x a m p l e s . We illustrate the general theory by considering specific applications. E x a m p l e 5.18. Let (e~) be a complete and orthonormal basis in H. Define operators A and Q by the relations: Aen = -anen,

Qen

= "Then,

n = 1,...

(5.44)

where am, ~f~ > 0, n E N. The operators S ( t ) , Qt, At = Q t l / 2 S ( t ) , A t Q 1/2 and A t A have exactly the same system of eigenvectors as A and Q and the corresponding sequences of eigenvalues are as follows:

an

--

~

\V~n(e2-'E~t -- 1 ) ) (' i

e 2 a2an ~ -

"

and

Consequently, if the positive sequence (an) is separated from 0, then +co

TrQt < +oo

iff

E'Y-2E < + o c r t : l O/n

and the range condition, as well as the requirement A t Q 1/2 E 14(H), hold under rather mild assumptions on the sequences (aN) and ('y~). It is interesting to note that the condition A t Q 1/2 E 14(H)

involves only the operator A and not the covariance operatorQ. Let in particular an

~

na,Tn

z

r t - ~ ' , ol,-y ~ 0 .

Then, Tr Qt < + c o iff a + 7 > 1.

If, a+7>l, then, for all t > 0, the range condition holds, the operator A t Q 1/2 belongs to 14(H) and the operator A t A is bounded. Consequently in that situation the results on the parabolic equation proved in the present chapter are applicable. We finally consider specific operators Q and A, commonly used in applications.

Parabolic equations on Hilbert spaces

173

Example 5.t9. Let H = L2(0, 1) and S(t), t > 0 be the heat semigroup d2

corresponding to the Dirichlet boundary condition. Thus A = ~ and the domain D(A) consists of all absolutely continuous functions x such that d2 x(0) = x(1) = 0, ~ x is also absolutely continuous with d~zx in H . Let Q be the following operator:

f

Qx(() =

q((, r/)x(r~) dr,

( C (0, 1), x e H ,

71-

where q(~, 7) = min(~, ~/) - ~ , ,

~, r/ e (0, 1).

Eigenfunctions and eigenvalues corresponding to Q are as follows: en(~) =v/2sin(n~r~) , ~ E (0,1) ,7n=(nTr) -2 n = l , . . . . Since the eigenfunctions form an orthonormal and complete basis in H and the eigenvalues are positive the operator Q is positive as well. Moreover:

Ae,~(~) = -(nzc)2e~(~),~ E (O, 1)n = 1 , . . . . and ~n=(n~)

2, n=l,

....

Therefore this specific example is of the previously considered form. In a similar way one can treat the case when the covariance operator is determined by the Brownian sheet, see w 2.3.

Example 5.20. Let H = L2(0,1) and S(t), t > 0 be the heat semigroup d2

corresponding to the mixed boundary conditions : A = ~ and the domain D(A) consists of all absolutely continuous functions x such that x(0) = d2 x(1) = 0, ~ x is also absolutely continuous with a-~-x in H . The operator Qx(~) =

f

q(~,v)x(v)@,

~ ~ (o,1), x e H ,

lr

has now a simpler kernel, q(~, 7) = min(~, ~/), ~, r] E (0, 1). Eigenfunctions and eigenvalues corresponding to Q are as follows: en(~) = v ~ s i n ( ( n + ~ ) ~ r ~ ) , ~ E ( 0 , 1 ) , T n = ( ( n + ) ~ r ) )

-2 n = l , . . . .

Again the eigenfunctions form an orthonormal and complete basis in H and the eigenvalues are positive and consequently the operator Q is positive . Moreover:

174

J. Zabczyk

Ae.~( () = - ( (n + 170 )2 en( ~) ,( E (0, 1), n = l , . . . Z,

and

1

a n = ((n+2)702, n = l , . . . . Thus also this example is of the previously considered form. Note that in both cases A = _ Q - 1 .

6. G e n e r a l

parabolic

equations.

Regularity

Regularity of the solutions of general second order parabolic equations on Hilbert spaces is sudied. Nonsmooth initial functions are also considered using the Bismut-Elworthy-Xe formula. 6.1 C o n v o l u t i o n t y p e a n d e v a l u a t i o n m a p s The present chapter is concerned with the Kolmogorov equation

~-[(t,Ou x) u(O,x)

=

-21Tr[(G(x)Q1/2).D2u(t,x)(a(x)Q1/2) ]

=

+(Ax + F(x),Du(t,x)}, ~(x), x C H.

(6.1)

t >_ 0, x E D(A)

The generalized solution to 6.1 is defined by the usual formula: =

=

Pt~(x),

t>_O, x c H

where the process X(t, x) t > O, x C H, satisfies the stochastic equation,

dX

=

( A X + F(X)) dt + G(X(t)) dW(t),

(6.2)

x(o)

=

xeH,

(6.3)

on H.

We first prove results on existence of solutions to 6.2 and on their dependence on initial data. To do so it will be convenient to treat 6.2, as a fixed point problem: X = K(x, X) , (6.4) where K(x, .) is a convolution like transformation defined by the right hand side of 2.28.We will use a functional analytic approach. The approach is based on implicit function theorem, postponed to the Appendix, and on properties of deterministic and stochastic convolutions, gathered in the present section. Denote by HP([0, T]) the space of all progressively measurable H-valued process r defined on [0, T], T > 0, p > 2, equipped with the norm:

Parabolic equations on Hilbert spaces

IIr

=

sup(Ellr

175 (6.5)

t
Similarly denote by ~P([0, T]) the space of all progressively measurable 7/valued processes ~ , see w167 2.3.2, defined on [0, T], equipped with the norm:

]I~']ln,[0,T] = sup(E[l~(t)II~)I/P.

(6.6)

The normed spaces HP([0, T]), 7-/P([0, T]) are Banach spaces denoted shortly by H p and ~P. We will need the following equivalent norms:

supe-~(E(llC(t)]lH)P) 1/~ t
IICIIp,~,T =

IIl~,Ilip,X,T = supe-Xt(EII~(t)lI~t) 1/p.

(6.7) (6.8)

t
Assume now that S(t), t > 0 is a C0-semigroup of bounded linear operators on H and define for r C HP([O,T]), ~ C 7/P([0,T]) two convolution type mappings: To(r

=

Tl(~)(t)

=

/o

S ( t - s)r

t e [0, T],

S(t- s)~(s)dW(s),

t 9 [0,T].

(6.9) (6.10)

Let MT = supt< T I[S(t)llL(H,H) 9 We have the following easy consequence of the Burkholder-Davis-Gundy inequalities 2.23. T h e o r e m 6.1. For arbitrary p >_ 2, T > O, the formulae 6.9 and 6.10, define linear operators from HP[0, T] into HP[0, T] and from 7-/P[0, T] into HP[0, T] respectively. In addition, for arbitrary (~ 9 1[, there exists ~ > 0 such that,

II%(r II~(~)iI,,~,T

< <

~IIr ~lll~Iilp,~,T.

(6.11) (6.12)

Proof. We will show only how to select A > 0 to fulfill 6.12. By the very definition:

ELITI(~)(t)II~

=

Eli

S ( t - sN'(s) dW(s)llPH

<_ M~cptp/2-1 ft eXtPE(e -x~ Ile(s)ll)~ ds Jo

176

J. Zabczyk

<_ M~cpt p/2-1

<

P

P (/o ' e ~tv ds ) (lll~lllp,~,T)

oXtp

M~cpt

p/2--1 ~

Ap

III~III,.:,,T. P

Therefore, 1 T p/2-1 p II%(~) ~,~,T < ~ M ~ C p - - ~ l l [ ~ ' l l l p , x , T

and it is enough to choose A<

M~cpTP/2-1

. []

(6.13)

P Assume now that F : H --+ H, G : H --+ ~ / a r e given transformations. They induce mappings Y and ~ on stochastic processes according to the formulae:

J=(r G(r

= =

F(r c(r

r 9 Hv[0, T], t 9 [0, T],

(6.14)

r 9 HP[0, T], t 9 [0, T].

(6.15)

They are often called evaluation maps. T h e o r e m 6.2. If F : H + H , G : H -+ 7-I are Lipschitz mappings with the Lipschitz constant 7 > 0 then also mappings .T" : HP([0, T]) -+ HP([0, T]), G : HP([O,T]) + ~/P([0, T]) are Lipschitz with the same constants in the norms II' II,,r.~, Ill'lllp,r,~ for any ~ > O. Proof. We consider for instance transformation ~. Let r162 then

eatPEIlG(r

-

G(r

=

e~PEllC(Cl(t))

< <

~Pe~tV~:llr "yPllr - r

-

C HP([0, T])

C(r

- r

Consequently

as required.

[]

T h e o r e m 6.3. Assume that Lipschitz mappings F : H --+ H , G : H -+ 7t have directional derivatives OkF(x;Yl,...,Yk),

okG(x;yl,...,Yk),

x E H, Y l , . . . , Y k E H

(6.16)

for k = 1 , . . . , n, continuous in all variables and such that for k = 1 , . . . , n

Parabolic equations on Hilbert spaces sup[

sup

xEH

Ilyill~l,i:l

sup [ 9 CH

IlOkF(x;yl,...,yk)llH]

'

i=1

(6.17)

..... k

sup

L [[yil[
< +CO,

177

Iloka(x;yl,...,yk)lLn] <

(6.18)

J

..... k

Then the transformations 2 r : HP([O,T]) -~ HP([0, T]), G : HP([O,T]) -+ NP([0, T]), have directional derivatives O k ' ( X ; Y 1 , . . . , Yk),

0k~(X; Y~,-.-, Yk),

k = 0, 1 , . . . , n

(6.19)

for each X 9 HP([0, T]) and Y I , . . . , Y k E HkP([O,T]). The derivatives are continuous from H p x H kp x ... x H kp into H p, and ~P respectively and k-times

sup [ sup ((Ok.Tz(X;Vl,...,Yk)(lH,] X~H~ II~llm,_
< +oo,

(6.20)

sup [ sup I[O~g(X;Yl,...,Yk)nn,] XeHp IIY~IIH~
< +o0. (6.21)

Moreover

0kT(X;h,...,Yk)(t)

akg(X;h,... ,Yk)(t)

= OkF(X(t);Yl(t),...,Yk(t)), = OkO(X(t);Yl(t),...,Y~(t)),

t 9 [0, T], (6.22) t 9 [O,T 1. (6.23)

Pro@ We will consider for instance the operator ~. Note that if the formula 6.22 holds then it defines a transformation with the properties specified in the theorem. The theorem is true if n = 1. If it is true for some k then for arbitrary X 9 H v, Y1,... ,Yk+l 9 H(k+I)P, a > 0: l [ok y ( x (t) + ~rYk+ 1 (t); ]I1 (t),. . . , Yk(t) ) - OkY( X ; Vl,. . . , Yk) (t)] = l[okF(X(t)

+ aYk+l (t); Y l ( t ) , . . . , Yk(t)) - O k F ( X ( t ) ; Y l ( t ) , . . . , Vk (t))]

o" P

1

= ] c9kF(X(t) + asYk+l (t); Y l ( t ) , . . . , Yk (t), Yk+l (t)) ds. Jo Therefore,

h

--~k-l[~l F(x (.) ~YI(.),..., <_ E

I 7.tp

F(X(t) + asYk+l(t); Yl(t),... ,Yk(t))

- O k + l F ( X (t); I/1 (t),..., Yk (t))]

--+ 0

P

Yk(.), Yk+l ())1

because a $ 0 []

178

J. Zabczyk

Let ~ : H -+ R 1 be a given function. If ( is an H-valued random variable defined on (~2,.T, P) then ~(~) given by the evaluation formula: 9

=

9

is a real-valued random variable provided ~ is a Borel function. The following classical result can be proved in a similar way as the previous theorems and therefore its proof will be omitted. T h e o r e m 6.4. i) If ~ is a bounded and continuous function then 9 is a continuous map from LP(~2; H) into Lq(~2; H) for any p, q > 1. ii) If ~ is bounded and continuous together with its first derivative then for arbitrary p > 2 and q 9 [1, ~-p_q], q~ is Frdchet differentiable from LP( ~2; H) into Lq(f2; H). iii) If ~ is bounded and continuous together with its first and second derivatives then for arbitrary p > 4 and q 9 [1, p--~2], 4~ is twice Frdchet differentiable from LP($2; H) into Lq($2; H). 6.2 S o l u t i o n s of s t o c h a s t i c e q u a t i o n s We go back to the equation 6.2 and using results of the w and of the Appendix we investigate existence of solutions to 6.2 and the character of their dependence on initial conditions x E H. T h e o r e m 6.5. Assume that F : H --+ H, G : H -+ 74 are Lipschitz mappings. Then for each p > 2, T > 0 and each x E H the equation 6.2 has a unique solution X ( . , x ) in HP([O,T]). Moreover the mapping x --+ X ( . , x ) from H into HP([0, T]) is Lipschitz. Proof. According to the definitions one has to show that the integral equation 2.28 has a unique solution in H~([0, T]). For each x 9 H and Y 9 HP([0, T]) define a mapping/C: h:(x,Y)(t)

=

S(t)x + +

/o

(6.24)

S(t- s)F(Y(s))ds

S(t - s ) G ( Y ( s ) ) d W ( s ) ,

t 9 [0, T],

from H • HP([0, T]) into HP([0, T]). By results of w the mapping/(: is well defined. It is linear in the variable x and is the sum of linear, integral mappings To and T1 composed with evaluation maps ~ and G- By Theorem 6.1 and Theorem 6.2 we can apply Theorem 10.1 from the Appendix with the function H replaced by /(:,the space A replaced by H and the space E replaced by HP([0, T]). This way we get existence of a solution X(., x) which depends continuously on x. The mapping x ~ X ( . , x ) is in fact Lipschitz and this follows from 10.3. [2

Parabolic equations on Hilbert spaces

179

T h e o r e m 6.6. In addition to assumptions of Theorem 6.5 assume that F E C 2 ( H , H ) , G 9 C~(H,7-l). Then for each p > 2 and T > 0 the mapping x --+ X(., x), from H into HP([0, T]) is 2-times continuously differentiable in x. Moreover for any y , z 9 H the processes ~Y(t) = O~X(t,x;y), ~Y'z(t) = O~X(t, z; y, z), t 9 [0, T] are the unique solutions of the following equations:

d~ y

:

(A~ y + O~F(X;~h))dt + O ~ G ( x ; ~ h ) d W ( t )

ey(o)

=

y,

&?y,z

=

(d~?y,z + O~F(X;vy,~))dt + O~G(X;7? y ' z ) d W ( t ) +02 (X; ey, ez) dt + 02G(X; ~ , z , ~?~,~)dW(t).

(6.25) (6.26)

Proof. To prove 6.25 we apply Theorem 10.2 from the Appendix. Its assumptions are satisfied by Theorem 6.3 and Theorem 6.1. Note that equation 6.25 in equivalent to the integral one:

Cy(t)

=

s(t)y+ +

(6.27)

S(t-s)O~f(X(s,z);(Y(s))ds

S(t - s ) O ~ G ( X ( s , x ) ; ~ Y ( s ) ) d W ( s ) ,

t 9 [0, T]

which has a unique solution by the contraction mapping principle. It is equivalent to equation 10.4. To derive 6.26 we apply Theorem 10.4 from the Appendix, taking E = HP([0, T]) and E0 = H2p([0, T]). By Theorem 6.3 and Theorem 6.1 assumptions of Theorem 10.4 from the Appendix are satisfied. Moreover the equation 6.26 is equivalent to 10.6. [] 6.3 Space a n d t i m e r e g u l a r i t y of g e n e r a l i z e d s o l u t i o n s As a first consequence of the previous results we have a theorem on differentiability of the generalized solution. T h e o r e m 6.7. Assume that mappings: ~ E H -+ R 1, F : H --+ H, G : H --+ 7{ belong respectively to C~ (H; R1), C 2 (H, H) and C 2 (H, 7-l). Then for arbitrary t >_ 0 the function u(t, .) is twice continuously differentiable.

Proof. The function x --+ u(t,x) can be regards as a composition of the mappings from H into LP(r2; H) and from Lp(~; H) into Lq(~; R1), given by x --4 X ( t , x), ~ --+ ~(~) and of the linear, integration operator 77 --4 E(~) from Lq(~2; R 1) into R 1. Therefore it is enough to apply Theorem 6.6 and Theorem 6.4. [] We will show now that the generalized solution u is continuous with respect to time and that the same is true for its first and second space derivatives. To do so we consider subspaces HP[0, T] and 7-/P[0, T] of HP[O, T] and 7-lP[O,T]

180

J. Zabczyk

consisting of those processes r and ~P which are continuous in t E [0, T]. Thus r 9 HP[0, T] if r 9 HP[O, T] and for each to 9 [0, T], lira E l r t--~ to

- r

= 0

In a similar way ~P C 7-/P[0, T] if ~P e nP[0, T] and for each to C [0, T], lim

t-+[0,T]

Ell~(t) - ~ ( t o ) l l ~ = 0

The spaces HP[0, T], ~/~[0, T] are linear, closed subspaces of HP[O,T], and NP[0, T] respectively and therefore are Banach spaces as well. We have also the following important proposition: P r o p o s i t i o n 6.8. The formulae 6.9 and 6.10 define continuous, linear mappings from HP[O, T] and from ~/P[0, T] into HP[0, T] respectively.

Proof. We prove the result for T1 only; the proof for To is similar. Let 0 _< s
<

- Ze(t)l"

2P-IE fo s S(s - ~)~(~) dW(~) - S(h) fo~ S(s - ~)g,(o) dW(~) p

+2P-IE f s+h S(s + h - o)~(~)dW(~)Ip "/8

(/:

)

Since ll(S(h) - l)~(cr)ll~ < ( M + I)Pl]O(<7)[I~, II(S(h) - l)~(a)ll~ --+ 0 as h -+ 0, I?-almost surely and E l : [l~P(a)[[~t da < +(x~, the first integral Ix converges to 0 as h --+ 0, (uniformly in t, s E [0, T]). It is clear that also /2 --+ 0 as h --+ 0 and the result follows. D

Taking into account Proposition 6.8 and the proofs of Theorem 6.6 and Theorem 6.7 we arrive at following time-regularity result: T h e o r e m 6.9. Under the assumptions of Theorem 6.7 the generalized solution u(t, x), t E [0, T], x E H is continuous with respect to both variables together with its first and second x-derivatives.

Parabolic equations on Hilbert spaces

181

6.4 S t r o n g F e l l e r p r o p e r t y We continue our study of the regularity properties of the generalized solution u determined by the stochastic equation, dx

=

(Ax + F ( X ) ) d t + G ( X ( t ) ) d W ( t ) ,

x(o)

=

x,

(6.28)

discussed in the previous sections. Let Pt be the corresponding transition semigroup: P v(x)

=

E(v(x(t,x))),

=

u(t,x),

t>_O, x e t t

We consider irregular initial functions ~o. For applications it is important to know when the transition semigroup (Pt) is strong Feller in the sense that for arbitrary t > 0, ~ E Bb(H), Pt~o E Cb(H). Necessary and sufficient conditions for the strong Feller property in the case F = 0 and G = I were given in Theorem 5.5, w It is easy to check that if, in particular, Q = I, then the condition 5.18 holds and the corresponding Ornstein-Uhlenbeck semigroup is strong Feller. In this section we consider rather general F and G , assume that Q = I and follow [PeZa]. T h e o r e m 6.10. Assume that F : H --4 H, G : H -+ L ( H ) are Lipschitz transformations and that for all x E H, G(x) is an invertible mapping such that for some K > O, [IG-l(x)ll < K,

x E H.

(6.29)

Assume in addition that for all t > O, S(t) E L H s ( H , H ) , (6.30)

f f IIS(o-)ll~sdo- < +oo.

Then for arbitrary T > 0 there exists a constant CT > 0 such that for all (p E Bb(H) and all t E [0, T], H.

(6.31)

Proof. We start from proving 6.31 in the case of smooth F, G and ~. L e m m a 6.11. Assume that the mappings F, G and the function ~ have bounded and uniformly continuous derivatives up to the second order. Then for each t > O, Pt~(') E UC2(H) and ~(X(t,x))

=

Pt~(x) +

/o'

(6.32)

( D P ~ _ s ~ o ( X ( t , x ) ) , G ( X ( s , x ) ) d W ( S ) , F-a.s.

182

J. Zabczyk

Proof. Sketch. Let {e~} be a complete orthonormal basis in H. For each n, let Xn(.) = Xn(', x) be the solution to the equation dXn

=

(A,~X,~+ F(X,~)) dt + G(X,~)Q~/2 dW,

x~(o)

=

~,

(6.33)

where A~ = nA(n - A) -1 is the Yosida approximation of A and Q~ is the orthogonal projection of H onto l i m { e l , . . . , e,~}, n 9 l~. It can be shown that the function

u~(t,x) = E(~o(X~(t,x))),(t,x) 9 [ 0 , + o o ) • H

(6.34)

is the classical solution of the Kolmogorov equation 0

~(t,x)

Tr[G(x)D2u~ (t, x)G* (x)]

=

(6.35)

+(A~x + F(x), Dun(t, x)l, un(0, x)

=

~o(x),

t _ 0, x 9 H

Applying ItO's formula to the process r x 9 H one obtains ]?-a.s.,

~ ( x n ( t , x ) ) = u~(t,~:)+

/o

(6.36)

= un(t - s, Xn(s, x)), s 9 [0, t],

(Du~(t-s,X,~(s,x))G(X~(x,s))dW(s)}.

(6.37)

Note that

~(s, x) Du~(s,x)

= =

E(~(x.(s,x))), E[(DXn(s,x))*D~o(Xn(s,x))],

~(s,x)

=

~.(~(X(s,x))),

Du(s,x)

=

E[(DX(s,x))*DT~(X(s,x))].

(6.38) (6.39)

Applying Theorem 10.6 and Theorem 10.7, and taking into account that the approximation procedure affects only the semigroup and the Wiener process, one can pass to the limit in 6.38 to get 6.39 and in 6.37 to arrive at 6.32. [] We derive now the so called Bismut-Elworthy-Xe formula, see [DaPrE1Za]. L a m i n a 6.12. Under the assumptions of Lamina 6.11, the directional deriva-

tive (OPts(x), h) is given by the formula, (DPt~)(x), h} 1

(6.40) t

= ~E{~a(X(t,x)) fo ( B - l ( X ( s ' x ) ) ( D X ( s ' x ) h ) ' d W ( s ) } } "

Parabolic equations on Hilbert spaces

Proof. Fix h E H and define both sides of 6.32 by

u(t,x) = Ptg~(x), t >_0,

183

x E H. Multiplying the

ot (B -1 (X(s, x))[DX(s, x)h], dW(s)), and taking expectations one gets,

E ((p(X(t,x)) foot(G-l(X(s,x))(DX(s,x)h),dW(s)}) = E

= E


[/o

,

]

ds ,

=

(D[E(Pt_s~(X(s,

=

.D

L e m m a 6.13.

x))),

h) ds,

Under the assumptions of Lemma

6.11,

the estimate

6.31

holds true. Pro@

Fix T > 0, then by Lemma 6.12, 1

[(Dpt(p(x),h}[2

t

< ~ll@l~)E{ foo IG-l(X(s,x)h]e ds } <-

K2

( rt

~-[l~][2E~J0

IDX(s'x)hl2

ds }

(6.41)

'

and we need an estimate on the process Y(t) = DX(t, x)h, t > O. By Theorem 6.6 in w the process Y is the mild solution of the equation:

dY(t) = [AY(t) + DF(X(t);Y(t))] dt + DV(X(t);Y(t))dW(s) Y(O) = h

(6.42)

The equation 6.42 is linear, with random coefficients, and equivalent to the integral equation:

Y(t)

= S(t)h + +

s(t-

S ( t - s)DF(X(S);Z(s))ds ~)Da(X(S);Y(s)dW(s)

By our assumptions, for arbitrary T > 0, there exists M > 0, such that:

184

J. Zabczyk

IDF(x;y)] IIS(cr)DG(x;y)IIHD

<_ MlYl, <_ Mly],

x,y E H

Applying the contraction mapping principle in 7/2[0, T] one easily obtains that 6.42 has a unique solution satisfying ElY(t)[ 2 _< Cih[ 2,

t E [0, T],

(6.43)

if T is sufficiently small. Reiterating the procedure one gets 6.43 for arbitrary T > 0, h E H. Applying 6.43 to 6.41 one has that

I(DPt~(x),h}t 2 <_

Ctl~It~lhl 2,

t E [0, T].

Let x, y be arbitrary elements in H. Then by the mean value theorem, for ~(y) E [0, T]:

Pt~(x) - Pt~(Y) = (DPt~(x + a(y)(x - y)), x - y}. Therefore

IP~(~) - P~(y)l

gv/-~

<_ - - ~ - I I ~ l l o l x

- yl,

as required.

[]

L e m m a 6.14. If 6.31 holds for arbitrary ~ E UC~(H), then it holds for all

c Bb(H). Proof. If ~ E UCD(H), then there exists a sequence (~n) of functions from UC~(H) such that limn ~n(x) = ~(x), limn II~nll0 = II~tl, and therefore 6.31 holds for all ~ E UCb(H). From elementary properties of measures on metric spaces one has the following estimate of the variation of the signed measure

P(t, x, .) - P(t, y, .) : Var(P(t, x, .) - P(t, y, .))

=

sup I1~,11o_<1 ~pEUOb(H)

<

--~tx-yl,

IPt~(x)

CT

- Ptv(y)l

x, y E H .

But then for ~ E BD(H): I P ~ ( x ) - P~(Y)I

<_

f ./ (z)[P(t,x, dz) - P(t,y, dz)]

<_ ]]~]]oVar(P(t,x,.) - P(t,y,.))

___ II~ll0~tt x - yI.D It follows from Lemma 6.14 that to prove the theorem one can assume that ~a E UC~ (H). It is possible, see [PeZa], to construct approximations F~ of F and G~ of G with the following properties:

Parabolic equations on Hilbert spaces

185

i) Fn, Gn, n E N are twice Fr~chet differentiable with bounded and continuous derivatives, ii) F,~, Gn, n E N satisfy the Lipschitz condition, iii) Operators Gn, n E N are invertible and lim [sup n-++oo

iv)

lim

n--+-Foo

IJGEI(x)II]

xEH

< sup -- xEH

]F~(x) - F(x)J = 0,

lim

n--++oo

tlG-~(x)ll,

JiGs(x) - G(x)l I = 0,

x EH

To complete the proof of the theorem denote by X,~(-, x), the solution of the equation

dX,~ = x~(o)

=

(AXn + Fn(X~)) dt + G~(Xn) dW(t) x

and let Pt~, t > 0 be the corresponding semigroup. Fix T > 0, ~ E UC~(H). By Lemma 6.13

CT ]P[~o(x) -P[~(Y)I <- ~ll~ollol=-yl,

x , y 9 H, t E [0,T].

However for fixed t > 0 and x 9 H there exists a subsequence (X,~k (t, x)) such that X,~(t,x) -+ X(t,x), F-a.s. as k -~ +c~. Since ~ is bounded and continuous function we have

P[~k~(x) = E(~o(X,~k (t, x))) --+ E(~o(X(t, x))) = Pt~o(x), ms k --+ + ~

and therefore

IP~(x) - P~(v)I < -C-~ll~llolx - yl,

-v~

x,y

H, t

[O,T]

as required.

[] d2

Example 6.15. Assume that H = L2(0,1r) and A = ~

with the Dirichlet boundary conditions. Eigenvalues and eigenfunctions of A are ( - n 2) and (V/-~7sinn() respectively. Consequently -J-oc

IIS(t)ll~s

=

~e

-2~,

n=l

follS(s)ll~sas t

=

+oo

~

1 - e-2n2 -2t'~ < +(x),

t > O.

Let F(x)(~) = f(X(~)), ~ E (O,~r), x E H where f is a given Lipschitz function. If in addition B - I then all conditions of the theorem are satisfied and the transition semigroup corresponding to the stochastic parabolic equation,

186

J. Zabczyk

OX x(o,~)

-

02X ~ (t,~) + f ( X ( t , ~ ) ) + ~OW - ( t ,~)

=

x(~),

~c(o,~),

t>o,

transforms bounded functions onto continuous and the estimate 6.31 holds.

7. G e n e r a l

parabolic

equations.

Uniqueness.

The problem of uniqueness of the solutions to the Kolmogorov equation is discussed. ItO formula and regularization schemes for stochastic evolution equations are used. 7.1 U n i q u e n e s s for t h e h e a t e q u a t i o n First we discuss uniqueness in the simplest case of the infinite dimensional heat equation. T h e o r e m 7.1. Assume that functions ~(x), u(t,x), D,u(t,x), D~u(t,x), D2u(t, x), t > O, x E H , are uniformly continuous on closed, bounded subsets of ]0, +oo[ • H and that equation:

-~( limu(t,x) t$o

=

~lTrD~u(t,x),

=

~(x),

t>OxEH

x E H,

holds. If, in addition, the solution u is continuous on [0, +oo) • H and for arbitrary T > 0 there exists M > 0 such that, lu(t,x)l + IID~u(t,x)l[ <_ M e MH~'I,

(t,x) e [0,T] x H,

then ~(t, x) = E(~(x + W(t))),

t > o, x e E.

Proof. Fix t > to > 0, x C H and define, r

= ~(t - s, x + W ( s ) ) ,

s e [0, to].

Due to the assumptions imposed on u one can apply ItO's formula to r r

+ +

= r

(7.1)

TrD~u(t-s,x+W(s))-~-[(t-s,x+W(s)) ( D ~ ( t - s, x + w ( ~ ) , d W ( s ) l

But u satisfies the heat equation and therefore,

ds

Parabolic equations on Hilbert spaces ~ ( t - to, x + W ( t o l ) = ~(t, x) +

(D~(t

- s, x + W ( s l , d W ( s l ) .

187 (7.2)

However, E

IlD~u(t - s, x + W(s)ll2ds <_ M2e 21[~11

ds < +co.

Applying expectation operator to the both sides of 7.2 one arrives at E ~ ( t - to, x + w ( t o ) ) = ~ ( t , x).

Since ]u(t - to, x + w(to))] < MeilxilesuP~<- ~ ]]w(s)]l and, by Fernique's theorem, see [DaPrZa2], E(eSUp~<~ IlW(s)ll) < +co, the Lebesgue theorem implies u(t, x) = lira Eu(t - to, x + w(to)) = E(~(x + w(t)). [3 to$O

7.2 U n i q u e n e s s in t h e g e n e r a l case We pass now to the general case and consider Kolmogorov equations of the form: Ou

1

=

-5-[ ( t, x )

-~ T r [ ( G ( x ) Q 1 / 2 ) D 2 u ( t , x ) ( G ( x ) Q 1 / 2 ) *] +(Ax + F(x),Du(t,x),

~(O,x)

=

~(x),

(7.3)

t > O, x 9 D ( A ) , )

x 9 H.

As before A denotes the infinitesimal generator of a C0-semigroup S(t), t >_ 0 on H , Q is a selfadjoint bounded, nonnegative operator on H, and F : H -+ H and G : H -+ 7/ are Lipschitz mappings. Equation 7.3 corresponds to the stochastic equation dX(t)

=

(AX(t) +F(X(t))dt

X(O)

=

x E H,

+ G(X(t))dW(t)

(7.4)

t>0

which has a unique solution X ( . , x ) in HP([O,T]), for any T > 0 and p > 2. Our aim is to prove the following theorem. T h e o r e m 7.2. A s s u m e that F and G are Lipschitz mappings and u ( t , x ) , t >_ O, x 9 H is a bounded, continuous function such that for each t > O, the first and second space derivatives D u ( t , x ) , D 2 u ( t , x ) , x 9 H , exist are bounded on (0, +co) • H and uniformly continuous on bounded subsets of [6, +co) • H , for any ~ > O. If u satisfies 7.3 then u(t,x) = E(v(X(t,x))), where X is the solution to 7.4.

t >__O, x 9 H,

(7.5)

188

J. Zabczyk

For the proof we will need some preparatory work. We can not repeat the proof from the previous section because of two reasons. To apply the I t 0 formula to the process:

u(t- s,X(s,x)),

s e [0, t0], to < t ,

the function u(t,x), t > O, x E H should have continuous the first time derivative, which exists, in general, only if x C D(A). Moreover the solution X of 7.4 is not given in the integral form:

X(t)

=

x + +

f

/0'

(AX(S) + F(X(S)))ds

G(X(S)) dW(S),

(7.6)

t >__O,

required by the formulation of the ItO formula, but it satisfies a convolution type version of the stochastic equation, see the w167 2.3.3. To overcome the difficulties we will approximate both the function u and the process X in such a way t h a t I t 0 ' s formula will be applicable and by passing to the limit in the formulae we will arrive at 7.5. We first need the concept of strong solutions of the stochastic equations. If a solution to 7.4 takes values in D(A),

fo

t IAX(S)[ds < +oo, for all t >_ O , P - a . s . ,

and 7.6 holds, then X is called a strong solution to 7.5. Since S(t), t >_ 0 is a C0-semigroup, there exist constants M > 0, w E R 1 such that, IS(t)l <_ M e ~*, t >_ 0 (7.7) and, for all A > w, the formula

Rxx =

e-XtS(t)x dt,

x e H

(7.8)

defines a bounded linear operator. It is well known and is easy to see that, II~R~II- ~-w

and

~Rxx--+xas~-++~.

We fix A > w and denote by T~ the operator AR~. Then T~ is a linear, bounded operator from H into the space D(A), equipped with the graph norm. For ;~ > w define

F),(x) = 7"~F(x),

G~(x) = "l-~G(x),

x e H.

It is easy to see that if F and G are Lipschitz mappings with respect to the spaces H , U then F~ and G~ are Lipschitz with respect to D(A), U.

Parabolic equations on Hilbert spaces

189

Consequently, for each initial T~x, x 9 H, there exists a unique solution X~ of the equation dX~

=

(AX~ + F~(X~)) dt + G~(X~) d W

x~(o)

=

~x

(7.9)

where A is the restriction of A to D(A). The operator .4 generates the same semigroup S(t), t ~ 0 but restricted D(A). L e m m a 7.3. If F and G are Lipschitz mappings from H into H and H into 7i respectively then, for arbitrary X > w, p > 2, T > O, the equation 7.9 has a unique solution X~ in HP([0,T], D(A)) which is a strong solution # 7 . 4 with F, G and x are replaced by F~, G~ and T~x. Moreover if )~ --+ +oo, X~ -+ X in HP([O,T], H). Proof. That the solution X~ exists has been already shown. The convergence property follows from Theorem 10.6 ii). Since X~ is a solution to 7.9 it is also a solution to 7.4 with F, G and x replaced by F~, G~ and T~x. By the very definition, for all t > O, f t IAX~(S)I ds < + ~ , IP-a.s. and this easily implies that X~ is the required strong solution. [] We pass now to the proof of Theorem 7.2 Proof. Assume that u satisfies equation 7.3 and has the properties formulated in Theorem 7.2. Fix any to 9 (0, t), A > 0 and define r

= u~(t - s,X~(s)), where u~(s,x) = u(s, Txx),

s 9 [0, to].

(7.10)

ds

(7.11)

Assumptions of the I t 0 formula are satisfied and

+
s 9 [0, to].

Here s

=

1 -~ Tr(G;~(x)Q1/2)*D2~o(x)(G:~(x)q 1/2) + ( A x + F~(x), D ~ ( x ) } ,

x C D(A),

is the value of the operator determined by the process Xx, on the function ~. Note that Oux Ot (s, x) = ~OU i (s, T~x),

D~ux(s,x) = 7"~*D~u(s, T~x),

D2u:~(s,x) = T~*D~u(s, Txx)T~, Therefore

s > O, x 9 H.

190

J. Zabczyk

.)(z)

~:~(s,

1

=

-~ Tr[(T~G~(x)Q1/~)*D~u(s,T~x)](T~Gx(x)Q ~/~) +(AT~x + T~F~(x),D~u(S,T~x)},

and ~S,X) + s

_

= --

0U~(s,%x) + ~ ( s , . ) ( ~ )

= ~:~(s,

(S,%X) +~U~(S,')(X)

+ ~(s,.)(x)

- ~(s, .)(~)

.)(x) - ~ ( ~ , . ) ( ~ x )

= ~ Tr{[(TxG)~(x)Q1/2)(T~G)~(x)Q1/2) * -(G(T~x)Q~/2)(G(T~x)Q~/2)*]D~u(s, T~x)} +(~F~(x)-F(T~x),D~u(s,T~x)}, s>0, xEH, where s is the operator determined by the process X. Since E(r = E(u~ (t - to, X~(to))) = E(u(t - to, T~X~(to)), E(r = u(t, T~x), one gets from 7.11 that E(u(t - to,

T~X~(to))

(7.12)

= u(t,T~x) + !2E fJot ~ Tr{[(%G~(X~(S))Q1/2))(T~G~(X~(S))Q1/2)* -(G(T~X~ (S) )Q~/2)(G(T~X:,(S) )Q~/2)*]D2~u(t - s, T~X~(s))} ds +E

=

( ~ F ~ ( X ~ ( s ) ) - F ( ~ X ~ ( s ) , D ~ ( t - ~, ~ X x ( ~ ) ) ) d s

~ ( t , ~ ( ~ ) ) + ~II(to)+I~,(to). L

Taking into account that the process X~ has continuous trajectories one gets, by the Lebesgue dominated convergence theorem, that

E(u(t, T~X~(t) ) = u(t, Txx) + ~ I~(t) + I~(t). It is therefore enough to show that I~(t) --> 0, I2(t) -+ 0 as A -+ +oo. We will prove for instance that l i m ~ + ~ I~(t) = 0. Note that if B and C are Hilbert-Schmidt operators then

ITrBB*-TrCC*I

~

IIBB*-CC*II1 II(g - C)B*II1 + IIC(B - C)*ll~

_< [ l U - CIIHs(IIBIIHs + IICIIHs) Let M =

sup{[lD2u(s,x)[]; s > 0, x E H}. By 7.13,

(7.13)

Parabolic equations on Hilbert spaces

II(t )

191

/o -C( ~ x ~,(S) ) )Q 1/2 IIHs(ll~ C ~,(X ~,(S) )Q 1/2 IIHs

<_ M E

II(T~G~,(X~,(S))

+llC(~X~,(S) )Q1/21[Hs)ds (L T <_ M

E

\1/2 .

II(T~G;~(X~(S)) - G(T~X~(S)))Q1/2112Hsds)

9 (ELT[,I~G~,(X~,(S))Q1/211HS+[,G(~X~,(S))Q1/21IHs]2ds) 1/2 Taking into account that G is Lipschitz form H into 7 / a n d that X~ -+ X as )~ --+ +c~ in H2[O,t;H], one sees t h a t I~(t) ~ 0 as )~ --~ +c~. []

8. P a r a b o l i c e q u a t i o n s in o p e n s e t s Heat equation with a linear first order term is considered in open subsets of a Hilbert space. Boundary conditions are of the Dirichlet type. Regularity of the generalized solution is studied. 8.1 I n t r o d u c t i o n Let O be an open subset of a separable Hilbert space H. The present chapter is devoted to the Kolmogorov equation in the set O, with the Dirichlet boundary condition:

~(t,Ou x) u(0, x)

-

21TrD~u(t,x) + (Ax, Du(t,x)}, x E O, t > 0,

(8.1)

=

~(x), x C O ,

(8.2)

u(t,y) = 0 , y E O O , t > 0 .

If O = H then the problem 8.1-8.2 is identical with the one considered in w 5. Its generalized solution was given by the formula:

u(t, x) = Pt~(x) = E ( ~ ( X ~ (t)), t _> 0, x G H , with X ~ the solution to the stochastic Ornstein-Uhlenbeck equation:

dX = AXdt + dW(t), X(O) = x.

(8.3)

The generalised solution to 8.1-8.2 is defined by a modified formula:

u(t,x) = E(~(X~(t)); T~ > t),

x C (9

(8.4)

where T5 = inf{t > 0; XX(t) e Oc},

(8.5)

192

J. Zabczyk

is the so called exit time of the process X ~ from (9. As a counterpart of the semigroup (Pt), it is convenient to introduce a family of linear operators (Rt), corresponding to the Dirichlet problem in (9, and acting on Borel functions defined on (9 in the following way

= E(

(X (t)); T$ > t),

x e (9.

(8.6)

One can show that the family (Rt) forms a semigroup: Rt+s = RtR~ which will be called the restricted semigroup. We will prove t h a t , under weak requirements, the function Rt~o is as regular in (9, for each t > O, as the function Pt~ in H. We limit our presentation to the differentiability of the generalized solution in the space variable and follow the paper [DaPrGoZa]. It is an open problem to characterize those initial functions for which the generalized solution is the classical one.

8.2 M a i n t h e o r e m Assume that ~(t), t > 0, is a decreasing positive function such that

Iin2Pt~(x)[] <_ ~/(t)]I~]10,

for all ~ E Sb(U), t > O, x E U

(s.7)

The main result of the present section is the following theorem taken from [DaPrGoZa]. T h e o r e m 8.1. Assume that 8.7 holds and that for a decreasing sequence (sn) -+ O, sl = t and an open set U C (9:

n I.xEU Then, for arbitrary ~o E Bb( (9), the function Rtqo is continuously differentiable in U. The proof will be based on several results of independent interest. The following proposition, due to Dynkin, see [Dy], is valid for general transition semigroups. If ~o C Bb (H) then /~tqo denotes a function on H given by the formula:

[tt~o(x) = { Rt~o(x) 0

if x C (9 if x E (9c.

P r o p o s i t i o n 8.2. For arbitrary t > O, s e [0, t], x E (9 and ~ C Bb((9): IR~o(x) - Ps(Rr162

~ I[~II0~(T~ ~ s).

Parabolic equations on Hilbert spaces

193

Proof. Rt~o(x)

=

E(~o(X~(t)); X=(r) E O

for all r E [0, t])

=

E(~p(XX(t)); X=(r) E O

for all r E [s,t])

=

E(~o(X=(t)); X=(r) E O

for some r E [0, s]

and X*(r) E O =

for all r E [s,t])

11 + / 2 -

By the M a r k o v property, see [DaPrZa2], /1

=

E(~(X~(t));

=

X~(r) E O for a l l r E [s,t]) X

(s) c O) =

Since Iz21

<__ [~lo~(XZ(r) E 0 c for some r E [0, s] and X=(r) E O for all r E [s,t]) <

[s

E O c for some r E [0, s])

< the p r o o f is complete. Proposition

[]

8.3. Assume that for ~o E BD(H), Pt~o is a continuous function

and that ~(~-~9 <- s) --+ 0 as s --+ 0 uniformly with respect to x from compact sets. Then for arbitrary ~ E Bb(O) and arbitrary t > O, Rt~o is a continuous function. Proof. T h e result follows i m m e d i a t e l y from Proposition 8.2. We need the following i n t e r p o l a t o r y result. 8.4. Assume that g is of class C 2 in an open set U and let d(x) = distance(x, U ~) A 2, x E U. Then

Proposition

IDg(x)l <

4

d--~ Iv"~u,,/lglv + IlD2gllv,

x E U,

where I" Iv and I1" IIv denote the suprema over the set U. Proof. A s s u m e t h a t the closure of the open ball, B(x, r) = {y E H ; Ily - xll < r }, is contained in U and Ih[ = r. By the m e a n value t h e o r e m , for each t E [0, 1] there exists z E B(x, r) such t h a t

g(x + th) = g(x) + t(Dg(x), h) + lt~(D2g(z)h, h).

(8.9)

194

J. Zabczyk

F r o m 8.9:

_< 211gllUtlh+~

(Dg(x), ~ }

~tlhlllD~gllu ~- + ,

<

and therefore, for all t 9 (0, 1)

<- (1Hg[l~u)l+t(~

IDg(x)l

<

1

~-+tt3.

-

t

are gets t h a t ~71 + t/3 _< 2V/C~(~ +/3). Consequently

Setting t =

IDg(x)l <_ 2~1/211gllU + ~rllD2gllu V

r

V

r

< ~IIIgllu+~r~llD~gilu Since r was an arbitrary, positive n u m b e r smaller t h a n d(x), the estimate follows. []

Proof. We pass to the proof of the main theorem. Note t h a t -}-oo

=

Ps,(R~-s,~)(~)

Y]~(P~+,R,-,~+,~)(=)

+

- Ps,,(R,-~o+,~)(~))

n,:l +~O

= r162162

x9

n:l

By 8.7 I[D2~bn+l(X) -

Der

_< 2~/(s~)l~lo,

x 9 H,

and by the lemma:

[DO~+l(X)

-

Dr

_< ~v/1r a(x)

- r162

- Cnl + [ID2r

-

D2r 9

However I1~n+l(X) -- l/)n(X)l ~ 2]~(T~ ~ Sn) ,

and the result follows.

[]

Parabolic equations on tlilbert spaces 8.3 E s t i m a t e s

195

of the exit probabilities

To apply Theorem 8.1 to specific examples one needs good estimates on the function V, and on the exit probabilities :

~(T$ < s), s > 0. Estimates on the function V can Uhlenbeck transition semigroups formula 8.8 for D 2 p t ~ ( x ) and the t h a t for interesting examples one

be deduced from results on the Ornsteinin w 5. In particular taking into account definition 5.19 of the function At one finds has that:

M 0 <_ 7(s) _< ~

(8.10)

for some positive 5. In particular if Q = I, then 5 = 1, see w 5. The following theorem plays an essential role in the estimates of the exit probabilities. It is an improved version of a result form [DaPrGoZa] with a simpler proof based on the arguments of [Pe] and on the properties of Gaussian measures. Theorem

8.5. A s s u m e t h a t / o r some 0 < a < 1, T > O, T

ot-"llS(t)Q1/2112HS dt < +c~.

(8.11)

Then the process Z ( t ) = f t S ( t - s) d W ( s ) , t C [0, T], where W is a Q- Wiener process, has continuous version and there exist positive constants C1 > O, C2 > 0 such that for all s C (O,T] and all r > O, F(sup NZ(t)II >_ r) <_ C l e -

C

~2

~.

(8.12)

t<s

Proof. We will use the following factorization:

Z(t)

-

sin~Tr ~ot ( t --

s)~-lS(t - s)Y(s)ds,

t > O,

(8.13)

7r

where Y(s) =

f

(s - r

- a)dW(r

s >_ 0

(8.14)

see [DaPrZa2] and [DaPrZa4], valid for fl E (0, 89 Define: = ~ 2'

no =

+ 1,

= 2n0,

=

1"

Then po 1 + qo i = 1. Moreover if n >_ no, n E N and q = 2n, p = q ( q - 1) -1 then by the HSlder inequality:

196

J. Zabczyk

[[Z(t)[[

<

sin~TrLt (t - s ) n - l l l S ( t 7r

<_

sin/37r (Tr

-

s)ll

IIY(s)llds

L t(t- 8)(~-l)P['~(t-s)HPds)I/p(Lt'[Y(s)"qds)1/q

Therefore, for all u E (0, T]:

supllZ(t)ll

< eu

t
2(B_:t)p+l(Lu

where c = ~{sin~r~2M2 ~ s , M = sup(llS(t)ll, n > no, u C [0, T], suPt<~, IIz(t)N 2

cAua

( supt<_,~ llz(t)ll2 ) '~ E \ cAu ~

)2/q

,

IIY(s)llqds

t _< T). For arbitrary A > O,

< (1/o

<

) ~)

,

-1LUE(HYT)H2)nds. U

Consequently:

E

o ~" \

)

c~u~

<_

1 -

L~t E(e (~)11~ IIV

U

By Theorem 2.7 from the Preliminaries, if EIIY(T)II 2 _< E(e~)HY(~)u2

~

1

ql

(8.16)

= C(A).

- -~EIIY(T)II 2

Note that E I I Y ( T ) I I 2 = fo~ t-<~llS(t)Qll211~sdt.

no - 1:

(8.15)

and s E (0, T],

1 q l - ~E[[Y(s)[[ 2

< -

)ds.

Moreover, for n = 0, 1 , . . . ,

E( supt<~iiZ(t)H2~n< ( E ( supt<-ullZ(t)ll2 ) )---or~ ~o

c l u '~

J

-

c~u,~

and

"~ lE(sup,<_~Hz(t)H2~ n~O

(~( .....<~,,~<~,,~)~o)~o~o <e

(8.17)

Parabolic equations on Hilbert spaces

197

Since, n 0 .' l ~

(supt<=-~ulIZ(t)]12~ n~

E\

c~u ~

/

u

< -

n O"11 u

-

u E ( ~ 01. r .(IIY)112/(s )

E e ~

no

)ds

ds

~_ no]C(A) then, taking into account 8.15, 8.16 and 8.17, // ~"Pt<. llZ(*)ll2 1

E [,e

-~o~

. <_ C(~) + e (n~176

Finally, by Chebyshev's inequality: P ( s u p l l Z ( t ) l l >_ r)

=

// supt< ~ llZ(t)ll2 ,.2 "~ ~ [e o~o~ > e rz~-

<

2 E e- ~--~-~--

]

\

t
~/

~e \

supt
/

~2 ( C ( ~ ) + e(no,C(A))l/~o ) <_ e o~" and the proof of the estimate 8.12 is complete. The proof of the continuity of Z is a consequence of the formula 8.13 and is left to the reader, see [DaPrZa2]. D The following theorem provides the required estimates on the exit probabilities. T h e o r e m 8.6. Assume that the condition 8.11 holds. Let a E (9, r > O, and T > 0, be such that,

B(a,r) C(9, and s u p l S ( t ) a - a [ < t
r

- - -4"

Define M -- sup(llS(t)N; t < T), andr0 = r / 4 M . Then there exist positive constants C1, C2 such that for s E [0, T], sup

x

C

~2

?(~-B(a,r) ~-- S) ~ Cle- 27~.

(8.18)

xEB(a,ro)

Proo]. To prove 8.18 notice that for x C B(a, r0): I[X~(t) - aN = <

[IS(t)x + Z(t) - all <_ HS(t)(x - a)ll + HS(t)a - all + z(t)H 1/2r + IlZ(t)ll.

Therefore, for x E B(a, ro) and s <_ T: P(TB(ax) -< s) -< P(sup 'IX~(t) - al[ >- r) <- P (sup "Z(t)[' >- 2 )

t<s

and it is enough to apply 8.12.

\t<_s

[]

198

J. Zabczyk

9. A p p l i c a t i o n s The theory of linear equations is applied to a class of nonlinear parabolic equations arising in stochastic control of infinite dimensional systems. An application to m a t h e m a t i c a l finance is given as well. 9.1

ttJB equation of stochastic control

Results on Kolmogorov equations, discussed in previous chapters, play an i m p o r t a n t role in the analysis of a class of nonlinear parabolic equations on Hilbert spaces. They appear in connection with control of stochastic infinite dimensional systems. We start from recalling basic control theoretic concepts. Consider a family of stochastic equations on a Hilbert space H :

dX x(o)

=

(AX + F(X,z))dt +dW(t)

(9.1)

=

parametrized by a p a r a m e t e r z from a metric space Z. Here A : D(A) C H --+ H is the infinitesimal generator of a C0-semigroup S(t), t _> 0 on H and F : H x Z -+ H is a fixed mapping. Moreover W(t), t > 0 is a Wiener process on H, with the covariance operator Q, defined on a probability space (~2, ~ , F). The probability space ($2, $-, F) is equipped with an increasing family of a-fields of events Yt C Y, t _> 0, which may happen up to time t > O. A strategy z(t), t > 0 is any progressively measurable Z - v a l u e d stochastic process . If a strategy z(.) and the initial state x E H are given then X~()(t,x), t > 0 denotes the solution to 9.1 in which the p a r a m e t e r z was replaced by the process z(.). One of the main problems in the stochastic control theory is to find a strategy z(.) which, for fixed T > 0 and x E H , minimizes the cost functional:

JT(x,z(-)) = E

,

(9.2)

In 9.2, f : H x Z ~ R 1, ~ : H ~ R 1 are given functions. There is a standard way to approach the problem of minimizing 9.2 under the constraints 9.1. One defines the so called value function V by the formula :

V(t, x) = inf Jr(X, z(.)), z(.)

t > O, x C H.

(9.3)

where the infimum is taken with respect to the all processes z(.). Denote by 12z, z E Z, the differential operator determined by the 9.1 with the p a r a m e t e r z fixed : s162

1 = ~ Tr QD2r

+ (Ax + F(x, z), De(x)),

x E H.

(9.4)

Parabolic equations on Hilbert spaces

199

One can show, by a heuristic argument, that the function V should satisfy the following equation:

O--U-U (t,5) Ot u(O,z)

= =

inf[s zez qo(x),

+ f(x,z)], x9

(9.5)

t>O.

The equation 9.4 is called a HamiIton-Jacobi-BeUman equation or shortly HJB equation. Assume that the equation 9.5 has a solution u, continuous on [0, +oo) x H with continuous first derivative Du(t,x), t > 0, x E H. Assume in addition that for arbitrary t > 0, x E H the infimum in 9.5 is attained at k(t, x) E Z. Then, in majority of cases, the solution u of 9.5 can be identified with the value function 9.3 and the optimal strategy ~(.) is of the form :

"Z(t) = k(T - t, X(t)),

t e [0, T],

(9.6)

where )~ is a process satisfying

d.~(t)

=

AX(t) + F(X(t),k(T-

2(0)

=

x,

t,.~(t)))dt + dW(t)

(9.7)

te(0,T],scH.

This means that given the function k(.,.), one solves first the stochastic equation 9.7, then the solution )( is an optimal process and corresponds to the strategy ~. We will not formulate a precise result on existence of an optimal control but only indicate the main steps of the proof that 9.6 defines the optimal control and that u = V. Let z(.) be any admissible control and XZ()(t, x), t > 0, the corresponding solution of 9.1. Applying ItO's formula to ~(t) = u ( T - t, X z() (t, x)), t E [0, T] and calculating expectations one gets that E(u(0, X ~() (T, x))) = u(T, x)

(9.8)

Since the function u solves 9.5 and the infimum in 9.5 is attained at k(t, x) E Z, therefore, for all t > 0, x E H and z E Z: Ou

~ ( t , x) + s

x) = [-s

x) - f(x, k(t, x)) + s

x) (9.9)

+f(x,z)] - f ( x , z ) = h(t,x,z) - f ( x , z ) . The function h defined in 9.9 is called the discrepancy function of the control problem. Note that,

h(t,x,z) > 0,

for all t > 0, x E H, z C Z,

(9.10)

200

J. Zabczyk

and h(t,x,z) =0,

ifz=k(t,x)

t>O, xEH,

z E Z.

(9.11)

But E(u(0, X z() (T, x))) = E(~(X z() (T, x)), and from 9 . 8 , 9 . 9 , E(~(X z() (T, x))) = u(T, x) +E

(9.12)

(J0"

[h(T - t, X Z ( ) ( t , x ) , z ( t ) ) - f ( X Z ( ) ( t , x ) , z ( t ) ) ] d t

)

.

Rearranging terms in 9.12 one arrives at: JT(X,Z(')) = u(T,x) + g

(so"

h ( T - t, X Z ( l ( t , z ) , z ( t ) ) d t

)

.

(9.13)

It follows now from 9.10 and 9.11 that for any admissible strategy z(-), JT(x,z(.))) > u ( T , x ) and if z(t) = k ( T - t, X~(')(t, x)), then z(.) is optimal. In addition we have the equality u(t,x) = V ( t , x ) , t_>0, x C H . The main identity 9.13 is usually obtained for suitable regularizations of u and X and then, by a limit argument, for the true solutions. For the control system 9.1 the HJB equation can be written in a more convenient way. The Hamittonian, related to the problem of minimizing 9.2, is a function J-l(x,p), x E H, p C H, given by the formula: 7t(x,p) = infz{(F(x , z),p) + f ( x , z)}.

(9.14)

Let s be the differential operator determined by the linear stochastic equation d X = A Z d t + dW(t),

x(O) = x.

(9.15)

Then, as we already know, 1 T r Q D 2 0 ( x ) + (Ax, DO(x)),

x C D(A),

(9.16)

and 9.5 becomes ~~u( t , x)

=

s

u(0, x)

=

~(x),

x) + Ji(x, Du(t, x)), xcH,

(9.17)

t>0.

It is important to notice that the HJB equation 9.17 is a perturbation of the linear parabolic equation:

Parabolic equations on Hilbert spaces

201

Ou 0--t- = s

u(0) = ~

(9.18)

by a nonlinear, first order term ~ ( x , Du(t, x)), x E H, t > 0. This remark will be used in the study of the HJB equation in the next section. Notice also that the discrepancy function h is related to the Hamiltonian in the following way: h(t, x, z) = [(F(x, z), p) + f ( x , z)] - 7/(x, p), (9.19) with p = Du(t, x). (9.20) Let us consider some examples.

Example 9.1. Assume that the system 9.1 is linear: d X = ( A X + Bz)dt + dW,

X(O) = x,

(9.21)

where B is a bounded linear operator from a Hilbert space Z into H. Let

JT(X,Z(.)) = E

(J0

[g(X(t,x)) + Iz(t)]2]dt + ~ ( x ( r , x ) )

/

.

Then F ( x , z ) = Bz, f ( x , z ) = g(x) + izl 2, x E H, z E Z, 7-l(x,p) = - 88 The corresponding HBJ is of the form:

Ou

N(t,x)

=

1

QD2u(t,x) + (Ax, D u ( t , x ) )

u(O,x) = ~(x), Moreover,

lIg*Du(t,x)12,

(9.22)

2. (9.23)

x C H, t > O.

1 , k(t,x) = - - ~ B D u ( t , x ) ,

t > O, x E H,

and the function h is of the form

h ( t , x , z ) = z + ~B*p 2 ,

p = D u ( t , x ) , t > O, x 9 H.

Example 9.2. Let again the control system be 9.21, but the space of the control parameters be the unite ball Z1 = {z 9 Z; ]z] <_ 1} and, JT(x,z(.)) = E

(J;

g(X(t,x))dt +~(X(T,x))

)

.

Then F ( x , z ) = Bz, f ( x , z ) = g(x), x 9 H, ]z I _< 1, and ~t(x,p) = -IB*pl. Therefore the corresponding HJB equation is of the form:

-~(t,x)OU u(0, x)

=

1_2Tr QD2u(t, x) + {Ax, Du(t, x)) - IB*Du(t, x)l, (9.24)

=

~(x),

x e H, t > 0 .

k(t,x) -

B*Du(t,x) IDu(t,x) [ '

Moreover

t > O, x e H.

202

J. Z•bczyk

9.2 S o l v a b i l i t y o f H J B e q u a t i o n Let (Pt) be the transition semigroup corresponding to the parabolic equation:

~t (t x)

=

21 TrQD2u(t,x) + (Ax, Du(t,x)),

u(O,x)

=

~(x),

(9.25)

x C H, t > O.

Applying the variation of constant formula to HJB equation 9.17 one arrives at an integral HJB :

+ We the can the We

/0

(9.26)

are concerned here with the equation 9.26 which is more convenient for analytical t r e a t m e n t than 9.17. If it has a unique solution u then one usually show, following the steps indicated in the previous section, t h a t function u is equal to the value function V. will assume that:

(A.1) For some 7 E (0,1) and M > 0, IIAtll

M

<

--

-

tT'

t > 0.

(9.27)

As far as the Hamiltonian 7-I is concerned we will require: (A.2) H is a continuous function of both variables and for a constant M > 0

[Tt(x,p) - ~t(x,q){ ~ M [ p - q [ , (A.3) For every r C CD(H), "//(.,r

x,p, q e H .

(9.28)

E Cb(H), and the transformation

r -+ 7-/(., r

(9.29)

maps bounded sets in Cb(H) into bounded sets of Cb(H). For fixed T > 0 denote by Co1'~ the space C~"r([O,T],H) of all bounded, continuous functions u : [0, T] • H --+ R 1 such t h a t for all t e (0, T] , u(t, .) C C~ (H) and the function: t~Du(t, x), (t, x) G ]0, T] • H , is measurable and bounded. The space C~ '~ with the norm:

{[u[[c~.~ = sup [lu(t,')[[o+ sup t~[IDu(t,')l[o, tC[0,T]

te]0,T]

where [[-[]0 denotes the supremum norm on Bb(H), is a Banach space. We will sketch now the proof of the following existence theorem from [CaDaPr]. The case of a linear Hamiltonian was treated in [DaPrZal]. T h e o r e m 9.3. /f the assumptions (A.1)-(A.3) are satisfied then the equation 9.26 has a unique solution u in the space C~ ''r.

Parabolic equations on Hilbert spaces

203

Proof. For any u E C~ ''y define, F(u)(t) =

Pt-s[7-l(.,Du(s,.))]ds,

t e [0, T].

Then F is a transformation which maps Co1'~ into C~'~. We show that it is a 1,'), 1,~ contraction on C O , provided T > 0, is sufficiently small. Let u, v E C o , then t

IIF(u)(t) - F(v)(t)l]o

<

M

fo

<< M <

M

mlDu(s,.)- Dv(s,.)l[ods s-~s~llDu(s ,.) - Dv(s,.)llods

(/o ) s -'yds

][u-v[[c~,~ <_ 1 - " /

Moreover,

I[DF(u)(t)

-

DF(v)(t)]lo <_

<

M

<_ M 2 <_ M 2

/o'

I[DPt_~[Tt(.,Du(s,.)) - ~l(.,Dv(s,.))]l[ods

(t-s)-~[l~(.,Du(s,.))-~(.,Dv(s,.))Hods

/oo

( t - s)-~llDu(s,.) - Dv(s,.)llods (t - s ) - ~ s - ~ s ~ l l D ~ ( s ,

.) - D v ( s , . ) l l o ds

< M2tl-2q'(fot(l-a)-'ra-'rdo),[u-Vllc~,~ <

M2Ktl-2~ll~-v[Ic],~

,

for some constant K . Consequently,

t~llDF(u)(t) - DF(v)(t)llo < M 2 K t l - ~ l l u - vllct,~ and,

IIc(~)- r(v)llCo~,~_< L

]

1 -.~

+ M2KTI-~

] Ilu-

vlicg,~ 9

Therefore, if the positive number T is sufficiently small, the mapping F is a contraction in C~ '7 and the equation 9.26 has a unique solution in CI"Y([O,T],H). By the standard step method the theorem is true for arbitrary T > O. []

204

J. Zabczyk

9.3 K o l m o g o r o v ' s

e q u a t i o n in m a t h e m a t i c a l

finance

Parabolic equations are traditionally used in the m a t h e m a t i c a l physics and describe a variety of physical phenomena. They are also important in chemistry and biology to model reaction and diffusion processes and the time evolution of populations. Recently they have appeared, see e.g. [MuRu], in the m a t h e m a t i c a l finance. We present here equations respectively of one and of infinite number of variables. We start from the so called Black-Scholes equation in the one state variable. Let X(t) denote the price of a share at m o m e n t t > 0. It is assumed t h a t X is a stochastic process defined on a probability space (52, ~ , P), equipped with the filtration of events ~t and that for each t > 0, the price X(t) is measurable with respect to 9~t. The time evolution of the price is often modeled by a stochastic ItO equation.

dX(t) = X(t)(rdt + adW(t)),

X(0) = x > 0,

(9.30)

in which r > 0 and u > 0 stand respectively for the interest rate and for the volatility of the share. The parameters r and u can be identified from the real d a t a with sufficient accuracy. An important property of the solution X is t h a t the discounted price of the share :

Z(t) = e-rtX(t),

t >_O,

(9.31)

is a fair process or, in more technical terms, a martingale. This means t h a t for arbitrary t > s > 0 and for arbitrary event A E ~-s :

fA

= fA

Let ~ : R+ -4 R be a given function. For t > 0, ~(X(t)) can be interpreted as the gain of a holder of an European option with the m a t u r i t y time t. Let X~(t) denote the solution to 9.30, with x denoting the price of the share at m o m e n t 0. Then the price V(t, x), which should be paid by the buyer of the option to the seller, see [MuRu] , is given by the formula:

v(t,z)

=

t > o, x > o

(9.32)

and solves the following equation

OV (t,x ) = Ot

v(o,x) =

1 2 2 02V. ~t ~a x -O--~(t,x)+rx (t,x)-rY(t,x),

(9.33)

v(t,o)=o, t>o,

The equation 9.33 is the famous Black-Scholes equation. It can be solved explicitly and the resulting formula is called the Black-Scholes formula for the price of the European option, [MuRu].

Parabolic equations on Hilbert spaces

205

There exists an infinite dimensional version of the Black-Scholes theory, related to the interest rate processes. It leads to the so called Heath-JarrowMorton equation, see [HeJaMo] and [MuRu]. Below we basically follow [Mu]. Denote by B(t, ~), t >__ 0, ~ > 0 the price at moment t of a bond maturing at time t + ~. At a current time t > 0 the function B(t, .) is known but its future values, B(s, ~), s > t, ~ > 0 can be only estimated and, in general, are of stochastic nature. The forward rate function X ( t , ~), t > 0, ~ > 0, related to the price process B , is defined by the relation,

B(t, ~) = e-f~o x(t,~)a~,

t >_ 0, ~ _> 0,

(9.34)

or, equivalently, by the formula:

Z(t,~) = -~(lnB(t,~)),

t > 0, ~ > 0.

(9.35)

In particular r(t) = X ( t , 0), t >_ 0 is the value of the interest rate at time t. Starting from D.Heath, R.Jarrow and A.Morton model [HeJaMo], M. Musiela in [Mu], proposed to represent the process X as a solution of an evolution equation: d

dX(t,~) = i=l

x ( o , ~) = x(~), where W = ( W : , . . . , Wd) is a d - dimensional Wiener process, a and ai are functionals to be identified from the data and x(.) is the forward rate function at the initial moment 0. The economical interpretation of X leads to the requirement that for each T > 0, the process:

e- fo x(s'~

T - t) = e- f: xls,o~%- f j - ' x(,,~l%

0 < t < T,

is a martingale. One can show, see [HeJaMo] and [Mu], that this is true if and only if the functional a can be expressed in terms of the volatility functionals ai as follows:

a(~,y) = ~ o ~ ( ~ , y )

o~(~,yld~,

i=1

for all ~ > 0 and all forward rate functions y. It is natural to assume that forward rates are absolutely continuous functions and take as the underling Hilbert space H the space H i consisting of absolutely continuous functions y defined on [0, +00), such that

[lyll 2 = ly(O)l ~ + ~0+c~ e -~n d~yy(r/) 2 d~ < +exp.

206

J. Zabczyk

Here V _> 0 is a parameter to be chosen. If now ~ is a functional on H i and X~(t), t >_ 0 is a solution to 9.36, then ~(X~(t)) is the gain of a holder of an European option, specified by ~, on interest rate derivative. The corresponding price of the option is given by a formula similar to 9.32:

V(t,x) = E ( e - fox'(~'~

,

t>0,

x E H 1.

(9.37)

Thus V should satisfy the following Kolmogorov equation on H i :

OV ~{(t,x)

=

1 d ~ E D ~ Y ( t , x ) + D ( ~ + a ) V ( t , x ) - x(O)V(t,x),

=

~(x),

i=1

V(0, x)

t>0,

xEH~,

2 where DbV and D,~ V denote the first and the second directional derivatives of V in the directions b and cri respectively, see [GoMu].

10. Appendix The Appendix is devoted to existence and regular dependence on a parameter of the solutions to a nonlinear equation on a Banach space. These results were used in the analysis of the stochastic evolution equations. 10.1 I m p l i c i t f u n c t i o n t h e o r e m s Let A be an open subset of a Banach space and E a Banach space and H a transformation from A • E into E. We will assume that

[]H(A,x)-H(A,y)[[E~a[[x-y[]E,

for all A E A, x, y E E

(10.1)

and consider the equation x=H(A,x),

(A,x) E A •

(10.2)

T h e o r e m 10.1. Assume that the transformation H satisfies 10.1 with a E [0, 1[. Then for arbitrary A E A there exists exactly one solution x = ~o(A)

of the equation 10.2. If in addition H is continuous with respect to the first variable then the function ~ is continuous on A. Proof. The first part of the theorem is a consequence of the contraction mapping theorem. Note that for arbitrary A E A ~(A) = H(A, 9~(A)). To prove the latter part fix A0 E A then:

Parabolic equations on Hilbert spaces ~()~) - ~()~o)

207

=

H()~,~(s

- H(~o,~O()~o))

=

[H(;~, 9~(;~)) - H()~, ~()~o))] + [H(;~, ~(~o)) - H()~o, ~o()~o))].

Therefore: II~(A) - ~(Ao)iIE < all~(A) - ~(Ao)IIE + IIH(A, ~(Ao)) - H(Ao, (P(Ao))IIF, and 1 II~(;~) - ~()~O)HE <_ _1- - Z ~ l l g ( x , ~ ( ~ o )

- g()~o,~()~o))llE,

(10.3)

so the result follows.

[]

If G is a mapping from E into F then its directional derivative at x and in direction y will be denoted by OxG(x; y). This convention, naturally extends to higher and partial derivatives. 10.2. Assume that H is continuous with respect to the first variable and satisfies 10.1 with a E [0, 1[. If, in addition, there exist directional derivatives OxH(A, x; #), OxH(X, x; y),

Theorem

and are continuous with respect to all variables, then for arbitrary A, # E A there exists directional derivative 0xqo(),; #), 0),r

= [I -- OxH()%~()~))]-IO~H(A,~(A);#),

(10.4)

and is continuous with respect to all variables. Proof. Let us fix )~o, #o E A and let t > 0 be a sufficiently small number. Then qo(Ao + t#o) - ~(Ao) = H()~o + t#o, ~o()~o + t#o)) - H(Ao, qO(Ao)) =

[H(Ao + t#o, qO(Ao + t#o)) - H(Ao, ~(~o + trio))] +[H()~o, 99(Ao + t#o)) - H(A0, ~(;~o))]

=

f01

O;,H(~o + at#o, ~o(~o +ttto); t~o) de

+

/o

0 = H I , o , ~(~o) + o(~(~o + t~o) - ~(~o)); ~(~o + t~o) - ~(~o)] de.

B o t h integrals have a well defined meaning. Moreover the strong integral

fo 10~H(Ao, a + eb) de exists for a r b i t r a r y a, b E E and its operator norm is not greater t h a n a. Consequently

208

J. Zabczyk + t#o) -

=

[/o I -

O~H(,~o,

~o(,~o) + a(~o(1o + t#o) - ~o(~o)) de

]'

(lO.5)

~o 10:,H()~o + (~t#o, ~(~o + t#o); #o) da.

To pass in 10.5 to the limit as t $ 0 we need the following lemma.

[]

L e m m a 10.3. A s s u m e that a sequence (B~) of bounded linear operators strongly converges to B. If IIBnll < ~, IIBII < c~ for some c~ E [0,1[ and all n = 1 , 2 , . . . then (I - B~) -1 -+ (I

-

B) -1

strongly as n -~ +oo

Proof. Let

x,~--+x

and

Yn = ( I -

Bn)-lXn,

y = (I-

B)-lx.

Since y,~ - B,~y,~ = x,~, y - B y = x, therefore y,~ - y = B,~(y~ - y) + (xn x) + (Bny - B y ) and IlY,~ - glIB "~ 1-~a[NXn -- XlIE + IIBny - BylIE], SO the result follows. [] By Lemma 1 one can pass to the limit in 10.5 as t $ 0 to get the formula 10.4. The continuity statement is also a consequence of Lemma 10.3. [D T h e o r e m 10.4. A s s u m e that Eo is a Banach space continuously embedded into E and that 10.2 holds both in E and Eo with the same constant a E [0, 1[. Let moreover assumptions of Theorem 10.2 be satisfied in E and Eo. If, in addition, there exist second directional derivatives, O~O~H,

O~O~H,

O~H,

02H,

in all points and in all directions of A • Eo and are continuous with respect to all variables as functions with values in E, then there exists the second directional derivative of ~o at any point )~ E A and any direction #o, no, continuous with respect to all variables. Moreover,

cg~(Ao; #o, "o) = [I - o~g(Ao, ~(Ao))] -1

(10.6)

{0~H(Ao, ~(Ao); O~(Ao; # o ) 0 ~ ( A o ; "o) + O~O~H (Ao, ~(Ao); 0 ~ ( A 0 ; #o), "o) + O~o~g(Ao, ~(Ao); #o, O~(Ao; Vo))

Remark 10.5. Note that by Theorem 10.1 and Theorem 10.2, functions ~o()~), 0 ~ ( / ~ ; # ) , A,# E A are continuous as Eo-valued functions, therefore, by Lemma 10.3 and assumptions of Theorem 10.4, the formula 10.6 defines a continuous function of all three variables. Formulation of the theorem and its proof follow the Appendix in [DaPrZa4], see also [DaPrZa2].

Parabolic equations on Hilbert

spaces

209

Proof. We start from the obvious identities ~o(A) = H()L ~o(A)),

A 9 A,

0~(~o + tuo; /to) - 0 x ~ ( ) ~ o ; / t o ) [0~H(Ao + tuo, ~O(Ao+ tuo);/to) + 0~H(Ao + tuo, ~(Ao + tuo); 0;~qO(Ao + tuo;/to))] - [0;~H(Ao, cp(Ao);/to) + 0,H(Ao, qO(Ao);0x~O(Ao;/to)] [0xH(Ao + tuo, ~O(Ao+ tuo);/to) - 0xH(Ao, ~O(Ao);/to)] +[0~H(Ao + tuo, ~O(Ao+ Uo); 0x~o(Ao + tuo;/to)) -0~H(Ao, ~O(Ao);0x~O(Ao + tuo;/to))] +0~H(Ao, ~O(Ao);0~O(~o + tuo; Po) - 0x~O(Ao;/to)) J~(t) + J2(t) + 0~H(Ao, ~O(Ao);0x~O(Ao + tuo;/to) -0x~O(Ao;/to)).

= =

=

They imply that 1

-

0~o(;~o; ~o)]

= [I-OxH(Ao,qO(Ao))]-l [~J~(t)+lj2(t)] Since 1

J1 (t)

=

y1 [0~H(Ao + tuo, ~O(Ao+ tuo); #o) - 0;~H(Ao, qO(Ao+ tuo); #o)]

+I[0xH(Ao, ~O(Ao+ tuo); #o) - 0~H(Ao, ~O(Ao);/to)] therefore, by the integral representations (as in the proof of Theorem 10.2), one has that: lim 1jl(t) = O~H(Ao, ~(Ao); #o, uo) + OxO~H(Ao, ~(Ao); #o, O~(Ao; Uo)) t$o In a similar way, 1

-~J2 (t)

=

~[0~H(Ao + tuo, cp(Ao + tuo); 0;~cp(Ao + uo; #o)) -0~H(Ao, qO(Ao+ tuo); 0~r

+ tuo; Po))]

+t[0~H(;~o, ~O(Ao+ tuo); 0x~O(Ao + tuo; #o))

-OzH(Ao, ~O(Ao);0~O(Ao + tuo; ~o)], lim 17J2(t) t$o

=

O~O~H(Ao, ~p(Ao); 0xcp(Ao; #o), uo)

+O~H(~o,~(~o);

0~(;~o; ~o)0~o(;~o; ~'o)]. []

210

J. Zabczyk

Assume now that we have a sequence of mappings Hn, n = 1,...; and a mapping H such that [[H(A,x) - H(A,Y)IIE I]Hn(A,x) - H,~(A,Y)I]E

<_ ~ l [ x - y l l ,

x, y E E, A e A,

<

n ~ N, x, y C E, A E A 0 0 . 8 )

~llx-y]l,

(10.7)

If c~ E [0, 1[ than for each A C A, equations x

=

H(A,z)

x

=

H ~ ( ~ , x)

(10.9) (10.10)

have unique solutions denoted by qo(A) and ~ (A) respectively. Thus ~(A) = H(A, ~(A)),

p~(A) = Hn(A,~o,~(A)),

A e A, n C N.

(10.11)

T h e o r e m 10.6. Assume that functions H, Hn, n C N, satisfy 10.7 and 10.8 with (~ C [0, 1[. i) If H~(A,x) --+ H(A,x), for all (A,x) E A x E then ~ ( A ) --+ qo(A), for all AEA ii) If H , Hn, n C N are continuous and Hn ~ H uniformly on compact sets then g)~ ~ qo uniformly on compact sets. Proof. i) Note that

IIHn(.X, ~n(A)) - H~(.X,'~(A))IIE +I[H~(A, ~(A)) - H(A, ~()'))llE

I]~n(;,) - ~(;,)ll~

+llHn(),, ~'(),)) - H(A, ~(;~))IIE Therefore IIv~(~) - ~(A)II~

1

<

- - { { H ~ ( A , ~ o ( A ) ) - H(A,~o(A))]{E

-

1-c~

(10.12)

and ~ ( A ) -+ ~(A) for each A C A ii) Let K C A be a compact set. Since ~o is a continuous function, the set {(%, qo(A)); A E K } is compact as well. Formula 10.12 implies now that ~,~ --+ ~ uniformly on K. [] The convergence result from Theorem 10.6 can be extended to directional derivatives of 9 as well. T h e o r e m 10.7. Assume that mappings H, H.,z, n E N satisfy conditions of Theorem 10.2. If H,~(A,x) -+ H ( A , x ) ,

O~Hn(A,x;y) --+ O~H(A,x;y)

O;~H,~()~, x; #) -+ O:~H(A, x; #)

as n --+ +oo

(10.13) (10.14)

Parabolic equations on Hilbert spaces

211

uniformly in (A, x) form compact sets and uniformly in y and It from bounded sets, then a ~ n ( ~ ; i t ) -~ 0 ~ ( ~ ; i t )

as n - ~ + ~

(10.15)

uniformly in A from compact sets and It from bounded sets. Proof. The result follows from the explicit formula 10.4 for the partial derivatives. D By the same reasons the following result holds true: T h e o r e m 10.8. Assume that mappings H, Hn n E N satisfy conditions of Theorem 10. 4 with the same space Eo. Assume that conditions of Theorem 10.7 are satisfied both in E and Eo. If, in addition: O:~O~Hn(A, x; y; #)

-+

O:~O~H()~,x; y; It),

as n --+ +oo

O~axHn(A,x;#,y)

-+

O~O~H(A,x;p~,y),

as n --+ +c~

02Hn()~, x; # , , )

-~

O~H()~, x; It, L,),

o~gn()~, x; y, z)

--~ 02H()~, x; y, z),

as n -+ +oo as n --+ +c~

uniformly in ( )~, x) from compact subsets of A x Eo and uniformly in (#, ~, y, z) from bounded sets of A x A x E x E, then 0~(~;

, , , ) -+ 0 ~ ( ~ ; #, ,),

as n ~ +c~

(10.16)

uniformly in )~ from compact subsets of A and uniformly in #, ~ from bounded subsets of A x A. References

[AhFuZa] N.U.Ahmed, M.b-~lhrman and J. Zabczyk, On filtering equations in infinite dimensions, J. Func. Anal., 143(1997), 180-204. [Bi] P. Billingsley Probability and Measure , John Wiley and Sons, 1979 [CaDaPr] P.Cannarsa and G.Da Prato,Second-order Hamilton-Jacobi equations in infinite dimensions, SIAM J.Controi and optimization, 29(1991), 474-492 . [CeGg] S.Cerrai and F.Gozzi,Strong solutions of Cauchy problems associated to weakly continuous semigroups, Differential and Integral Equations, 8(1994), 465-486. [ChojGol] A. Chojnowska-Michalik and B. Goldys , On regularity properties of nonsymmetric Ornstein-Uhlenbeck semigroup in L p spaces, Stochastics Stochastics Reports, 59 (1996), 183-209. [ChojGo2] A. Chojnowska-Michalik and B. Goldys, Nonsymmetric Ornstein- Uhlenbeck semigroup as a second quantized operator, submitted. [Chow] P.L.Chow,Infinite-dimensional Kolmogorov Equations in Gauss -Sobolev spaces, Stochastic Analysis and Applications,(to appear). [ChowMel] P.L.Chow and J.L.Menaldi, Variational inequalities for the control of stochastic partial differential equations,Proceedings of the Trento Conference 1988,LN in math.,1390(1989).

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[CrLi] M.G.Crandall and P.L.Lions.Hamilton -Jacobi equations in infinite dimensions, Part IV, Unbounded linear terms,J.Functional Analysis,90(1900),237283. [Dal] Yu. Daleckij, Differential equations with functional derivatives and stochastic equations for generalized random processes, Dokl. Akad. Nauk SSSR, 166(1966), 1035-38. [DaPrl] G.Da Prato,Some results on elliptic and parabolic equations in Hilbert spaces,Rend. Mat.Acc.Lincei,9 (1996 ), 181- 199. [DaPr2] G. Da Prato,Parabolic equations in Hilbert spaces, Lecture Notes, May 1996. [DaPr3] G. Da Prato,Stochastic control and Hamilton-Jacobi equations, Lecture Notes, February 1998. [DaPrE1Za] G. Da Prato, D. Elworthy and J. Zabczyk Strong Feller property for stochastic semilinear equations, Stochastic Analysis and Applications, 13(1995), no.l, 35-45. [DaPrGoZa] G.Da Prato, B. Goldys and J. Zabczyk , Ornstein- Uhlenbeck semigroups in open sets of Hilbert spaces, Comptes Rendus Acad. Sci. Paris, Serie I, 325(1997), 433-438. [DaPrZal] G.Da Prato and J.Zabczyk, Smoothing properties of transition semigroups in Hilbert Spaces,Stochastic and Stochastics Reports,35 (1991),63-77. [DaPrZa2] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992. [DaPrZa3] G.Da Prato and J.Zabczyk,Regular densities of invariant measures in Hilbert spaces ,J.Functional Analysis,130(1995), 427 -449. [DaPrZa4] G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996. [Da] E.B. Davis, One-parameter Semigroups, Academic Press, 1980. [Do] R. Douglas On majoration, factorization and range inclusion of operators on Hilbert spaces , Proc. Amer. Math. Soc., 17,413-415. [DS] N. Dunford and J.T. Schwartz Linear Operators. Part II: Spectral Theory, Interscience Publishers, New York, London 1963. [Dy] E.B. Dynkin, Markov Processes, Vol I, Springer Verlag, 1965. [GoMu] B. Goldys and M. Musiela, On partial differential differential equations related to term structure models, Working paper, University of New South Wales, 1996. [Go] F.Gozzi,Regularity of solutions of a second order Hamilton- Jacobi equation and application to a control problem~ Commun. in Partial Differential Equations,20(1995), 775-826 . [Gu] P. Guiotto, Non-differentiability of heat semigroups in infinite dimensional Hilbert space, forthcoming in Semigroup Forum. [Fr] A. Friedman, Partial Differential Equations of Parabolic Type, Englewood Cliffs, New York, Prentice Hall, 1964 [GiSk] I.I. Gikhman and A.V. Shorokhod, The Theory of Stochastic Processes, Vols. I, II and III. Springer-Verlag, 1974, 1975, 1979. [Gr] L. Gross, Potential theory on Hilbert spaces, J. Funct. Anal., 1(1967), 123181. [HeJaMo] D. Heath, R. Jarrow and A. Morton, Bond pricing and term structure of interest rates: a new methodology for contingent claim valuation, Econometrica, 60(1992), 77-105. fit] K.It0, On a stochastic integral equation, Proc.Imp.Acad., Tokyo, 22(1946), 32-35. [Ko] A.N.Kolmogorov, Uber die analitischen methoden in der Wahrscheinlichkeitsrechnung,Math. Ann. 104(1931),415-458 .

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N.V.Krylov,Lectures on elliptic and parabolic equations in Holder spaces,

AMS, Providence, 1996. [Ku] H.H. Kuo, Gaussian measures in Banach spaces, Lect. Notes in Math., 463(1975), Springer Verlag. [Le] P.Levy Problemes Concrets d'Analyse Fonctionelle, Gauthier-Villar, Paris, 1951. [MaRa] Z.M.Ma and M.Rockner,Introduction to the Theory of (Non Symmetric) Dirichlet Forms Springer Verlag, 1992. [Mu] M. Musiela, Stochastic PDEs and the term structure models , Journees Internationales de Finance, IGR-AFFI. La Baule, June 1993. [MuRu] M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling , Springer, 1997. [NeZa] J.M.A.M. Neerven and J. Zabczyk, Norm discontinuity of OrnsteinUhlenbeck semigroups, forthcoming in Semigroup Forum. [NuUs] D.Nualart and A.S. Ustunel, Une extension du laplacian sur l'espace de Wiener et la formule d'ItO associee, C.R.Acad.Sci. Paris, t.309, Serie I, p. 383-386, 1984. [Pa] K.R. Parthasarathy, Probability Measures on Metric Spaces, Academic Press, 1967. [Pe] S.Peszat, Large deviation principle for stochastic evolution equations, PTRF, 98 (1994), pp. 113-136. [PeZa] S. Peszat and J. Zabczyk , Strong Feller property and irreducibility for diffusions on Hilbert spaces, Annals of Probability, 23(1995), 157-172. [Ph] R.R. Phelps Gaussian null sets and differentiability of Lipschitz map on Banach spaces,Pac. J. Math. 77(1978), 523-531. [Pie] A. Piech, A fundamental solution of the parabolic equation on Hilbert space, J.~nc.Anal.,3 (1969), 85-114. [Pr] E. Priola, 7r -semigroups and applications , Preprint 9, Scuola Normale Superiore, Pisa, 1998. [Sw] A.Swiech,"Unbounded" second order partial differential equations in infinite dimensional Hilbert spaces, Commun.in Partial Differential Equations, 19(1994), 1999- 2036. [TeZa] G.Tessitore and J. Zabczyk Comments on transition semigroups and stochastic invariance , Preprint, Scuola Normale Superiore, Pisa, 1998. [Yo] K. Yosida ,Functional Analysis, Springer, 1965. [Va] T. Vargiolu, Invariant measures for a Langevin equation describing forward rates in an arbitrage free market, forthcoming in Finance Stochastic. [Zal] J.Zabczyk, Linear stochastic systems in Hilbert spaces: spectral properties and limit behavior, Report No. 236, Institute of Mathematics, Polish Academy of Sciences, 1981. Also in Banach Center Publications, 41(1985), 591-609. [Za2] J.Zabczyk, Mathematical Control Theory. An Introduction, Birkhauser, 1992. [Za3] J.Zabczyk,BeUman's inclusions and excessive measurcs,Preprint 8 Scuola Normale Superiore, March 1998. [Za4] J.Zabczyk,Infinite dimensional diffusions in Modelling and Analysis,, Jber.d Dt.Math.- Verein., (to appear).

LIST O F C.I.M.E.

1954

-

SEMINARS

i. A n a l i s i

funzionale

2. Q u a d r a t u r a 3. E q u a z i o n i

1955

-

4. T e o r e m a 5. T e o r i a

Publisher

delle

C.I.M.E.

superficie

e questioni

differenziali

n o n lineari

di R i e m a n n - R o c h

e questionl

connesse

connesse

dei n u m e r i

6. T o p o l o g i a 7. T e o r i e

1956

-

non

1958

in elasticitY,

proiettivo-differenziale

9. E q u a z i o n i

alle d e r i v a t e

i0.

Propagazione

delle

Ii.

Teoria

funzioni

della

funzioni

1957

linearizzate

8. G e o m e t r i a

onde

a caratteristiche

reali

elettromagnetiche

di pi~ v a r i a b i l i

complesse

e delle

automorfe

- 12.

Geometria

aritmetica

13.

Integrali

singolari

14.

Teoria

della

- 15. V e d u t e

parziali

idrodinamica,aerodinamica

turbolenze

e problemi

16.

Problemi

17.

II p r i n c i p i o

e algebrica e questioni (2 vol.)

attuali

di g e o m e t r i a

(2 vol.) connesse

in r e l a t i v i t ~

differenziale

di m i n i m o

generale

in g r a n d e

e le sue a p p l i c a z i o n i

alle

equazioni

funzionali

1959

1960

- 18.

1962

e statistica

Teoria

20.

Gruppi,

anelli

- 21.

Sistemi

dinamici

22.

1961

Induzione

19.

- 23.

Forme

Geometria Teoria

25.

Onde

e teoremi

del

delle

Topologia

calcolo

delle

distribuzioni

differenziale e autosoluzioni

Magnetofluidodinamica

automatici della

(2 vol.)

coomologia

ergodici

e loro i n t e g r a l i

superficiali

27. A u t o v a l o r i 28.

dei m e c c a n i s m i

di Lie e teoria

differenziali

24.

- 26.

algebrica

variazioni

(2 vol.)

215 1963 - 29. E q u a z i o n i d i f f e r e n z i a l i

astratte

30. F u n z i o n i e v a r l e t & c o m p l e s s e 31

1964

- 32 33

P r o p r i e t ~ di m e d i a e teoremi di c o n f r o n t o

in Fisica M a t e m a t i c a

Relativit& generale D i n a m i c a dei gas r a r e f a t t i

34. A l c u n e q u e s t i o n i

di analisi n u m e r i c a

35. E q u a z i o n i d i f f e r e n z i a l i

non lineari

1965 - 36. N o n - l i n e a r c o n t i n u u m theories 37. Some a s p e c t s of ring t h e o r y 38. M a t h e m a t i c a l

optimization

in e c o n o m i c s

1966 - 39. C a l c u l u s of v a r i a t i o n s

Ed. Cremonese,

40. E c o n o m i a m a t e m a t i c a 41. Classi c a r a t t e r i s t i c h e

e q u e s t i o n i connesse

42. S o m e a s p e c t s of d i f f u s i o n t h e o r y

1967 - 43. M o d e r n q u e s t i o n s of celestial m e c h a n i c s 44. N u m e r i c a l a n a l y s i s of partial d i f f e r e n t i a l e q u a t i o n s 45. G e o m e t r y of h o m o g e n e o u s b o u n d e d domains

1968 - 46. C o n t r o l l a b i l i t y 47.

and o b s e r v a b i l i t y

Pseudo-dlfferential

operators

48. A s p e c t s of m a t h e m a t i c a l

1969 - 49. P o t e n t i a l

logic

theory

50. N o n - l i n e a r

c o n t i n u u m theories in m e c h a n i c s

and p h y s i c s

and their a p p l i c a t i o n s 51. Q u e s t i o n s of a l g e b r a i c v a r i e t i e s

1970 - 52. R e l a t i v i s t i c

fluid d y n a m i c s

53. T h e o r y of g r o u p r e p r e s e n t a t i o n s 54. F u n c t i o n a l 55. P r o b l e m s

and Fourier a n a l y s i s

e q u a t i o n s and i n e q u a l i t i e s

in n o n - l i n e a r analysis

1971 - 56. S t e r e o d y ~ a m i c s 57. C o n s t r u c t i v e a s p e c t s of functional a n a l y s i s 58. C a t e g o r i e s and c o m m u t a t i v e a l g e b r a

(2 vol.)

Firenze

216 1972 - 59. N o n - l l n e a r m e c h a n i c s

1973

60

Finite g e o m e t r i c s t r u c t u r e s and their a p p l i c a t i o n s

61

Geometric measure

- 62

theory and minimal surfaces

Complex analysis

63. N e w v a r i a t i o n a l

techniques

in m a t h e m a t i c a l p h y s i c s

64. S p e c t r a l a n a l y s i s

1974

~ 65. S t a b i l i t y p r o b l e m s 66. S i n g u l a r i t i e s 67. E i g e n v a l u e s

1975 - 68. T h e o r e t i c a l 69. Model

of a n a l y t i c spaces

of non linear p r o b l e m s

c o m p u t e r sciences

t h e o r y and a p p l i c a t i o n s

70. D i f f e r e n t i a l

o p e r a t o r s and m a n i f o l d s

Ed Liguori,

1976 - 71. S t a t i s t i c a l M e c h a n i c s

Napoli

72. H y p e r b o l i c i t y 73. D i f f e r e n t i a l

topology

1977 - 74. M a t e r i a l s w i t h m e m o r y 75. P s e u d o d i f f e r e n t i a l 76. A l g e b r a i c

1978

o p e r a t o r s with a p p l i c a t i o n s

surfaces

- 77. S t o c h a s t i c d i f f e r e n t i a l 78. D y n a m i c a l

systems

equations Ed Liguori,

1979 - 79. R e c u r s i o n t h e o r y and c o m p u t a t i o n a l 80. M a t h e m a t i c s

Napoli and Birh~user Verlag

complexity

of b i o l o g y

1980 - 81. Wave p r o p a g a t i o n 82. H a r m o n i c a n a l y s i s and g r o u p r e p r e s e n t a t i o n s 83. M a t r o i d t h e o r y and its a p p l i c a t i o n s

1981 - 84. K i n e t i c T h e o r i e s and the B o l t z m a n n E q u a t i o n

(LNM 1048)

85. A l g e b r a i c T h r e e f o l d s

(LNM

947)

86. N o n l i n e a r F i l t e r i n g and S t o c h a s t i c Control

(LNM

972)

(LNM

996)

1982 - 87.

Invariant Theory

88. T h e r m o d y n a m i c s 89. Fluid D y n a m i c s

and C o n s t i t u t i v e E q u a t i o n s

Springer-Verlag

(LN Physics 228) (LNM 1047}

"

217 1983 - 90. C o m p l e t e

Intersections

(LNM 1092)

91, B i f u r c a t i o n T h e o r y and A p p l i c a t i o n s

(LNM I057)

"

92. N u m e r i c a l M e t h o d s

(LNM i127)

"

in Fluid Dynamics

1984 - 93. H a r m o n i c M a p p i n g s and M i n i m a l

Immersions

1986 -

(LNM 1161)

94. S c h r 6 d i n g e r O p e r a t o r s

(LNM 1159)

95. B u i l d i n g s and the G e o m e t r y of D i a g r a m s

(LNM 1181)

1985 - 96. P r o b a b i l i t y and A n a l y s i s

(LNM 1206)

97. Some P r o b l e m s in N o n l i n e a r D i f f u s i o n

(LNM 1224)

98. T h e o r y of M o d u l i

(LNM 1337)

(LNM 1225)

99. Inverse Problems i00. M a t h e m a t i c a l

1987 - 102. R e l a t i v i s t i c 103. Topics

(LNM 1330)

Economics Optimization

(LNM 1403)

Fluid D y n a m i c s

(LNM 1385)

i01. C o m b i n a t o r i a l

(LNM 1365)

in C a l c u l u s of V a r i a t i o n s

(LNM 1429)

1988 - 104. Logic and C o m p u t e r Science 105. Global G e o m e t r y and M a t h e m a t i c a l

Physics

1989 - 106. M e t h o d s of n o n c o n v e x a n a l y s i s 107. M i c r o l o c a l A n a l y s i s and A p p l i c a t i o n s

1990 - 108. G e o m e t r i c Topology: 109. H

Recent D e v e l o p m e n t s

Control Theory

a ii0. M a t h e m a t i c a l M o d e l l i n g of Industrial

(LNM 1451)

(LNM 1446) (LNM 1495)

(LNM 1504)

"

(LNM 1496)

"

(LNM 1521)

Processes

1991 - iii. T o p o l o g i c a l M e t h o d s Differential

for O r d i n a r y

(LNM 1537)

Equations

112. A r i t h m e t i c A l g e b r a i c G e o m e t r y

(LNM 1553)

113. T r a n s i t i o n

(LNM 1589)

to Chaos in Classical and

Quantum Mechanics

1992 - 114. D i r i c h l e t 115. D-Modules,

(LNM 1563)

Forms R e p r e s e n t a t i o n Theory,

(LNM 1565)

and Q u a n t u m G r o u p s 116. N o n e q u i l i b r i u m Systems

Springer-Verlag

Problems in M a n y - P a r t i c l e

(LNM 1551)

218 1993 - 117. Integrable

Systems and Quantum Groups

(LNM 1620)

118, Algebraic Cycles and Hodge Theory

(LNM 1594)

119. Phase Transitions

(LNM 1584)

and Hysteresis

1994 - 120, Recent Mathematical

Methods

in

(LNM 1640)

Nonlinear Wave Propagation 121. Dynamical Systems 122. Transcendental

(LNM 1609)

Methods

in Algebraic

(LNM 1646)

Geometry

1995 - 123. Probabilistic Models for Nonlinear PDE's

(LNM 1627)

124. Viscosity Solutions and Applications

(LNM 1660)

125. Vector Bundles on Curves, New Directions

(LNM 1649)

1996 - 126. Integral Geometry,

Radon Transforms

(LNM 1684)

and Complex Analysis 127. Calculus of Variations

and Geometric

LNM 1713

Evolution Problems 128

1997 - 129

Financial Mathematics

Mathematics

LNM 1656

Inspired by Biology

130. Advanced Numerical Approximation

LNM 1714 of

LNM 1697

Nonlinear Hyperbolic Equations 131. Arithmetic Theory of Elliptic Curves

LNM 1716

132. Quantum Cohomology

to appear

1998 - 133. Optimal Shape Design

to appear

134. Dynamical Systems and Small Divisors

to appear

135. Mathematical

to appear

Problems

in Semiconductor

Physics 136. Stochastic PDE's and Kolmogorov Equations

LNM 1715

in Infinite Dimension 137. Filtration

in Porous Media and Industrial

to appear

Applications

4 ~ ~

- 138. Computional Mathematics

driven by Industrual

Applicationa

to appear

139. Iwahori-Hecke Algebras and Representation Theory

to appear

Springer-Verlag

219 140. Theory and Applications of Ramiltonian Dynamics

Springer-Verlag to appear

141. Global Theory of Minimal Surfaces in Flat Spaces

to appear

142. Direct and Inverse Methods in Solving Nonlinear Evolution Equations

to appear

11

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Computational

Mathematics driven by Industrial Applications"

is the subject of the first 1999 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionah detle Ricerche (C.N.R.), the Mini~ero dell'UniversitA e della Ricerca Scientifica e Tecnologica (M.U.tLS.T.) and the European Community, will take place, under the scientific direction of Professors Vincenzo CAPASSO (Universit~ di Milano), H~inT.W. ENGL (Johannes Kepler UniversitaeL Linz) and Doct. Jacques PERIAUX (Dassault Aviation) at the Ducal Palace of Martina Franca (Taranto), from 21 to 27 June, 1999. Courses

a) Paths, trees and flows: graph optimisation problems with industrial applications (5 lectures in English) Prof. Ruiner B U R J C k R D (Technische Universit/ItGraz) Abstract Graph optimisation problems play a crucialrolein telecommunication, production, transportation, and many other industrial areas. This seriesof lectures shall give an overview about exact and heuristicsolution approaches and their inherent cli~culties. In particular the essential algorithmic paradigms such as greedy algorithms, shortest path computation, network flow algorithm% branch and bound as well as branch and cut, and dynamic programming will be outlined by means of examples stemming from applications.

References 1) ~ K. Ahuja, T. L. Magnanti & J. B. Ortin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, 1993 2) R. K. Ahuja, T. L. Magnanti, J.B.Orlin & M. R. Reddy, Applications of Network Optimization. Chapter 1 in: Network Models (Handbooks of Operations Research and Management Science, Vol. 7), ed. by "iV[.O. Ball et al., North Holland 1995, pp. 1-83 3) R. E. Burkard & E. Cela, Linear Assignment Problems and Extensions, Report 127, June 1998 (to appear in Handbook of Combinatorial Optimization, Kluwer, 1999). Can be downloaded by anonymous ftp from ftp.tu-graz.ac.at, directory/pub/papers/math 4) R. E. Burkard, E. Cela, P. M. Pardalos & L. S. Pitsoulis, The Quadratic Assignment Problem, Report 126 May 1998 (to appear in Handbook of Conibinatorial Optimi~,.~tion, Kluwer, 1999). Can be downloaded by anonymous ftp from ftp.tugraz.ac.at, directory/pub/papers/math. 5) E. L. Lawler, J. K. Lenstra, A. H. G.lqinnooy Kan & D. B. Shmoys (Eds.), The Travelling Salesman Problem, Wiley, Chichester, 1985. b) New Computational Concepts, Adaptive Differential Equations Solvers and Virtual Labs (5 lectures in English) Prof. Peter DEUFLHARD (Konxad Zuse Zentrum, Berlin).

221

Abstract The series of lectures will address computational mathematical projects that have been tackled by the speaker and his group. In all the topics to be presented novel mathamatical modelling, advanced algorithm developments, and efficient visualisation play a joint role to solve problems of practical relevance. Among the applications to be exemplified are: I) Adaptive multilevel F E M in clinical cancer therapy planning; 2) Adaptive multilevel F E M in optical chip design; 3) Adaptive discrete Galerkin methods for countable ODEs in polymer chemistry; 4) Essential molecular dynamics in RNA drug design.

Relerences 1) P. Deuflhaxd & A Hohm~m, Numerical Analysis. A first Course in Scientific Comlrutatio~ Verlag de Gruyter, Berlin, 1995 2) P. Deuflhard et al A nonlinear multigrid eigenproblem solver for the complex Helmoltz equation, Konrad Zuse Zentrum Berlin SC 97-55 (1997) 3) P. Deuflhard et al. Recent developments in chemical computing, Computers in Chemical Engineering, 14, (1990),pp.1249-1258. 4) P. Deuflhard et al. (eds) Computational molecular d~mamics: challenges, methMs, ideas, Lecture Notes in Computational Sciences and Engineering, vol.4 Springer Verlag, Heidelberg, 1998. 5) P.Deuflhard & M. Weiser, Global inexact Ne,~ton multilevel FEM for nonlinear elliptic problerrts, Konrad Zuse Zentrum SC 96-33, 1996. c) Computational Methods for Aerodynamic Analysis and Design. (5 lectures in English) Prof. Antony JAMESON (Stanford University, Stanford). Abstradt The topics to be discussed will include: - Analysis of shock capturing schemes, and fast solution algorithm~q for compressible flow; - Formulation of aerodynamic shape optimisation based on control theory; - Derivation of the adjoint equations for compressible flow modelled by the potential ELder and Navies-Stokes equations; - Analysis of alternative numerical search procedures; - Discussion of geometry control and mesh perturbation methods; - Discussion of numerical implementation and practical applications to aerodynamic design. d) Mathematical Problems in Industry (5 lectures in English) Prof. JacquesLouis LIONS (Coll6ge de France and Dassault Aviation, France). Abstract 1. Interfaces and scales. The industrial systems are such that for questions of reliability, safety, cost no subsystem can be underestimated. Hence the need to address problems of scales, both in space variables and in time and the crucial importance of modelling and numerical methods. 2. Examples in Aerospace Examples in Aeronautics and in Spatial Industries. Optimum design. 3. Comparison of problems in Aerospace and in Meteorology. Analogies and differences 4 Real time control. Many melfnods can be thought of. Universal decomposition methods will be presented. References 1) J. L. Lions, Parallel stabilization hyperbolic and Petrowsk y syster~, WCCM4 Conference, CDROM Proceedings, Buenos Aires, June 9.9- July 2, 1998.

222

2) W. Annacchiarico & M. Cerolaza, Structural shape optimization of ~-D finite elements models using Beta-splines and genetic algorithms, WCCM4 Conference, CDROM Proceedings, Buenos Aires, June 29- July 2, 1998. 3) J. Periaux, M. Sefrioui & B. Mantel, Multi-objective strategies for complex optimization problems in aerodynamics using genetic algorithms, ICAS '98 Conference, Melbourne, September '98, ICAS paper 98-2.9.1

e) Wavelet t r a n s f o r m s a n d Cosine T r a n s f o r m in Signal a n d I m a g e Processing (5 lectures in English) Prof. Gilbert STRANG (MIT, Boston). Abstract In a series of lectures we will describe how a linear transform is applied to the sampied data in signal processing, and the transformed data is compressed (and quantized to a string of bits). The quantized signal is transmitted and then the inverse transform reconstructs a very good approximation to the ori~nM signal. Our analysis concentrates on the construction of the transform. There are several important constructions and we emphasise two: 1) the discrete cosine transform (DCT); 2) discrete wavelet transform (DWT). The DCT is an orthogonal transform (for which we will give a new proof). The DWT may be orthogonal, as for the Daubechies family of wavelets. In other cases it may be biorthogonal - so the reconstructing transform is the inverse but not the transpose of the analysing transform. The reason for this

possibility is that orthogonal wavelets cannot also be symmetric, and symmetry is essential property in image processing (because our visual system objects to lack of symmetry). The wavelet construction is based on a "bank" of filters - often a low pass and high pass filter. By iterating the low pass filter we decompose the input space into "scales" to produce a multiresolution. An infinite iteration yields in the limit the scaling function and a wavelet: the crucial equation for the theory is the retinement equation or dilatation equation that yields the scaling function. We discuss the mathematics of the refinement equation: the existence and the smoothness of the solution, and the construction by the cascade algorithm. Throughout these lectures we will be developing the mathematical ideas, but always for a purpose. The insights of wavelets have led to new bases for function spaces and there is no doubt that other ideas are waiting to be developed. This is applied mathematics. References 1) I. Daubechies, Ten lectures on wavelets, SIAM, 1992. 2) G. Strang & T. Nguyen, Wavelets and filter banks, Wellesley-Cambridge, 1996. 3) Y. Meyer, Wavelets: Algorithms and Applications, SIAM, 1993.

Seminars Two hour seminars will be held by the ScientificDirectors and Professor IL Matttheij. 1) Mathematics of the crystallisation process of polymers. Prof. Vincenzo C A P A S S O (Un. di Milano). 2) Inverse Problems: Regularization methods, Application in Industry. Prof. H. W. E N G L (JohAnnes Kepler Un., Linz). 3) Mathematics of Glass. Prof. R. MAT'rHEIJ (TU Eindhoven). 4) Combining game theory and genetic algorithms for solving multiobjective shape optimization problems in Aerodynamics Engineering. Doct. J. PERIAU-X (Dassault Aviation).

223

Applications Th(ze w h o want to attend the Session should fillin an application to C.I.NLE Foundation at the address below, not later than April 30, 1999. A n important consideration in the acceptance of applications is the scientificrelevance of the Session to the field of interestof the applicant. Applicants are requested, therefore,to submit, along with theirapplication,a scientificcurriculum and a letterof recommendation. Participation will only be allowed to persons who have applied in due time and have had theirapplicationaccepted. C I M E willbe able to partiallysupport some of the youngest participants. Those who plan to apply for support have to mention it explicitelyin the applicationform.

Attendance No registration fee is requested. Lectures will be. held at Martina Franca on June 21, 22, 23, 24, 25, 26, 27. Participants are requested to register on June 20, 1999. Site a n d lodging Martina Franca is a delightful baroque town of white houses of Apulian spontaneous architecture. Martina Franca is the major and most aristocratic centre of the "Murgia dei Trulli" standing on an hill which dominates the well known Itria valley spotted with "Ttulli" conical dry stone houses which go back to the 15th century. A masterpiece of baroque architecture is the Ducal palace where the workshop will be hosted. Martina ~ a n c a is part of the province of Taranto, one of the major centres of Magna Grecia, particularly devoted to mathematics. Taranto houses an outstanding museum of Magna Grecia with fabulous collections of gold manufactures.

Lecture Notes Lecture notes will be published as soon as possible after the Session. Arrigo CELLINA CIME Director

Vincenzo VESPRI CIME Secretary

Fondazione C.I.M.E. c/o Dipartimento di Matematica ?U. Dini? Viaie Morgagni, 67/A - 50134 FIRENZE (ITALY) Tel. +39-55-434975 / +39-55-4237123 FAX +3955-434975 / +39-55-4222695 E-mail [email protected] Information on CIME can be obtained on the system World-Wide-Web on the file HTTP://WWW.MATH.UNIFI.IT/CIME/WELCOME.TO.CIME

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Iwahori-Hecke

Algebras

and Representation

Theory"

is the subject of the second 1999 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.R.), the Ministero dell'Universit&e della Ricerca Scientificae Tecnologica (M.U.R-S.T.) and the European Commu.llity~will take place, under the scientificdirection of Professors Velleda B A L D O N I (Universit& di R o m a "Tot Vergata") and Dan B A R B A S C H (Cotnell University) at the Ducal Palace of Martina Franca (Taranto), from June 28 to July 6, 1999. Courses

a) Double I-IECKE algebras a n d applications (6 lectures in English) Prof. Ivan CHEREDNIK (Un. of North Carolina at Chapel Hill, USA) Abstract: The starting point of many theories in the range from arithmetic and harmonic analysis to path integrals and matrix models is the fbrmula: f

F ( k + l / 2 ) -- 2Jo

oo

2

e-X x2kdx"

Recently a q-generalization was found based on the Hecke algebra technique, which completes the 15 year old Macdonald program. The course will be about applications of the double AfR.e Hecke algebras (mainly one-dimensional) to the Macdonald polynomials, Verlinde algebras, Gauss integrals and s.m.~. It will be understandable for those who are not familiar with Hecke algebras and (hopefuLly) interesting to the specialists. 1) q-Gauss integrds. We will introduce a q-analogue of the classical integral formula for the gamma-function and use it to generalize the Gaussian s,ms at roots of

unity. 2) U/trasphericalpolynomials. A connection of the q-ultrasphericalpolynomials (the Rogers polynomials) with the one-dimensional double arlene Hecke algebra will be established. 3) Duality.The duality for these polynomials (which has no classicalcounterpart) will be proved via the double Hecke algebras in full Jetails. 4) Verlinde algebras.W e will study the polynomial representation of the l-dim. D H A at roots of unity, which leads to a generalization and a simplificationof the Verlinde algebras. 5) PSL2(Z)-action. The projective action of the PSL2(Z) on D H A and the generalized Verlinde algebras will be considered for At and arbitrary root systems. 6) Fourier transform of the q-Gaussian. The invariance of the q-Gaussian with respect to the q-Fourier transform and some applications will be discussed. R e f etlg nce3 :

225

1) ~orn double Hecke algebra to analysis, Proceedings of ICM98, Documenta Mathematica (1998). 2) Difference Macdonald-Mehta conjecture, IMRN:10, 449--467 (1997). 3) Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, MSJ Memoirs (1997).

b) Representation theory of atfine Hecke algebras Prof. Gert HECKMAN (Catholic Un., Nijmegen, Netherlands) Abstract. 1. The Gauss hypergeometric equation. 2. Algebraic aspects of the hypergeometric system for root systems. 3. The hypergeometric function for root systems. 4. The Plancherel formula in the hypergeometric context. 5. The Lauricella hypergeometric function. 6. A root system analogue of 5. I will assume that the audience is familiar with the classical theory of ordinary differential equations in the complex plane, in particular the concept of regular singular points and mortodromy (although in my first lecture I will give a brief review of the Gauss hypergeometric function). This material can be found in many text books, for example E.L. Ince, Ordinary differential equations, Dover Publ, 1956. E.T. Whittaker and G.N. Watson, A course of modem analysis, Cambridge University Press, 1927. I will also assume that the audience is familiar with the theory of root systems and reflection groups, as can be found in N. Bourbaki, Groupes et alg~bres de Lie, ClL 4,5 et 6, Masson, 1981. J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge University Press, 1990. or in one of the text books on semisimple groups. For the material covered in my lectures references are W.J. Couwenberg, Complex reflection groups and hypergeometric functions, Thesis Nijmegen, 1994. G.J. Heckman, Dunld operators, Sere Bourbaki no 828, 1997. E.M. Opdam, Lectures on Dunld operators, preprint 1998. e) Representations of affine Hecke algebras. Prof. George LUSZTIG (MIT, Cambridge, USA) Abstract Aflqne Hecke algebras appear naturally in the representation theory of p-adic groups. In these lectures we will discuss the repvc3entation theory of atfine Hecke algebras and their graded version using geometric methods such as equivariant Ktheory or perverse sheaves. References. 1. V. Ginzburg, Lagrangian construction of representations of Heeke algebras, Adv. in Math. 63 (1987), 100-112. 2. D. Kazladan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras., Inv. Math. 87 (1987), 153-215. 3. G. Lnsztig, Cuspidal local systems and graded Hecke algebras, I, IHES Publ. Math. 67 (1988),145-202; II, in "Representation of groups" (ed. B. Allison and G. Cliff), Conf. Proc. Canad. Math. Soc.. 16, Amer. Math. Soc. 1995, 217-275. 4. G. Lusztig, Bases in equivariant K-theory, Represent. Th., 2 (1998). d) Afflne-like Hecke Algebras and p-adie representation theory Prof. Roger HOWE (Yale Un., New Haven, USA) Abstract

226

AfRne Hecke algebras first appeared in the study of a special class of representations (the spherical principal series) of reductive groups with coefficients in p-adic fields. Because of their connections with this and other topics, the structure and representation theory of aSne Hecke algebras has been intensively studied by a variety of authors. In the meantime, it has gradually emerged that affme Hecke algebras, or slight generalizations of them, allow one to understand far more of the representations of p-adic groups than just the spherical principal series. Indeed, it seems possible that such algebras will allow one to understand all representations of p-adic groups. These lectures will survey progress in this approach to p-adlc representation theory. Topics: I) Generalities on spherical function algebras on p-adic groups. 2) lwahori Hecke algebras and generalizations. 3) - 4) AtOne Hecke algebras and harmonic analysis 5) - 8) ~Ai~ne-like Hecke algebras and representations of higher level.

~fere~ces: J. Adler, Refined minimal K-types and supercuspidal representations, Ph.D. Thesis, University of Chicago. D. Barbasch, The spherical dual for p-adic groups, in Geometry and Representation Theory of Real and p-adic Groups, J. Tirao, D. Vogan, and J. Wolf, eds, Prog. In Math. 158, Birkhauser Verlag, Boston, 1998, 1 - 20. D. Barbasch and A. Moy, A unitarity criterion for p-adic groups, Inv. Math. 98 (1989), 19- 38. D. Barbasch and A. Moy, Reduction to real infinitesimal character in a.O~neHecke algebras, J. A. M. S.6 (1993), 611- 635. D. Barbasch, Unitary spherical spectrum for p-adic classical groups, Acta. Appl. Math. 44 (1996), 1 - 37. C. Bushnell and P. Kutzko, The admissible dual of GL(N) via open subgroups, Ann. of Math. Stud. 129, Princeton University Press, Princeton, NJ, 1993. C. Bushnell and P. Kutzko, Smooth representations of reductive p-adic groups: Structure theory via types, D. Goldstein, Hecke algebra isomarphisms for tamely ramified characters, R. Howe and A. Moy, Harish-Chandra Homomorphisms for p-adic Groups, CBMS Reg. Conf. Ser. 59, American Mathematical Society, Providence, RI, 1985. tL Howe and A. Moy, Hecke algebra isomorphisms for GL(N} over a p-adic field, J. Alg. 131 (1990), 388 - 424. J-L. Kim, Hecke algebras of classical groups over p-adic fields and supercuspidal representations,I, II, III, preprints, 1998. G. Lusztig, Classification of unipotent representations of simple p-adic groups, IMRN 11 (1995), 517 - 589. G. Lusztig, Afflne Hecke algebras and their graded versior~ J. A. M. S. 2 (1989), 599 - 635. L. Morris, Tamely ramified supereuspidal representations of classical groups, I, II, Ann. Ec. Norm. Sup 24, (1991) 705 - 738; 25 (1992), 639 - 667. L. Morris, Tamely ramified intertwining algebras, Inv. Math. 114 (1994), 1 - 54.

A. Roche, Types and Hecke algebras for principal series representations of split reductive p-adie groups, preprint, (1996). J-L. Waldspurger, Algebres de Hecke et induites de representations cuapidales pour GLn, J. reine u. angew. Math. 370 (1986), 27 - 191. J-K. Yu, Tame construction of supereuspidal representations, preprint, 1998.

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Appfications Those who want to attend the Session should fill in an application to the Director of C.I.M.E at the address below, n o t l a t e r t h a n April 30, 1999. An important consideration in the acceptance of applications is the scientific relevance of the Session to the field of interest of the applicant. Applicants are requested, therefore, to submit, along with their application, a scientific curriculum and a letter of recommendation. Participation will only be allowed to persons who have applied in due time and have had their application accepted. CIME will be able to partially support some of the youngest participants. Those who plan to apply for support have to mention it explicitely in the application form. Attendance No registration fee is requested. Lectures will be held at Martina F~anca on June 28, 29, 30, July 1, 2, 3, 4, 5, 6. Participants are requested to register on June 27, 1999. Site a n d lodging Martina Franca is a delightful baroque town of white houses of Apulian spontaneous architecture. Martina Franca is the major and most aristocratic centre of the Murgia dei Trulli standing on an hill which dominates the well known Itria valley spotted with Trulli conical dry stone houses which go back to the I5th century. A mp-~terpiece of baroque architecture is the Ducal palace where the workshop will be hosted. Martina Franca is part of the province of Taranto, one of the major centres of Magna Grecia, particularly devoted to mathematic.s. Taranto houses an outstanding museum of Magna Grecia with fabulous collections of gold manufactures. L e c t u r e Notes Lecture notes will be published as soon as possible after the Session. A.rrigo CELLINA CIME Director

Vincenzo VESPRI CIME Secretary

Fondazione C.I.M.E. c/o Dipartimento di Matematica U. Dini Viale Morgagni, 67/A - 50134 FIRENZE (ITALY) Tel. +39-55-434975 / +39-55-4237123 FAX +39-55-434975 / +39-55-4222695 E-mail CIME~UDINI.MATH.UNIFI.IT Information on CIME can be obtained on the system World-Wide-Web on the file I-rI~p://WWW.MATH.UNIFI.IT/CIME/WELCOME.TO. CIME.

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Theory and Applications of Hamiltonian Dynamics" is the subject of the third 1999 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.IL), the Minl.qtero dell'Universit~ e della Ricerca Scientifica e Tecnologica (M.U.R.S.T.) and the European Community, will take place, under the scientific direction of Professor Antonio GIORGILLI (Un. di 1Milano), at Grand Hotel San Michele,Cetraro (Cosenza), from July 1 to July 10, 1999.

Courses a) Physical applications of Nekhoroshev theorem and exponential estimates (6 lectures in English) Prof. Giancarlo BENETTIN (Un. di Padova, Italy)

Abstract The purpose of the lectures is to introduce exponential estimates (i.e., construction of normal forms up to art exponentially small remainder) and Nekhoroshev theorem (exponential estimates plus geometry of the action space) as the key to understand the behavior of several physical systems, from the Celestial mechanics to microphysics. Among the applications of the exponential estimates, we shall consider problems of adiabatic invariance for systems with one or two frequencies coming from molecular dynamics. We shall compare the traditional rigorous approach via canonical transformations, the heuristic approach of Jeans and of Landau-Teller, and its possible rigorous implementation via Lindstet series. An old conjecture of Boltzmann and Jeans, concerning the possible presence of very long equilibrium times in classical gases (the classical analog of "quantum freezing") will be reconsidered. Rigorous and heuristic results will be compared with numerical results, to test their level of optimality. Among the applications of Nekhoroshev theorem, we shall study the fast rotations of the rigid body, which is a rather complete problem, including degeneracy and singularities. Other applications include the stability of elliptic equilibria, with special emphasis on the stability of triangular Lagrangian points in the spatial restricted three body problem.

References: For a general introduction to the subject, one can look at chapter 5 of V.I. Arnold, W . Kozlov and A.I. Neoshtadt, in Dynamical Systems III, V.I. Arnold Editor (Springer, Berlin 1988). An introduc~on to physical applications of Nekhorshev theorem and exponential estimates is in the proceeding of the Noto School "NonLinear Evolution and Chaotic Phenomena", G. Gallavotti and P.W. Zweifel Editors (Plenum Press, New York, 1988), see the contributions by G. Benettin, L. Galgani and A. Giorgilli. General references on Neldaoroshev theorem and exponential estimates: N.N. Nekhoroshev, Usp. Mat. Nauk. 32:6, 5-66 (1977)[Russ. Math. Surv. 32:6, 1-65

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(1977)]; G. Benettin, L. Galgani, A. GiorgiUi, Cel. Mech. 37, 1 (1985); A. Giorgilli and L. Galgani, Cel. Mech. 37, 95 (1985); G. Benettin and G. Gallavotti, Journ. Star. Phys. 44, 293-338 (1986); P. Lochak, Russ. Math. Surv. 4T, 57-133 (1992); J. P~schel, Math. Z. 213, 187-216 (1993). Applications to statistical mechanics: G. Benettin, in: Boltzmann's legacy 150 years alter h/s b/rth, Atti Accad. Nazionale dei Lincei 131, 89-105 (1997); G. Benettin, A. Carati mad P. Sempio, Journ. Star. Phys. 73, 175-192 (1993); G. Benettin, A. Carati and G. Gallavotti, Nonlinearity 10, 479-505 (1997); G. Benettin, A. Carati e F. Fhss6, Physica D 104, 253-268 (1997); G. Benettin, P. Hjorth and P. Sempio, Exponent]al/y long equdlibHum times in a one dimensional collisional model o / a classical gas, in print in Journ. Stat. Phys. Applications to the rigid body: G. Benettin and F. FassS, Nonlinearity 9, 137-186 (1996); G. Benettin, F. Fass6 e M. Guzzo, Nonlinearity 10, 1695-1717 (1997). Applications to elliptic equilibria (recent nonisochronous approach): F. Fass6, M. Guzzo e G. Benettin, Comm. Math. Phys. 197, 347-360 (1998); L. Niederman, Nonlinear stability around an elliptic equilibrium point in an Hamiltonian system, preprint (1997). M. Guzzo, F. Fasso' e G. Benettin, Math. Phys. Electronic Journal, Vol. 4, paper 1 (1998); G. Benettin, F. Fass6 e M. Guzzo, Netdmrosliev-stability of IA and L5 in the spatial restricted three--body problem, in print in Regular and Chaotic Dynamics. b) K A M - t h e o r y (6 lectures in English) Prof. Hakan ELIASSON (Royal Institute of Technology, Stockholm, Sweden) Abstract Quasi-periodic motions (or invariant tori) occur naturally when systems with periodic motions are coupled. The perturbation problem for these motions involves small divisors and the most natural way to handle this difficulty is by the quadratic convergence given by Newton's method. A basic problem is how to implement this method in a particular perturbative situation. We shall describe this difficulty, its relation to linear quasi-periodic systems and the way given by K.AM-theory to overcome it in the most generic case. Additional difficulties occur for systems with elliptic lower dimensional tori and even more for systems with weak non-degeneracy. We shall also discuss the difference between initial value and boundary value problems and their relation to the Lindstedt and the Poincar6-Lindstedt series. The classical books Lectures in Celestial Mechanics by Siegel and Moser (Springer 1971) and Stable and Random Motions in Dynamical Systems by Moser (Princeton University Press 1973) are perhaps still the best introductions to KAM-theory. The development up to middle 80's is described by Bost in a Bourbaki Seminar (no. 6 1986). After middle 80'sa lot of work have been devoted to elliptic lower dimensional tori, and to the study of systems with weak non-degeneracy starting with the work of Cheng and Sun (for example "Existence of KAM-tori in Degenerate Hamiltonian systerr~", J. Diff. Eq. 114, 1994). Also on linear quasi-periodic systems there has been some progress which is described in my article "Reducibility and point spectrum far quasi-periodic skew-products', Proceedings of the ICM, Berlin volume II 1998. c) The Adiabatic lnvariant in Classical Dynamics: Theory and applications (6 lectures in English). Prof. Jacques HENRA.RD (Facult6s Universitaires Notre Dame de la Pain, Namur, Belgique). Abstract

230

The adiabatic invariant theory applies essentially to oscillating non-autonomous Hamiltonian systems when the time dependance is considerably slower than the oscillation periods. It describes "easy to compute" and "dynamicaly me~ninLfful" quasiinvariants by which on can predict the approximate evolution of the system on very large time scales. The theory makes use and may serve as an illustration of several cla~ical results of Hamiltonian theory. I) ClassicalAdiabatic InvariantTheory (Includingan introductionto angle-action variables) 2) ClassicalAdiabatic InvariantTheory (continued) and some applications (including an introductionto the "magnetic bottle") 3) Adiabatic Invariantand SeparatrixCrossing (Neo-adiabatictheory) 4) Applicationsof Neo-Adiabatic Theory: Resonance Sweeping in the Solar System 5) The chaotic layerof the "Slowly Modulated Standard Map"

Referemes: J.R. Cary, D.F. Escande, J.L. Tennison: Phys.Rev. A, 34, 1986, 3256-4275 J. Henrard, in "Dynamics reported" (n=B02- newseries),SpringerVerlag 1993; pp 117-235) J. Henrard: in "Les m~thodes moderne de la m~.anique c~leste" (Benest et Hroeschle eds), Edition [~ontieres, 19901 213-247 J. Henrard and A. MorbideUi: Physica D, 68, 1993, 187-200. d) S o m e aspects of qualitative t h e o r y of Hamiltonian P D E s (6 lectures in English). Prof. Sergei B. KUKSIN (Heriot-Watt University, Edinburgh, and Steklov Institute, Moscow) Abstract. I) Basic properties of Hamiltoaian PDEs. Symplectic structures in scales of Hilbert spaces, the notion of a Hami!tonian PDE, properties of flow-maps, etc. II) Around Gromov's non-squeezing property. Discussions of the finite-dimensional Gromov's theorem, its version for PDEs and its relevance for mathematical physics, infinite-dimensional symplectic capacities. III) Damped Hamiltonian PDEs and the turbulence-limit. Here we establish some qualitative properties of PDEs of the form <non-linear Hamiltonian PDE>+<small linear damping> and discuss their relations with theory of decaying turbulence Parts I)-II) will occupy the first three lectures, Part III- the last two. References [1] S.K., Nearly Integrable Infinite-dimensional Hamiltonian Systems. 1.2XlM1556, Springer 1993. [2] S.K., Infinite-dimensional symplectic capacities and a squeezing theorem for Hamiltonian PDE's. Comm. Math. Phys. 167 (1995), 531-552. [3] Hofer H., Zehnder E., 8ymplectic invariants and Hamiltonian dynamics. Birkhauser, 1994. [4] S.K. Oscillations in space-periodic nonlinear Schroedinger equations. Geometric and ~ c t i o n a l Analysis 7 (1997), 338-363. For I) see [11 (Part 1); for H) see [2,31; for III) see [4]." e) A n o v e r v i e w on some problems in Celestial Meehmtlcs (6 lectttres in English) Prof. Caries SIMO' (Universidad de Barcelona, Spagna) Abstract 1. Introduction. The N-body problem. Relative equilibria. Collisions.

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2. The 3D restricted three-body problem. Libration points and local stability analysis. 3. Periodic orbits and invariant tori. Numerical ,rod symbolical computation. 4. Stability and practical stability. Central manifolds and the related stable/unstable mar~ifolds. Practical confiners. 5. The motion of spacecrafts in the vicinity of the Earth-Moon system. Results for improved models. Results for full JPL models. P,efer~nces: C. SimS, An over~iew of some problems in Celestial Mechanics, available at http://www-mal.upc.es/escorial . Click of "curso completo" of Prof. Caries Sim6

Applications Deadline for application: May 15, 1999. Applicants are requested to submit, along with their application, a scientific curriculum and a letter of recommendation. CIME will be able to partially support some of the youngest participants. Those who plan to apply for support have to mention it explicitely in the application form. Attendance No registration fee is requested. Lectures will be held at Cetraro on July 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Participants are requested to register on June 30, 1999. Site and lodging The session will be held at Grand Hotel S. Michele at Cetraro (Cosenza), Italy. Prices for full board (bed and meals) are roughly 150.000 italian liras p.p. day in a single room, 130.000 italian liras in a double room. Cheaper arrangements for multiple lodging in a residence are avalaible. More detailed information may be obtained fTom the Direction of the hotel (tel. +39-098291012, Fax +39-098291430, email: [email protected]. ~lrther information on the hotel at the web page www.sanmichele.it

Arrigo CELLINA CIME Director

Vincenzo VESPRI CIME Secretary

Fondazione C.I.M.E. c/o Dipartimento di Matematica U. Diai Viale Morgagni, 67/A - 50134 FIRENZE (ITALY) Tel. +39-55-434975 / +39-55-4237123 FAX +39-55-434975 / +39-55-4222695 E-mail [email protected] Information on CIME can be obtained on the system World-Wide-Web on the file H'YrP://WWW.MATH.UNIFI.IT/CIME/WELCOME.TO.CIME.

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Global

Theory

of Minimal

Surfaces

in Flat Spaces"

is the subject of the fourth 1999 C.I.M.E. SessionThe session, sponsored by the Consiglio Nazioaale delle Ricerche (C.N.IL), the Ministero dell'Unlversitt~ e della Ricerca Scientifica e Tecnologica (M.U.ILS.T.) and the European Community,will take place, under the scientific direction of Professor Gian Pietro PIROLA (Un. dl Pavia), at Ducal Palace of Martina Franca (Taranto), from July 7 to July 15, 1999. Courses a) A s y m p t o t i c g e o m e t r y of p r o p e r l y e m b e d d e d minimal surfaces (6 lecture in English) Prof. William H. MEEKS, III (Un. of Massachusetts, Amherst, USA).

Abstract: In recent years great progrms has been made in understanding the asymptotic geometry of properly embedded minimal surfaces. The first major result of this type was the solution of the generalized Nitsch conjecture by P. CoUin, based on earlier work by Meeks and Rosenberg. It follows from the resolution of this conjecture that whenever M is a properly embedded minimal surface with more than one end and E C M is an annular end representative, then E has finite total curvature and is asymptotic to an end of a plan or catenoid. Having finite total curvature in the case of an ~mnular end is equivalent to proving the end has quadratic area growth with respect to the radial function r. Recently Collin, Kusner, Meeks and Rosenberg have been able to prove that any middle end of M, even one with infinite genus, has quadratic area growth. It follows from this result that middle ends are never limit ends and hence M can only have one or two limit ends which must be top or bottom ends. With more work it is shown that the middle ends of M stay a bounded distance from a plane or an end of a catenoid. The goal of my lectures will be to introduce the audience to the concepts in the theory o f properly embedded minimal surfaces needed to understand the above results and to understand some recent classification theorems on proper minimal surfaces of genus 0 in flat three-manifolds.

~ferences 1) H. Rosenberg, Some recent developments in the theory of properly embedded minimal surfaces in E, Asterisque 206, (19929, pp. 463-535; 2) W. Meeks & H. Rosenberg, The geometry and canformal type of properly embedded minimal surfaces in E, Invent.Math. 114, (1993), pp. 625-639; 3) W. Meeks, J. Perez & A. Ros, Uniqueness of the Riemanrt minimal examples, Invent. Math. 131, (1998), pp. 107-132; 4) W. Meeks & H. Rosenberg, The geometry of periodic minimal surfaces, Comm. Math. Helv. 68, (1993), pp. 255-270; 5) P. Collin, Topologie et courbure des surfaces minimales proprement plongees dans E, Annals of Math. 145, (1997), pp. 1-31;

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6) H. Ro~nberg, Minimal surfaces of finite type, Bull. Soc. Math. France 123, (1995), pp. 351-359; 7) Rodriquez & H. Rosenberg, Minimal surfaces in E with one end and bounded cu~ature, Manusc. Math. 96, (1998), pp. 3-9. b) P r o p e r l y e m b e d d e d m i n i m a l surfaces with finite t o t a l c u r v a t u r e (6

lectures in English) Prof. Antonio R O S (Universidad de Granada, Spain) Abstact: Among properly embedded minimal surfaces in Euclidean 3-space, those that have finite total curvature form a natural and important subclass. These surfaces have finitely many ends which are all parallel and asymptotic to planes or catenoids. Although the structure of the space .A,t of surfaces of this type which have a fixed topology is not well understood, we have a certain number of partial results and some of them will be explained in the lectures we will give. The first nontrivial examples, other than the plane and the catenoid, were constructed only ten years ago by Costa, Hoffman and Meeks. Schoen showed that if the surface has two ends, then it must be a catenoid and Ldpez and Ros proved that the only surfaces of genus zero are the plane and the catenoid. These results give partial answers to an interesting open problem: decide which topologies are supported by this kind of surfaces. Ros obtained certain compactness properties of .M. In general this space is known to be noncompact but he showed that Ad is compact for some fixed topologies. P~rez and Ros studied the local structure of ~ around a nondegenerate mnXa~ceand they proved that around these points the moduli space can be naturally viewed as a Lagrangian submanifold of the complex Euclidean space. In spite of that analytic and algebraic methods compete-to solve the main problems in this theory, at this moment we do not have a satisfactory idea of the behaviour of the moduli space A4. Thus the above is a good research field for young geometers interested in minimal surfaces. References 1) C. Costa, Example of a compete minimal immersion in R 3 of genus one and three embedded ends, Bull. SOc. Bras. Math. 15, (1984), pp. 47-54; 2) D. Hottman & H. Karcher, Complete embedded minimal surfaces of finite total curvature, I t Osserma~ ed., Encyclopedia of Math., vol. of Minimal Surfaces, 5-90, Springer 1997; 3) D. Hoffznan & W. H. Meeks III, Embedded minimal surfaces of finite topology, Ann. Math. 131, (1990), pp. 1-34; 4) F. J. I_~pez & A. Ros, On embedded minimal surfaces of genus zero, J. Differential Geometry 33, (1991), pp. 293-300; 5) J. P. Perez & A. Ros, Some uniqueness and nonexistence theorems for embedded minimal surfaces, Math. Ann. 295 (3), (1993), pp. 513-525; 6) J. P. Perez & A. Ros, The space of prvperly embedded minimal surfaces with finite total curvature, laxdiana Univ. Math. J. 45 1: (1996), pp.177-204. e) Minimal surfaces o f finite t o p o l o g y p r o p e r l y e m b e d d e d in E (Euclidean 3-space).(6 lectures in English) Prof. Harold ROSEN'BERG (Univ. Paris VII, Paris, France) Abstract: We will prove that a properly embedded minimal surface in E of finite topology and at least two ends has finite total curvature. To establish this we first prove that each annular end of such a surface M can be made transverse to the horizontal planes

234

( after a possible rotation in space ), [Meeks-~nberg]. Then we will prove that such an end has finite total curvature [Pascal CoUin]. We next study properly embedded minimal surfaces in E with finite topology and one end. The basic unsolved problem is to determine if such a surface is a plane or helicoid when simply connected. We will describe partial results. We will prove that a properly immersed minimal surface of finitetopology that meets some plane in a finitem,mher of connected components, with at most a finitenumber of singularities,is of finiteconformal type. If in addition the curvature is bounded, then the surface is of finite type. This means M can be parametrized by meromorphic data on a compact Riemann surface. In particular, under the above hypothesis, M is a plane or helicoidwhen M is also simply connected and embedded. This is work of Rodriquez- Rosenberg, and Xavier. If time permits we will discuss the geometry and topology of constant mean curvature surfaces properly embedded in E.

Refere.ces 1) H. Rosenberg, Some recent developments in the theory of properly embedded minimal surfaces in E, Asterique 206, (1992), pp. 463-535; 2) W.Meeks & H. Rosenberg, The geometry and conformal type of properly embedded minimal surfaces in E, Invent. 114, (1993), pp.625-639; 3) P. CoUin, Topologie et courbure des surfaces minimales proprement plong~es darts E, Annals of Math. 145, (1997), pp. 1-31 4) H. Rmenberg, Minimal surfaces of finite type, Bull. Soc. Math. France 123, (1995), pp. 351-359; 5) Rodriquez & H. Rosenberg, Minimal surfaces in E with one end and bounded curvature, Manusc. Math. 96, (1998), pp. 3-9. Applications Those who want to attend the Session should fillin an application to the C.I.M.E Foundation at the address below, not later than May 15, 1999. A n important consideration in the acceptance of applications is the scientificrelevance of the Session to the fieldof interest of the applicant. Applicants are requested, therefore, to submit, along with their application, a scientificcurriculum and a letter of recommendation. Participation will only be allowed to persons who have applied in due time and have had their application accepted. C I M E will be able to partially support some of the youngest participants. Those who plan to apply for support have to mention it explicitelyin the application form Attendance

No registration fee is requested. Lectures will be held at Martina Franca on July 7, 8, 9, 10, 11, 12, 13, 14, 15. Participants are requested to register on July 6, 1999. Site a n d lodging Martina Franca is a delightful baroque town of white houses of Apulian spontaneous architecture. Martina Franca is the major and most aristocratic centre of the Murgia dei Trulli standing on an hill which dominates the well known Itria valley spotted with Trulli conical dry stone houses which go back to the 15th century. A masterpiece of baroque architecture is the Ducal palace where the workshop will be

235

hosted. Martlna bqrmacais part of the province of Taranto, one of the major centres of Magna Grecia, particularly devoted to mathematics. Taranto houses an outstanding museum of Magna Grecia with fabulous collections of gold mantffactures. Lecture Notes Lecture notes will be published as soon as possible after the Session. Arrigo C E L L I N A C I M E Director

Vincenzo V E S P R I C I M E Secretary

Fondazione C.I.M.E. e/o Dipaxtimento di Matematica U. Dini Viale Morgagni, 67/A - 50134 FIRENZE (ITALY) Tel. +39-55-434975 / +39-55-4237123 FAX +39-55-434975 / +39-55-4222695 E-mall CIME~UDINI.MATH.UNIFI.IT Information on C I M E can be obtained on the system World-Wide-Web on the file HTTP://WWW.MATH.UNIFI.IT/CIME/WELCOME.TO.CIME.

FONDAZIONE C.I.M.E. CENTRO INTERNAZIONALE MATEMATICO ESTIVO INTERNATIONAL MATHEMATICAL SUMMER CENTER "Direct and Inverse M e t h o d s in Solving Nonlinear Evolution Equations" is the subject of the fifth 1999 C.I.M.E. Session. The session, sponsored by the Consiglio Nazionale delle Ricerche (C.N.tL), the Mini.qtero dell'Universita e della Ricerca Scientifica e Tecnologica (M.U.tLS.T.) and the European Comrnunity, will take place, under the scientific direction of Professor Antonio M. Greco (UuiversitA di Palermo), at Grand Hotel San Michele,Cetraro (Cosenza), from September 8 to September 15, 1999.

a) Exact solutions of nonlinear PDEs by singularity analysis (6 lectures in Enghsh) Prof. Robert CONTE (Service de physique de l'6tat condens6, CEA Saclay, Gifsur-Yvette Cedex, Prance) Abstract 1) Criteria of integrability : Lax pair, Darboux and Bttcklund transformations. Partial integrabihty, examples. Importance of involutions. 2) The Painlev~ test for PDEs in its invariant version. 3) The "truncation method" as a Darboux transformation, ODE and PDE situations. 4) The one-family truncation method (WTC), integrable (Korteweg-de Vries, Bonssinesq, Hirota-Satsuma, Sawada-Kotera) and partially integrable (Kuramoto-Sivashinsky) cases. 5) The two-family truncation method, integrable (sine-Gordon, mKdV, BroerKaup) and partially integrable (complex Ginzburg-Landau and degeneracies) cases. 6) The one-family truncation method based on the scattering problems of Gambier: BT of Kaup-Kupershrnidt and Tzitz6ica equations. References References are divided into three subsets: prerequisite (assumed known by the attendant to the school), general (not assumed known, pedagogical texts which would greatly benefit the attendant if they were read before the school), research (research papers whose content will be exposed from a synthetic point of view during the course). Prerequisite bibliography. The following subjects will be assumed to be known : the Painlev~ property for nonlinear ordinary differentialequations, and the associated Palnlev6 test. Prerequisite recommended texts treating these subjects are [P. 1] E. HiUe, Ordinary differential equations in the complex domain (J. Wiley and sons, New York, 1976). [P.2] R. Conte, The Painlev~ approach to nonlinear ordinary differential equations, The Painlevd property, one century later, 112 pages, ed. P,. Conte, CRM series in mathematical physics (Springer, Berlin, 1999). Solv-int/9710020.

237

The interested reader can find many applications in the following review, which should not be read before [P.2] : [P.3] A. Ramani, B. Grammaticos, and T. Bountis, The Painlevd property and sin-

tpdarity analysis of integrable and nonintegruble systems, Physics Reports 180 (1989) 159-245. A text to be avoided by the beginner is Ince's book, the ideas axe much clearer in iqille's book. There exist very few pedagogical texts on the subject of this school. A general reference, covering all the above program, is the course delivered at a Carg~se school in 1996 :

[G.1] M. Musette, Painlev~ analysis for nonlinear partial differential equations, The Painlev~ property, one century later, 65 pages, ed. R. Conte, CRM series in mathematical physics (Springer, Berlin, 1999). Solv-int/9804003. A short subset of [G.1], with e/lYphasis on the ideas, is the conference report [G.2] R. Conte, Various truncations in Painlev~ analysis of partial differential equations, 16 pages, Nonlinear dynamics : integrability and chaos, ed. M. Daniel, to appear (Springer? World Scientific?). Solv-int/9812008. Preprint $98/047. Research papers. [R.2] J.? Weiss, M. Tabor and G. Carnevale, The Painlev~ property for partial differentiafl, equations, J: Math. Phys. 24 (1983) 522-526. [R.3] Ndmerous articles of Weiss, from 1983 to 1989, all in J. Math. Phys. [singular manifold method]. [tL4] M. Musette and R. Conte, Algorithmic method for deriving Lax pairs from the invariant Painlev~ analysis of nonlinear partial differential equations, J. Math. Phys. 32 (1991) 1450-1457 [invariant singular manifold method]. [R.5] P,. Conte and M. Musette, Linearity inaido nonlinearity: exact solutions to the complex Ginz-burg-Landau equatior~ Physica D 69 (1993) 1-17 [Ginzburg-Landau]. [tL6] M. Musette and R. Conte, The two-singular manifold method, L Modified KdV and sine-Gordon equations, J. Phys. A 27 (1994) 3895-3913 [Two-singular manifold method]. [R.7] R. Conte, M. Musette and A. Pickering, The two-singular manifold method, II. Classical Boussinesq system, J. Phys. A 28 (1995) 179-185 [Two-singular manifold method]. [tL8] A. Pickering, The singular manifold method revisited~ J. Math. Phys. 37 (1996) 1894-1927 [Two-singular manifold methodI. [1L9] M. Musette and tL Conte, B~icklund transformation of partial differential equations from the Painlev~-Gambier classification, I. Kaup-Kupershmidt equation, J. Math. Phys. 39 (1998) 5617-5630. [Lecture 61. [R.10] R. Conte, M. Musette and A. M. Grundland, Biicklund transformation of partial differential equations from the Painlev~-Gambier classification, II. Tzitz~ica equation, J. Math. Phys. 40 (1999) to appear. [Lecture 6]. b) Integrable Systems and Bi-H~miltonian Manifolds (6 lectures in English) Prof. Franco MAGRI (Universit~ eli lk4ilano,Milano, Italy)

Abstract 1) Integrable systems and bi-hamiltonian manifolds according to Gelfand and Zakharevich. 2) Examples: KdV, KP and Sato's equations.

238 3) The rational solutionsof K P equation. 4) Bi-hamiltonian reductions and completely algebraicallyintegrable syst~a. 5) Connections with the separabiltytheory. 6) The r function and the I:Iirota'sidentitiesfrom a bi-hamiltonian point of view. Refere,w.es 1) tL Abraham, J.E. Marsden, Foundations of Meehan/cs, Benjamin/C-mmln~, 1978 2) P. Libermann, C. M. Marie, Symplectie Geometry and Analytical Mechanics, Reidel Dordrecht, 1987 3) L. A. Dickey, Soliton Equations and Hamiltonian Systems, World Scientific, Singapore, 1991, Adv. Series in Math Phys VoL 12 4) 1. Vaisman~ Lectures on the GeornePry of Poisson Manifolds, Progress in Math, Birkh~tuser, 1994 5) P. Casati, G. Falqui, F. Magri, M. Pedroni (1996), The KP theory revisited. I, II, III, IV. Technical Reports, SISSA/2,3,4,5/96/FM, SISSA/ISAS, Trieste, 1995 c) Hirota Methods for non Linear Differential and Difference Equations (6 lectures in English) Prof. Junkichi S A T S U M A (Universityof Tokyo, Tokyo, Japan) Abstract 1) Introduction; 2) Nonlinear differential systems; 3) Nonlinear differential-
d) Lie Groups and Exact Solutions of non Linear Differential and Difference Equations (6 lectures in English) Prof. Pavel WINTERNITZ (Universit~ de Montreal, Montreal, Canada) 3J7 Abstract 1) Algorithms for calculating the symmetry group of a system of ordinary or partial differential equations. Examples of equations with finite and in'hire Lie point symmetry groups; 2) Applications of symmetries. The method of symmetry reduction for partial differential equations. Group classification of differential equations; 3) Classification and identification of Lie algebras given by their structure constants. Classification of subalgebras of Lie algebras. Examplesand applications; 4) Solutions of ordinary differential equations, f.~wering the order of the equation First integrals, painlev~ analysis and the singularity structure of solutions; 5) Conditional symmetries. Partially invariant solutions.

239

6) Lie symmetries of difference equations.

References. 1) P. J. Olver, Applications of Lie Groups ~o Differential Equations, Springer,1993, 2) P. W'mternitz, Group Theory and Exact Solutions of Partially [ntegrable Diffe,-e,-~tial Systems, in Partially Integrable Ewlution Equations in Physics, Kluwer, Dordrecht, 1990, (Editors R.Conte and N.Boccara). 3) P. Winternitz, in "Integrable Systems, Quantum Groups and Quantum Field Theories", Kluwer, 1993 (Editors L .A. Ibort and M. A. Rodriguez).

Applications Those who want to attend the Session should fill in an appLication to the C.I.M.E Foundation at the address below, not later t h a n May 30, 1999. An important consideration in the acceptance of appLications is the scientific relevance of the Session to the field of interest of the applicant. Applicants are requested, therefore, to submit, along with their application, a scientific curriculum and a letter of recommendation. Participation will only be allowed to persons who have applied in due time and have had their application accepted. CIME will be able to partially support some of the youngest participants. Those who plan to apply for support have to mention it explicitely in the appLication form. Attendance

No registration fee is requested. Lectures will be held at Cetraro on September 8, 9, 10, 11, 12, 13, 14, 15. Participants are requested to register on September 7, 1999. Slte and lodging The session will be held at Grand Hotel S. Michele at Cetraro (Cosenza), Italy. Prices for hill board (bed and meals) are roughly 150.000 italian Liras p.p. day in a single room, 130.000 italian liras in a double room. Cheaper arrangements for multiple lodging in a residence are avalaible. More detailed informations may be obtained from the Direction of the hotel (tel. +39-098291012, Fax +39-098291430, email: sanmichele~antares.it. Further information on the hotel at the web page www.sanmichele.it

Lecture Notes Lecture notes will be published as soon as possible after the Session. Arrigo CELLINA CIME Director

Vincenzo VESPRI CIME Secretary

Fondazione C.I.M.E. c/o Dipartimento di Matematica U. Din] Viale Morgagni, 67/A - 50134 FIRENZE (ITALY) Tel. +39-55-434975 / +39-55-4237123 FAX +39-55-434975 / +39-55-4222695 E-mail [email protected] Information on CIME can be obtained on the system World-Wide-Web on the file HTTP://WWW.MATH.UNIFI.IT/CIME/WELCOME.TO.CIME.