t Dimensional Infinite Stochastic Analysis In Honor of Hui-Hsiung Kuo
QP-PQ: Quantum Probability and White Noise Analysis
Managing Editor: W. Freudenberg Advisory Board Members: L. Accardi, T. Hida, R. Hudson and K. R. Parthasarathy
QP-PQ: Quantum Probability and White Noise Analysis VOl. 22:
Infinite Dimensional Stochastic Analysis eds. A. N. Sengupta and P. Sundar
Vol. 21:
Quantum Bio-Informatics From Quantum Information to Bio-Informatics eds. L. Accardi, W. Freudenberg and M. Ohya
Vol. 20:
Quantum Probability and Infinite Dimensional Analysis eds. L. Accardi, W. Freudenberg and M. Schurmann
Vol. 19:
Quantum Information and Computing eds. L. Accardi, M. Ohya and N. Watanabe
Vol. 18:
Quantum Probability and Infinite-Dimensional Analysis From Foundations to Applications eds. M. Schurmann and U. Franz
Vol. 17:
Fundamental Aspects of Quantum Physics eds. L. Accardi and S. Tasaki
Vol. 16:
Non-Commutativity, Infinite-Dimensionality, and Probability at the Crossroads eds. N. Obata, T. Matsui and A. Hora
Vol. 15:
Quantum Probability and Infinite-Dimensional Analysis ed. W. Freudenberg
Vol. 14:
Quantum Interacting Particle Systems eds. L. Accardi and F. Fagnola
Vol. 13:
Foundations of Probability and Physics ed. A. Khrennikov
QP-PQ
Vol. 11:
Quantum Probability Communications eds. S. Attal and J. M. Lindsay
VOl. 10:
Quantum Probability Communications eds. R. L. Hudson and J. M. Lindsay
VOl. 9:
Quantum Probability and Related Topics ed. L. Accardi
Vol. 8:
Quantum Probability and Related Topics ed. L. Accardi
VOl. 7:
Quantum Probability and Related Topics ed. L. Accardi
QP-PQ Quantum Probability and White Noise Analysis Volume XXII
Infinite Dimensional Stochastic Analysis In Honor of Hui-Hsiung Kuo
Editors
Ambar N. Sengupta P. Sundar Louisiana State University, USA
N E W JERSEY
-
v
World Scientific
LONDON * SINGAPORE * BElJlNG * SHANGHAI
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HONG KONG * TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
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Library of Congress Cataloging-in-Publication Data Infinite dimensional stochastic analysis : in honor of Hui-Hsiung Kuo / edited by Ambar N. Sengupta & P. Sundar. p. cm. -- (QP-PQ, quantum probability and white noise analysis ; v. 22) Includes bibliographical references and index. ISBN-13: 978-981-277-954-0 (hardcover : alk. paper) ISBN-10 981-277-954-X (hardcover : alk. paper) 1. White noise theory. 2. Stochastic analysis. 1. Kuo, Hui-Hsiung, 194111. Sengupta, Ambar N. 111. Sundar, P. (Padmanabhan) QA274.29.154 2008 519.22-dc22 200705 1862
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V
PREFACE We are pleased t o present this collection of papers honoring Professor Hui-Hsiung Kuo. Several of the papers are authored by speakers at a special session of the American Mathematical Society Annual Meeting held in New Orleans in January 2007. The papers in this collection cover diverse aspects and contexts in stochastic analysis: white noise analysis, stochastic partial differential equations, non-commutative geometry, integral transforms, and applications to finance. Some of the articles are written with an expository slant so that they would be accessible t o new researchers and specialists in other areas. This volume will be of interest t o professional researchers and graduate students who will gain a perspective on current activity and background in the topics covered. We thank the anonymous referees for evaluating each paper in this collection. We also thank Ms Chionh of World Scientific for her patient assistance through the preparation of this volume. We wish to also acknowledge the support and encouragement we have received from our university and, especially, Guillermo Ferreyra, Dean of the College of Arts and Sciences.
Ambar N. Sengupta and P. Sundar Department of Mathematics
Louisiana State University
4th December, 2007
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vii
CONTENTS Preface
V
Complex White Noise and the Infinite Dimensional Unitary Group
1
T. Hida Complex It6 Formulas
13
M. Redfern White Noise Analysis: Background and a Recent Application
24
J . Becnel and A . N . Sengupta Probability Measures with Sub-Additive Principal Szego-Jacobi Parameters
42
A . Stan Donsker’s Functional Calculus and Related Questions P.-L. Chow and J. Potthoff
53
Stochastic Analysis of Tidal Dynamics Equation U. Manna, J. L. Menaldi, and S. S. Srztharan
90
Adapted Solutions to the Backward Stochastic Navier-Stokes Equations in 3D P. Sundar and H. Yin Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
A . Barhoumi, A . Rzahi, and H. Ouerdiane
114
135
...
viii
Contents
An Infinite Dimensional Fourier-Mehler Transform and the L6vy Laplacian K. Saito and K. Sakabe The Heat Operator in Infinite Dimensions B. C. Hall
149
161
Quantum Stochastic Dilation of Symmetric Covariant Completely Positive Semigroups with Unbounded Generator D. Goswami and K. B. Sinha
175
White Noise Analysis in the Theory of Three-Manifold Quantum Invariants A . Hahn
201
A New Explicit Formula for the Solution of the Black-MertonScholes Equation J. A . Goldstein, R. M. Mininni, and 5’.Romanelli
226
Volatility Models of the Yield Curve V. Goodman
236
Author Index
245
Infinite Dimensional Stochastic Analysis In Honor of Hui-Hsiung Kuo
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1
COMPLEX WHITE NOISE AND THE INFINITE DIMENSIONAL UNITARY GROUP Takeyuki Hida
Nagoya University, Emeritus Professor We discuss the infinite dimensional unitary group U(E,) in connection with complex white noise. Some important subgroups of U(E,) enjoy a beautiful structure expressed in terms of Lie algebras, and they play significant probabilistic roles.
Keywords: Complex white noise, Infinite-dimensional Unitary Group.
1. Introduction What we are going to present are in line with white noise analysis. Hence, before we come to the main topic, it is better t o remind the reader of the ideas and characteristics (in fact, advantages) of white noise theory. In this note we shall study a harmonic analysis of functionals of complex white noise by using the infinite dimensional unitary group. White noise analysis has developed extensively during the past three decades, and it seems a good time t o have the white noise variable complexified and to develop the theory of functionals of complex white noise. We are pleased t o describe three big advantages of white noise theory. (1) Use of generalized white noise functionals. Our idea starts out with reduction of the given random system, so that a system of idealized elemental random variables would be given.' They are independent and are taken to be the variables of functionals that describes the random system in question. Once the variables are given, it is the standard way to start with defining basic functions12 namely polynomials in those variables. If the innovation is Gaussian white noise, the standard system of variables are B ( t ) , tE R. Then, we need a clever trick t o define polynomials, which are most basic functions, and more general nonlinear functions of the B(t).Namely, we use the renormalization technique, in addition to the orthogonalization that is done in the classical
2
T.Hida
stochastic calculus. So, our calculus is different from, and in fact beyond the classical stochastic analysis over the Wiener space. We give an identity to B(t) and is used without smearing it by smooth functions of t. We are, therefore led to introduce the class of generalized white noise functionals, so that our harmonic analysis becomes systematic and further enjoys much freedom to find directions of further development. For instance, a good domain of the LBvy Laplacian can be defined and we can discuss applications to path integrals (Feynman and Chern-Simons, for instance) which heavily use those generalized functionals; similarly infinite dimensional Dirichlet forms, and so forth. (2) The infinite dimensional rotation group and the unitary group appear as the invariance of the white noise measure p and complex white noise measure v , respectively. (3) Since the innovation is defined as the basic notion of the analysis, this idea of using the innovation can be extended to the study of random fields that are parametrized by higher dimensional (space-time) manifold. Needless to say, much fruitful results can be obtained. We may say that this advantage comes from the reduction theory. 2. Complex white noise Let ( E * ,B, p) be a white noise space. We take two copies of this measure space and form a direct product to define a complex white noise in the following manner. Let ( E ; x E,*,B1 x B2, p1 x p2) be the product measure space, where bk, k = 1,2, are white noise measures with variance %.Set
and denote the vector space spanned by the z by EF. Naturally we can form a complex white noise (EE,v), where v = p1 x p ~ . 3. Infinite dimensional unitary group
The complex Hilbert space L2(E,*, v) is the basic space on which we shall carry out the analysis. In fact, a unitary group is defined in what follows, so that harmonic analysis can be done on L2(EE,v). Let g be a linear homeomorphism of E, = El iE2 such that
+
for all c' E Ec. 11~C11= IlCll collection U(E,)of such g's forms a group under
The and is called the infinite dimensional unitary group.
the usual product,
Complex White Noise and Infinite Dimensional Unitary Group
3
The group U(E,) is viewed as a complexification of the rotation group O ( E ) .On the Hilbert space L2(EZlv) we can define unitary representation of the group U(E,), hence we can discuss the harmonic a n a l y ~ i s .Finer ~ results on the analysis can be seen by observing significant subgroups of
U(&). 4. Subgroups of U ( E , )
1) Finite dimensional unitary group U ( n ) ,n 2 1 and its limit.
As in the case of G, c O ( E ) ,we choose an orthonormal system &, k 2 1, in E,. The first n members determine an n-dimensional subspace En and the unitary group U ( n ) acting on En is defined. The group U ( n ) can be embedded in the group U(E,) as its subgroup. Further, the projective limit U(m) of the U ( n ) is also introduced. A member of U(o0) may be considered to be approximated by finite dimensional transformations. 2) Another generalization technique is as follows. Take members in E , as many as nk such that they form an orthonormal system. They span a subspace En,. We take En, , k = 1 , 2 , .. , m, which are mutually orthogonal. Thus a subspace @ Enk c E, is given. Hence, we can form a product U(n) = @ u ( n k ) of the unitary groups U ( n k ) defined as above, where n = (n1,n2, . . . , n,). By letting rn + 00 we are given an analogue of the windmill subgroup W , which is essentially infinite dimensional. We then come to subgroups which are coordinate-free. Namely, we take one-parameter subgroups that come from diffeomorphisms of the parameter space. We call such a one-parameter group a whisker. 3) Conformal group. If, in the Rd-parameter case, the basic nuclear space E is taken to be DO,which is isomorphic to Cm(Sd),then we are given the conformal group C ( d ) which is a subgroup of O ( E ) .Hence, the complex form of C ( d ) , denote it by C,(d), acting on the complexified space Do,,, is a subgroup of U(DO,,). We call it a complex conformal group. It is locally isomorphic to the (real) linear group SO(d + I l l ) and is generated by one-parameter groups including whiskers as many as (d+1)2(d+2). Their generators of shift, isotropic dilation , rotation S O ( d ) and special conformal transformations are, respectively, as follows: d s = --
dui
, i = 1 , 2 , ..., d ,
4
T.Hida
The one-parameter subgroup r t , t E R, with generator r is an operator such that rtC(u) = C(ue-t)e -td/2 .
It is called an isotropic dilation. Remark 4.1. In order to have a finite dimensional Lie algebra (group), we should have the isotropic dilation instead of general dilations. 4) Heisenberg group. From now on, one can see the effective use of complex white noise with emphasis of the role of the Fourier transform, actually in many ways. The basic nuclear space E, is now specified t o be the complex Schwartz space
s, = s + i s . 4.1) The gauge transformation
-
It,t E
It : ~ ( u )
R, is defined by
It<(u) = eit<(u).
Obviously It is a member of U(E,), and { I t } forms a continuous one parameter subgroup. It is periodic with period 2n. I t I s = It+sr
t ,s E R,
The group {It,t E R } is called the gauge group. Actually, we have an abelian gauge group that is isomorphic t o U(1) 2 S1. Indeed. Let the unitary operator Ut be defined by Ul, , which forms a one-parameter unitary group acting on L 2 ( E , * , v )This . group has only point spectrum on the subspace Hn involving complex multiple Wiener integrals of degree n. The
C o m p l e x W h i t e N o i s e and Infinite D i m e n s i o n a l U n i t a r y Group
eigenspace belonging to the eigenvalue -n following proposition is proved:
+ 2k
is
H(n-k,k).
5
Hence, the
Proposition 4.1. The space H n , n > 1, is classified, according to the action of It, into its subspaces H(*-k,k) f o r which eigenvahe -n 2k is associated. The infinitesimal generator of the gauge group is iI, where I is the identity operator.
+
4.2) The shift Si. For the case d = 1, the shift St is the complex form of the shift belonging to O ( E ) . We can easily extend to the Rd-parameter case. The generators are
4.3) Multiplication 7rilj= 1,2,. . . , d. Let them be defined to be the conjugate to the shifts via the Fourier transform E 7r:
=Fsp-?
Actual expressions are (7r:<)(u) =
eZ"j<(u), u E Rd.
Its infinitesimal generator is denoted by "j
rj
and it is expressed as
=iUj.
Definition 4.1. The subgroup of the U(S,) generated by the gauge group, the shifts and the multiplication is called the Heisenberg group.
It should be noted that we have the commutation relation 7rtss=
IStSS"t.
In terms of the generators
[", s] = ZI, which is a most significant relation (actually, it is nothing but the uncertainty principle),
6
T. Hida
Gauge transformations (continued) We can extend the Heisenberg group which shows an invariance of complex white noise. The Heisenberg group introduced above involves three kinds of oneparameter groups. In addition to them, we have a new class of continuous linear operators acting on E, as a generalization of the { I t } . noindent 4.4) Define I , by
where a is a member of the (real) Schwartz space S. The operator I , is a unitary operator on L2(R)and it is called 5'-gauge transformation. The infinitesimal generator of Iat is ia. The collection {I,, a E S } forms a group under the usual product and is a subgroup of U ( S c ) .The group is called S-gauge transformation group. Obviously it is an abelian subgroup of U ( S C ) . We recall the infinitesimal generators of the whiskers obtained so far.
where a E S. Non-trivial commutation relations are [r,a] = 0, [s, a] = a' (E S ) .
The adjoint I: acts on z E S, in such a way that e-ia(u)z(u),and the
1; = ,-w4
aES,
7
is an Sx-gauge transformation. The collection
{I; = e-4") a 7
w
is called S*-gauge transformation. The following theorem can be proved without any difficulty.
Theorem 4.1. i ) The generators listed above f o r m a Lie algebra under the Lie product [ , 1. It is also a n algebra ( i n the ordinary sense). ii) The S*-gauge transformation acts on z = (x,y)-space, and u measure is invariant under the action of the adjoint I:.
Complex White Noise and Infinite Dimensional Unitary Group 7
5) The Fourier-Mehler transforms .To. Since particularly important roles of the Fourier transform can be seen in the study of complex white noise, we shall further proceed to the fractional powers of the ordinary Fourier transform. Actually, we define a oneparameter system of unitary operators Fe,8E [0,27r]such that Fe is viewed as the :-th (fractional) power of the ordinary Fourier transform, where 8 is considered mod 2 ~ . The operator in question is defined by the integral kernel Ke(u,v): i(u2+u2) Ke(U,u)= ( ~ (-1e ~ p [ 2 i 8 ] ) ) - ~ / ~ e x p
iuv
It defines an operator Fe by the formula
where B # i k ~ Ic, E 2. We now have some observations about the actions of Fe. Set
Then, it is proved that .Fetn(U> = einecn((u), 72
With this relationship we can prove that interpolation), and further
3 0
FeFe, = 3 e + p = Feu, 8 + 8' 3 0 +I
, as 8
---f
2 0.
is well defined for every 8 (by = 8" (mod 2 ~ ) .
0.
Particular choices of 8 give
Thus, we have obtained a periodic one-parameter unitary group including the Fourier transform and its inverse. The kernel function Ke illustrates this fact. Moreover one can see that the 3 0 defines a fractional power of the Fourier transform as is shown in what follows.
8
T. Hada
First we note an identity
a
vKe(u,v) = (ucose- i-sinO)K~(u,v).
au
Namely, the multiplication operator “u.is transformed by .Fe by the following formula
u
ia
-+ ucosB+
--sine.
i
au
This comes from the above identity. Similar computation proves that we are given, under Fe,
ia
--
i au
-usin0
H
ia + --cosO. i du
Symbolically writing, we have established
(--
(
( )
++ cOse sine) -sinBcose 7x au We are happy to see that there appeared the group SO(2) again. The infinitesimal generator of 30is denoted by if and is expressed in the form 3 0 :
a
1 . d2 if = - -1 ( 7 - u2 + 1) du
2
Observing the commutation relations of the generators, so as to have a finite dimensional Lie algebra, either real form or complex form, we are naturally given a new generator 0’ which is expressed in the form fff
5
= 1
(@+2). d2
We are particularly interested in the probabilistic roles of this operator (generator) in quantum dynamics. An easy and formal interpretation of of is that
1 -at* i
= U’Q
is the Schrodinger equation for the repulsive oscillator. For our purpose, it is convenient to take
(T
= d 4- $ I , namely we introduce
u = -1( , f ud22 t i I ) . 2
du
Complex White Noise and Infinite Dimensional Unitary Group 9
A one-parameter group with the generator u can be defined locally in spacetime only. It is, however, interesting to discuss the operator u in connection with the dynamics having a potential of repulsive force.
Lie algebras of infinitesimal generators. We have so far various infinitesimal generators. For simplicity we first consider the case d = 1. The Lie algebra of the conformal group is d du’ d 1 = u-+ -, du 2
s = -7
2 d
-+u,
K = U
du
Then, we come to the complex world, where one can find not only complexification of the real Lie algebras that have already been given, but also new members having close connection with the Fourier transform. The Heisenberg group defines the Lie algebra generated by
in
= iu,
Hereafter, the one-parameter subgroups I,t will be treated separately. One reason lies in the fact that we wish t o remain within finite dimensional Lie algebras (Lie subgroups). There are two interesting generators related to the Fourier transform. They are
o = - (1- 1 +d2 u2+iI).
2
du
We are now ready to consider the structure of the Lie algebras. Recall the Lie algebra c ( d ) of the conformal group. The c(1) is
c(1) = { S , T ,
K}.
10 T.Hida
Note. In the multi-dimensional case, say d-dimensional, we add the rotations Y j , k , 1 5 j , k d, to have c(d):
<
Behind the construction of the algebra c(d), the reflection R is involved. Thus, in order to define the conformal group it is recommended to take the basic nuclear space to be Do,not to be an arbitrary nuclear space E . The Lie algebra of the d-dimensional Heisenberg group is denoted by h(d). For the present case d = 1 we have
h(1) = {iI,S,T = iu}. The group and the algebra are based on complex white noise. Keep the following two concepts in mind. 1. There the ordinary Fourier transform F plays a key role. 2. It is a member of the unitary group U ( S c ) .
Then we are naturally led to the Fourier-Mehler transform Fe,the fractional power of the Fourier t r a n ~ f o r mThe . ~ generator if of Fe is - (&-u2+ I ) . Remind that we have introduced the operators, first a',then a: in fact, by hand for a moment, and later some interpretation is given so that the commutation relations appear in good shape. We have therefore had
i
(7 =
12 ("du2 + u2 + 21)
Summing up
Proposition 4.2. Based o n the set of operators
{iI,S , T , 7, f, we have 6-dimensional complex Lie algebra g . This algebra has the real form as is easily seen.
Table of commutation relations. For c(l), [T, S] = -S,
,
Complex White Noise and Infinite Dimensional Unitary Group
[T, 61
11
= K,
[ s , K ] = -27.
For h(l),
[ r , s ]= I . For the algebra g,
[f,r] = -20
+iI,
[a,T ] = 2 f - I .
The algebra h(1) is an ideal of g and is the maximum solvable Lie subalgebra. In short we may state Proposition 4.3. The ideal h(1) is the radical o f g . Proof is given by the actual and rather easy computations. It seems necessary to give some interpretation, from various viewpoints, to the generator 6 ,a member of the special conformal transformation. 1. The reason why the
6
has been taken.'
(i) Obviously the 6 is a good candidate to be invited to a class of possible generators expressed in the the form u ( u ) & id(.). We see that when the basic nuclear space E is taken to be Do,the K is well acceptable. Having had the K in our class, we have formed the algebra generated by those admissible generators and we see that the algebra is isomorphic to sZ(2, R). This is a beautiful result. (ii) Similar to s, the 6 is transversal to T. This is a significant property, and in fact, it defines a flow of the Ornstein-Uhlenbeck process. (iii) The K is related to the reflection with respect to the unit sphere. Hence, the parameter space must be RdU{m}.It is, however, useful when variational calculus is applied for random fields parameterized by a smooth simple surface.
+
2. On the other hand, there are good reasons why 6 should not be involved in the algebra g.
12
T.Hada
(i) From our viewpoint that the Fourier transform is particularly emphasized. So the complex Schwartz space is fitting for the complex analysis. Namely, the Schwartz space S,, which is invariant under the Fourier transform, is more significant. While, in order to introduce the IC. we need another space like Do,instead of (ii) Under the Fourier transform we have a formal adjoint which is not suitable for our purpose.
s,.
So far we have seen the beautiful structure of generators in terms of the Lie algebra. Important note is that their probabilistic roles are in cooperation with the beauty of the algebra. References 1. T. Hida and Si Si, Lectures on white noise functionals. World Scientific Pub. Co. 2007. 2. P. LBvy, Problhmes concrets d’analyse fonctionnelle. Gauthier-Villars, 1951. 3. T. Hida, Brownian motion. Springer-Verlag. 1980. 4. H.-H. Kuo, White noise distribution theory. CRC Press. 1996.
13
COMPLEX I T 0 FORMULAS Mylan Redfern
Department of Mathematics and Computer Science Valdosta State University Valdosta, G A 31698-0040 m.redfernOvaldosta. edu This paper continues the development of complex white noise analysis in of complex Wiener distributions. For a cerearlier w 0 r k s ~ 7on ~ the space, (D;), tain class of functions F : C m --f C and a vector @ = (@I,. . . , am)of complex Wiener integrals , formulas for $ F ( @ ) are developed. For F : Rm -+ R, and a vector 4~ of real Wiener integrals, the It6 formula for F(4w) is obtained in the complex setting.
Key Words: Complex multiple Wiener integrals, Complex white noise analysis, Complex Wiener distributions, It6 formulas. 1. Introduction The complex version of white noise analysis has its foundations in It6's 1953 paper,l and further treatment appears in the book of Hida.2 Recent applications of Hida's treatment can be found in Ref. 3. In Ref. 4 we generalized Hida's approach to allow for multiparameter time and complex Brownian motion &(t) with state space C M . For this we use the probability space:
R = R 1 XR1, where 01 = S * ( R d , C M is ) the dual of the space of CM-valued rapidly decreasing functions on Rd.The probability measure on R is V=PxP,
where 1-1Gaussian measure on 01. Our treatment incorporates a general covariance operator D into the measure: u = VD. We began the development of the white noise calculus on the space D*(R)c of complex Hida-Wiener distributions in.5 For a class of processes taking values in this space generalized derivatives and integrals are naturally defined. We continue this development here obtaining several It6 formulas.
14
M. Redfern
2. Background and Notation
The basic probability space R1 for the real theory is the infinite dimensional space of Schwartz distributions R1 = S*(R,R), endowed with a Minlos measure p~ on R1 with characteristic functional:
Here E S1, while (w1,&) E wl(&) denotes the canonical bilinear pairing of dual spaces S; and S1. We also use the notation: $(rl) for the random variable on 01 given by $ ( & ) ( w l ) = ( w l , & ) . The space of real Wiener functionals is: (L2)D = L2(Ql,dpD,R).
The Fock space (chaos) decomposition of this space of real Wiener functionals leads naturally to various spaces of generalized Wiener functionals,6-8 and in particular to the space (D*)O of real Wiener distributions (see Refs. 9,lO) which is also a special case of complex white noise. The probability space for the complex white noise theory is 52 = 0 1 x 0 1 , endowed with the product measure U D = p~ x p ~Since . 0 1 = S*(R, W) is a space of real-valued linear functionals, we can identify its complexification: R = R1 x R1,with a corresponding space of complex-valued linear functiona l ~i.e., , with S * ( R d , C M )For . w = ( ~ 1 ~ E~ 52, 2 define ) a complex-valued linear functional on S(Rd,C M )by:
( w , O= b J 1 , 5 1 )
- (w2,52)
+ [(w1,J2) + (W2,Il)li
(2)
+
Here E = &i, with (1, & E S(Rd,RM). From this we get the complex random variable $(<) on R, where $(E)(w)E (w,<). Note that the definition in (2) is different than Hida's [2, p.2381, and is specifically chosen so that $, as a map from S(Rd,C M ) Lo(R) is complex linear. Two additional random variables are needed: --f
@(Lr ] ) = $(t)+ S ( V ) The maps
(c,r ] )
c-)
(4)
$ and CP are both complex linear transformations and the map
a(<,r ] ) leads to the Fock space (chaos) decomposition of the space: (Li)2D
L2(R, d v D , C ) ,
of complex Wiener functionals. We use the following notation: V
Cp(Wd,V),
SIC =
= CM @ C M , D ~ c = LfC = L2(Wd,V). An element w E V S(Rd,V),
Complex It6 Formulas
15
has the form: ‘u = (z,w) with z , w E C M , while the elements in any of the last three spaces have the form: f = ( E , r ] ) , with J,r] in CF(RD,CM), S(Rd,C M ) or CM),respectively. A conjugation operation on Lfc is defined by f = (E,rl) E ( f j , f ) . Note also that the inner product on Lf@ is given by (f7.f’) = ((E,r]),(J‘,77/)) = (LE’) (V,r]?, where (tit’)= J~d(c(x),/t(x))dx= JRd Ej(x)J;(z)dx. The action of the covariance operator D is naturally extended t o the complex domain: DE = D t j +iD& and Df= D(<,r ] ) = ( D S ,Dq), We also use the notation:
q2((IWd,
+
EL
q5D(E)
= q5(D-1/2E)
3. Complex White Noise Analysis
The material in this section reviews the framework for the calculus. Proofs and a more complete discussion can be found in Ref. 4. The relevant characteristic functional in the complex case is given by
+
d2D(r]) is complex linBecause the map ( E , r ] ) H @ 2 D ( [ , r ] ) = q52D(E) ear, one gets from formula (5) that q52D(E) and $ 2 D ( r ] ) are complex Gaussian random variables with variances llE112 and 1[r]1(2, and thus the families: {420(c)IcE SIC}and { i 2 D ( r ] ) 1 r ] E SIC}are complex Gaussian systems. As a particular case of [4, Proposition 11, one has
and consequently the map f extends to an isometry (into):
H
@2D(f)
= I;t(f) defined for f
The image of this isometry, denoted by K;: complex 1-tuple Wiener integrals.
E
E
Sic
I;t(Lfc), is the space of
16
11.1. Redfern
To get the other spaces K:@" of complex multiple Wiener integrals we use the following spaces:
vn -- v@'"= ( @ M @ p ) @ n
(6)
Sn@= S(R"d, K ) (&@)@'" Dn@= C,.o(R"d, Vn)?x (Dl@)@'n LiC = P ( R n d ,Vn)!2 (L;,)@n
(7)
(8) (9)
In (7)-(9) the tensor product @ stands for the completion of the algebraic tensor product with respect t o an appropriate topology. Below we use the notation Bate denote the algebraic tensor product of vector spaces and we denote the symmetrization of the various objects in the theory by a hat -. For fj = (<j,$) E ST@, j = 1,...,m, and for a subset S (1, ...,m } with an even number of elements: IS1 = 2p, let
c
(P,. . f")s = 7
( p J j 1 ) .
.. ( P " f f j P ) .
(10)
pairs on S ti1 , j l } ~ ~ ~ { i P > j P )
Let P denote the polynomial subalgebra of Lo(R) generated by the random variables: { Q Z D ( f ) I f E SIC}. The nth multiple Wiener integral map: I:@" : S g " -+ P is defined by:
c c
b/ZI
I ; g ( f l b . . . hfn) =
(-1)h
k=O
( f l , . . . , f")s
nF y f f j ) .
(11)
jes
(SI=2k
00
This map extends linearly t o the coproduct S& =
(S~C)@~". n=O
Remark We find it convenient t o sometimes use the following alternative notation for the multiple Wiener integral defined in (11): P(f1)b '.
. h@(f") = I ; g ( f l b .
* *
&If").
(12)
One should aIso recognize that the tensor product notation used for the expression (12) is alternatively denoted in the literature by: : @D
(f1 ) . . . P (f") :
and is called the Wick product of these random variables. The nth multiple Wiener integral map I:.@"defined by (11) on SFcn c L& extends by continuity to a map 1:; : Lie + (LZ). For n # m,
Complex It6 Formulas
17
J , ~ $ ( f ) ~ & % ( g ) d v D = 0. While for n = m, J,I;,"(f)~;,"(g)dvD = n ! ( f , g ) . The image of this extension is denoted by
and is called the space of n-tuple complex multiple Wiener integrals. We thus arrive a t the complex chaos decomposition of the space of complex Wiener functionals 00
(L;)2D= @K:; n=O 00
The space L 2 ( 0 , d v , @ )is isomorphic t o the space L i
E
@n&, n=O
via the Wiener integral map I . Note that Lg is a subspace of the space 00
Vc F n D t c l with the usual identification of L& with a subspace of Vge. n=O
We denote an element T E Dc by T = CTnlthought of as a formal sum. The space V& endowed with the Cartesian product topology is naturally 00
identified with the dual of the space V@ z U D n e ,which is the coproduct of
n=O
the respective spaces Dnc of C" functions with compact support. Elements f of the coproduct are also denoted by formal sums f = C f n l but now f n = 0 for all but finitely many n. By endowing the coproduct Vc with the inductive topology, we see that it is continuously included in Lg, and thus we get a Gelfand triple:
vc c Lg c D;. By using the Wiener integral map I , its restriction, and dual we get the space of complex Wiener distributions:
Definition 3.1 (Complex Hida-Wiener Distributions). For each n let
D(R)iE = (Dnc)2D = I;"(&@). Then the space of complex Wiener distributions is: 00
n=O Hence,
18
M. Redfern
The elements of (2)c)2Dact on the elements of ( 2 ) ~via ) ~ the~ multiple in ( 2 ) ~ ) ~ ~ Wiener integral map I . More specifically, for P = C,"==, and Y = C,"==, I:,"(Un) in (2)c)2Dthe pairing is given by
n=O
Note that the Wick product naturally extends to ( 2 ) c ) 2 D :For X C,"==, I:e(Tn) and Y = C,"==, I,?$(Un) in (2)c)2Ddefine
=
For the remainder of this section we drop the D from all notation, it being understood that all formulas and constructions naturally involve it. Suppose F : C" -+ C is a Borel measurable function, and Q,1, ..., a, E K1c. Let Q, = ( @ I ,..., a,), be the corresponding random vector: Q, : R + Cm. We say that F(Q,) defines a n element of (Vc), and write F ( @ ) E (2);) if for every n = 0,1, ..., the mapping Pn c) d v , defines a continuous linear functional on (Dnc). In the following we use the multi-index notation a = (011, ...,a,) E Nr, with la1 = a1 . . . a,, and a! = a l ! .. . a,!. Also for z = ( z 1 , ...,z,) E C m , and Q, as above, let z a = zff' . . . zgm and @ha= Q1, h a l ~.. . We use the following standard partial differential operators acting on functions F ( z ) of m complex variables: For z = ( 2 1 , ..., z,) E C", with z j = xj yji let
+ +
&@em.
+
Suppose that $(
Theorem 3.1. Let $1, ...,,+, (4j = +(<j)) be linearly independent and A be the m x m matrix with Aij = (4j, &). Assume that F : @" + @ is Borel measurable. T h e n F ( $ ) E (276)ij and only if F ( z ) a ( & ) p r A is integrable f o r all a,p. W h e n this is the case F ( 4 ) E K L C and has Fock space decomposition
Complex It6 Formulas
19
4. Calculus of (D;)-Valued Processes In this section we recall the definitions of derivatives and integrals of generalized processes 8 : R + (23;) (Ref. 5)and obtain several It6 formulas. The definitions here are analogous to those for real white noise given in Ref. 10. The proof of Theorem 2 appeared in Ref. 5 but we include it here for completeness.
Definition 4.1. A function 8 : IR + (23;) is called (1) differentiable if the function t + ( 8 ( t ) , $ ) is differentiable for every $E Pc) (2) integrable if the function t -+ ( 8 ( t ) ,$) is integrable for every $ E (3%). It follows that
$ ( t )and J 8(t)dt are elements in (23;)
which satisfy
for every II,E ( D c ) Because of the structure of the test function space (Dc) the function 8 is differentiable (integrable) if and only if each of its components 8, is differentiable (integrable), and then
d8 = dt R
O0
n=O
%(On) d
and /B(t)dt =
00
/&(t)dt.
n=O
With respect to the product &I there is also the product rule. If C,8 : ( D c ) are differentiable then so is (68 and
4
Warning: In the theorems that follow we will be using the implied summation notation on repeated indices.
Theorem 4.1. ( I t 6 Formula 1)Suppose 4j(t) E 4(cj(t))E K ~ c j, = 1,.. , , m are linearly independent and differentiable o n a n interval of times and also that Aij(t) are differentiable f o r every t . Assume that F : C" --+ C is Bore1 measurable with F ( 4 ) ( t )E F(#(t)) in (D;) f o r each t . T h e n F ( 4 ) i s differentiable and
20
M. Redfern
Proof:Differentiating (13) gives us
= A;aiajjrA to the first series and using the Applying the identity product rule for 6 in the second, we get, after simplification,
Adjusting the index in the second and third series, we see the result.
Corollary 4.1. Suppose $ j ( t ) 3 4(cj(t)),$,3 4(fj,(t)),j = 1 . .. , m , r = 1 . . . q are in K1c. Also assume that & ( t ) ,. . . ,q5m(t),$1(t),. . . ,$q(t)are linearly independent on an interval of times and have a differentiable covariance matrix A ( t ) .If F : Cm+q -+ C is Bore1 measurable and F ( 4 , 4)(t) is in ( V c ) f or each value o f t then
+
+
where A is the m q by m q matrix with Aj, = (#,,$j) = (&,,&), Aj, = (fjTlcj),A,j = (<jIfj,),A,, = (fjslfj,). Here a,,j = 1 . . .m and D,, r = 1 . . .q are the derivatives with respect to 21,.. . , zm and Z m + l r . . . ,Z m + q , respectively. Proof: Note that F ( $ , $ ) = G(+,$) where G : Cm+q -+ C is defined by G ( ~ 1 , . . . , z m , ~ 1 , . . . , 3F(zl,...,zm,Wl,...,Wq). ~q)
Complex It6 Formulas
21
Corollary 4.2. (It6 Formula 11) Let @ j = a(&, qj), j = 1,.. . ,m, be elements of K1@with ~ ( C I ) .~. ., 4(Em),d(jil), . . . ,$(rim) linearly independent on an interval of times with differentiable covariance matrix as in Corollary 1. Suppose F : Cm 4C is Borel measurable and F ( @ ) ( t )is in (DE)for each value o f t . Then F ( @ ) is differentiable and
d -F(@) dt
d@j dt
= ajJ’(@)@-
+ a j F ( @ ) @dGj dt A
5. Real Case We will now see that the complex It8 formula contains the real case. For this we observe that there is the isometry -: ( L 2 ) O
defined by F
H-
F
= L2(R1,pUo,R) ( L i where F ( w ) = F(w1). --f
p5P(n,vo,C)
N
Example 5.1. For [I in S(R,R) let L2(R1, p ~R), defined by
4jfS(&)
be the element in
4 a t l ) ( ~ l=) (wl,D-1’2h) We have that
Corollary 5.1. (Real Case, D=I) Let F : R” 4 R be Borel measurable and assume that 4as(I:) ( t ) ,. . . 4as(
-
22
M. Redfern
where
Proof: Let F : Cm -+ R be defined by F(z1,. where xi = Re(zi).Then
. . , z,)
=
F (z 1 ,. . . , x m )
F(4ws(E:),. . * ,4ws(E;"))(w) = F(4ws(E:),. . . , 4 w s ( E Y ) ) ( w ) = F(4ws(t:)(wl),.. . 1 4 w s ( E ; " ) ( w d ) = F(-
-
4 R & : ) w , . . . 4ws(tY)(w)) = F M t : ) , . . 4(E;")) I
-
. ?
Setting 4 = (4(E;), . . . ,$([I"))we see t h a t (F(4ws))= F(q5).Denoting the covariance matrix of q5ws by Aws and t h a t of 4 by A@ we find ). for all a and t h a t A@ = 2Aws and thus r A c ( z ) = r A R s ( ( 2 ) r A R s ( yHence, p, F ( z ) (%)a (2) = F ( x ) (&)Q(&)PrAR ( x ) r A ,(y) is integrable. Therefore, F ( & ) ( t ) is in (D5)f.r each value o f t . By Theorem 2 ,
-
Combining the first two terms and observing t h a t have the conclusion.
$i
+ & = 24w(Ef) we
References 1. K. It6, Complex multiple Wiener integrals, Japan. J. Math. 22 (1953) 63-86. 2. T. Hida, Brownian Motion, Springer, New York, 1980 3. N. Obata, A note on Hida's whiskers and complex white noise, Analysis on Infinite Dimensional Lie Groups and Algebras, World Scientific, 1998, 321336. 4. M. Redfern, Complex White Noise Analysis, Infinite Dimensional Analysis, Quantum Probability, and Related Topics, 4, No.3, (2001) 347-375. 5. M. Redfern, Complex Stochastic Calculus, Contemporary Mathematics 317 (2003), H.H. Kuo and A. Sengupta (Eds.), 193-202.
Complex It6 Formulas
23
6. T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, (Eds), White Noise Analysis Mathematics and Applications (1990), World Scientific. 7. T. Hida, H.-H. Kuo, J. Potthoff, L. Streit, White Noise - an Infinite Dimensional Calculus, Kluwer, 1993. 8. H.-H Kuo, White Noise Distribution Theory, CRC Press, 1996. 9. D. Betounes, Trace operators, Feynman distributions, and multiparameter white noise, J. Theoretical Probability, 8 (1995) 119-137. 10. D. Betounes and M. Redfern, Wiener distributions and white noise analysis, Appl. Math Optim. 26 (1992) 63-93.
24
WHITE NOISE ANALYSIS: BACKGROUND AND A RECENT APPLICATION Jeremy J. Becnel Department of Mathematics and Statistics Stephen F. Austin State University Nacogdoches, Texas 75962-3040 e-mail:
[email protected] Ambar N. Sengupta Department of Mathematics Louisiana State University Baton Rouge, LA 70803 e-mail:
[email protected] We present a description of the framework of white noise analysis as an infinitedimensional distribution theory. We then describe some recent developments in the context of an application arising from quantum computing.
1. Introduction
In this paper we present some of the main ideas, results and techniques leading t o white noise analysis and then describe a recent application. This is addressed primarily t o new researchers in the field and experts in other fields who wish to see a short overview of the subject. We also include some recent developments in the context of an application of white noise analysis to a continuum form of the Shor algorithm in quantum computing. 2. Background: The Schwartz Space as a Nuclear Space
In this section we shall present a rapid description of Hermite polynomials and those aspects of the Schwartz space S(R) of rapidly decreasing functions which help motivate the infinite dimensional theory of white noise distributions. Then we will describe a natural abstraction of the structure of the Schwartz space to the infinite-dimensional setting, along with a construction of Gaussian measure on this space.
W h i t e N o i s e Analysis: Background a n d a R e c e n t A p p l i c a t i o n
25
2.1. Hermite polynomials, creation and annihilation operators We shall summarize the definition and basic properties of Hermite polynomials (our approach is essentially from Hermitel). We refer to the paper of Hall' in this volume, and to [3, page 1551, for more on Hermite polynomials in a closely related context. A central role is played by the Gaussian kernel
Properties of translates of p are obtained from ezy-$
- P(" - Y) P(X)
Expanding the right side in a Taylor series we have
where the Taylor coefficients, denoted H n ( x ) , are
This is the n-th Hermite polynomial and is indeed an n-th degree polynomial in which xn has coefficient 1, facts which may be checked by induction. The simple identity
W n 1 K n ) L Z ( p ( s ) d z ) = n!Snnl
(6)
Because these are orthogonal polynomials, the n-th one being exactly of degree n, their span contains all polynomials. It can be shown that the span is in fact dense in L'(R,p(x)dx). Thus the polynomials above constitute an orthonormal basis of L'(R,p(lc)dx). Next, consider the derivative of H,:
HA (x)= (- l)np(x)-1p("fl) (x)- (- l),p(z) -1p'(x)p(x) -1p(")(x)
26
J. Becnel and A. N. Sengupta
(-;
+z) h,(z) = m h , + l ( z ) .
2+
We refer t o (z) as the creation operator in L2(R; p ( z ) d z ) Next, from the fundamental generating relation (3) we have :
Using the relation (3) again on the left we have
Letting y = 0 allows us to equate the n = 0 terms, and then, successively, the higher order terms. From this we see that H A ( z ) = nHn-l(z) where H-1 = 0. Thus:
2
The operator is the annihilation operator in L2(R; p ( z ) d z ) . Let us now translate the concepts and results back to the space L2(R; d z ) . To this end, consider the unitary isomorphism:
u : P ( R , p ( z ) d z )+ L2(R,d z ) : f
I-+
4.f
(11)
Then the orthonormal basis polynomials h, go over to the functions given by
&
The family {$,},20 forms an orthonormal basis for L2(R,dz). We will now determine the annihilation and creation operators on L2(R,dz). If f E L2(R,dz)is differentiable and has derivative f’ also in L2(R, d z ) , we have:
So, on L2(R,d z ) , the annihilator operator is
A = ddx+ L 2z
and satisfies
A& = ,,hi&-l
(14)
White Noise Analysis: Background and a Recent Application
27
for all n 2 0, where 4-1 = 0. For the moment, we proceed by taking the domain of A to be the Schwartz space S(R)of rapidly decreasing functions. Next,
Thus the creation operator is
The reason we have written A* is that, as is readily checked, we have the adjoint relation (Af, g) = (f, 45)g) with the inner-product being the usual one on L2(R,ch)Again, . for the moment, we take the domain of C to be the Schwartz space S(R) (though, technically, in that case we should not write C as A * , since the latter, if viewed as the L2-adjoint operator, has a larger domain). For this we have
(-& +
A straightforward calculation shows that
[A,C]= A C - C A = I ,
(17)
the identity. Next observe that
CA& = f i f i $ n = n&
(18)
and so C A is called the number operator N :
N = A*A = C A =
-& + .” Ndn
-
1
the number operator
= nd,
(19)
(20)
Integration by parts shows that (f,9’) = -(f’,g) for every f,g E S(R), and so also (f,g”)= (f”,g). It follows that the operator N satisfies
(Nf,9 ) = ( f 7 N g ) for every f , g
E
S(R).
(21)
28
J . Becnel and A . N. Sengupta
2 . 2 . The Schwartr space as a nuclear space The operator
x2 + -1 T = N + 1 = --d2 + dx2 4 2 plays a very useful role in working with the Schwartz space. As we have seen from (20) the &, are eigenfunctions of T , i.e.
T&
=
(n
+ l)&
for all n E W = { 0 , 1 , 2 , ...}.
Let B be the bounded linear operator on L2(R) given on each f E L2(R) by
It is readily checked that the right side does converge and, in fact,
IIB~II$
=
C
(n+1)-21(f,4n)12L2
5
C I(f,4n)I:z
=
I I ~ I I ~ (24) ~
n EW
n EW
Note that B and T are inverses of each other on the linear span of the vectors &. That is, T B f = f and BTf = f for all f E C, where C is the linear span of the vectors &, for n E W . For any p 2 0, the image of BP consists of all f E L2(R) for which En€& 1)2pl(f,4n)12 < 03. Let
+
Sp(R)= BP (L2(R)).
(25)
This is a subspace of L2(R), and on Sp(R) there is an inner-product (., . ) P given by
(f,g ) p efC ( n+ l ) 2 p ( f , $n)(dn,9) = ( B - p f ,B - P g ) L 2
(26)
nEW
which makes it a Hilbert space, having L. Furthermore, an orthonormal basis of Sp(R) is formed by the elements ( n l)--P& E Sp(R),and
+
showing that the inclusion map Sp+l (R) 4 SP(R) is Hilbert-Schmidt. The structure of the Schwartz space as a nuclear space is encapsulated in the following chain of inclusions (the first of which is an important equality!):
S(R) = nPEWSP(R) c . . . c &(R)
cs l (c ~s~(R) ) =L~(R)
(27)
White Noise Analysis: Background and a Recent Application
29
and the fact that the topology on S(R) generated by the n o r m s II.Ilp coincides with the standard topology. The standard topology is the one generated by the family of seminorms on S(R) given by
for all a , b E {0,1,2, ...}. Proofs may be found in Simon4 and in Ref. 5 2.3. The abstract formulation
As before, we use the notation W = {0,1,2, ...}. We work with a real separable Hilbert space Ho, and a positive Hilbert-Schmidt operator B on Ho. Thus HO has an orthonormal basis { U , } ~ ~ Wof eigenvectors of B, with Bun = Xnun and CnloIXnI2 < 00, with each An The example to keep in mind is
> 0.
Ho = L2(R) un = 4 n
B = (--
d2
dx2
x2 -+ + A)-' 4 2
with eigenvalues An
=
(n
+ 1)-l.
On Fp we have the inner-product (., . ) p given by
(a,b),
=
C XiZPanbn n EW
This makes Fp a real Hilbert space, unitarily isomorphic to L2(W,p p ) where p p is the measure on W specified by p p ( { n } )= XG2p. Moreover, we have
F
def
=
n p E w F p C ' . . F 2 ~ F l ~ F o = L 2pa) (W,
Each inclusion Fp+l
-+
Fp is Hilbert-Schmidt.
(30)
30
J . Becnel and A . N . Sengupta
Now we pull all this back to L2(R). First set
H, = I-'(F,)
= {Z E
Ho : ~ X , z p l ( ~ , ~
(31)
n>O
It is readily checked that H, = BP(H0). On H , we have the pull back inner-product (., .), which works out to
(f,d
p =
W
P
f
1
B-,S)
(32)
Then we have the chain def
H = npEWHp C . . . H Z c H I C Ho,
(33)
with each inclusion Hp+l 4 H, being Hilbert-Schmidt. Equip H with the topology generated by the norms 1. 1, (i.e. the smallest topology making all inclusions H 4 H, continuous). Then H is, more or less by definition, a nuclear space. The vectors u, all lie in H and the set of all rational-linear combinations of these vectors produces a countable dense subspace of H . Consider a linear functional on H which is continuous. Then it must be continuous with respect t o some norm I .. , 1 Thus the topological dual H' is the union of the duals Hh. In fact, we have:
H'=UpcwH, 2 . . . H ; IIH i 2 HA N_ Ho,
(34)
where in the last step we used the usual Hilbert space isomorphism between Ho and its dual HA. Going over t o the sequence space, H; corresponds t o
F-,
ef
{(Z,),~W
X~Z:
:
< m}
(35)
n EW
The element y E F-, corresponds to the linear functional on F, given by
which, by Cauchy-Schwarz, is well-defined and does define an element of the dual FL with norm equals to lyl-,. 2.4.
Gaussian measure in infinite dimensions
Consider now the product space R W , along with the coordinate projection maps
xj : RW
4
R :X
H Zj
White Noise Analysis: Background and a Recent Application
31
for each j E W . Equip Rw with the product a-algebra, i.e. the smallest a-algebra with respect to which each projection map X j is measurable. Kolmogorov's theorem on infinite products of probability measures provides a probability measure u on the product a-algebra such that each function X j , viewed as a random variable, has standard Gaussian distribution. Thus, 3
L W
d v = e-"12
eitx'
for t E R, and every j E W . The measure v is the product of the standard Gaussian measure e - z 2 / 2 ( 2 n ) - 1 / 2 d x on each component IR of the product space Rw. Since, for any p 2 1,we have
it follows that V(F-~) = 1 for all p >_ 1. Thus v(F') = 1. We can, therefore, transfer the measure I/ back to H', obtaining a probability measure p on the a-algebra of subsets of H' generated by the maps
fij : H'
4
R :f
H
f(uj),
where {uj}jEw is the orthonormal basis of HO we started with (note that each uj lies in H = n,>,,H,). This is clearly the a-algebra generated by the weak topology on H' (which happens to be equal also to the a-algebras generated by the strong/inductive-limit topology6). Specialized to the example Ho = L2(IR),and B = (-& ;)-', we have the standard Gaussian measure on the distribution space S'(R). The above discussion gives a simple direct description of the measure p. Its existence is also obtainable by applying the well-known Minlos theorem. There is also the useful standard setting of Abstract Wiener Spaces for Gaussian measures introduced by L. Gross (see the account in Kuo7). To summarize, we can state the starting point of much of infinitedimensional distribution theory (white noise analysis): Given a real, separable Hilbert space Ho and a positive Hilbert-Schmidt operator B on Ho, we have constructed a nuclear space H and a unique probability measure p on the Bore1 a-algebra of the dual H' such that there is a linear map
+5+
Ho
L 2 ( H ' , p ) : 2 ++ 2 ,
satisfying
cite dp = e-t2~d,2/2
32
J . Becnel and A . N . Sengupta
for every real t and 2 E Ho. This Gaussian measure p is often called the white noise measure and forms the background measure for white-noise distribution theory. 3. White Noise Distribution Theory
We can now develop the ideas of the preceding section further to construct a space of test functions over the dual space H’, where H is the nuclear space related to a real separable Hilbert space HO as in the discussion in subsection 2.3. We will use the notation, and in particular the spaces H p l from subsection 2.3. The symmetric Fock space F s ( V )over a Hilbert space V is the subspace of symmetric tensors in the completion of the tensor algebra T ( V )under the inner-product given by
c 00
(a,b ) T ( V ) =
n!(an1bn)ven1
(1)
n=O
where a = { ~ ~ } ~ 2=0{b,},>o , b - are elements of T ( V )with a n l b n in the tensor power V B n .Then we have def
Fs(H)= np>oFs(Hp)C ’ . . C Fs(H2) C Fs(H1) c Fs(H0). (2) Thus, the pair H c HO give rise to a corresponding pair by taking symmetric Fock spaces:
F s ( H ) c FS(H0)
(3)
3.1. Wiener-It6 isomorphism In infinite dimensions the role of Lebesgue measure is played by Gaussian measure p. There is a standard unitary isomorphism, the Wiener-It6 isomorphism or wave-particle duality map, which identifies the complexified Fock space Fs(Ho)cwith L2(H’,p ) . This is uniquely specified by
I : Fs((Ho)c 4 L2(H’,p ) : Exp(z) H
~
1
ex-Tx
2
(4)
where z E H , x 2 = Izlg, and Exp(z) =
1 C -xBn. n!
n=O
Indeed, it is readily checked that I preserves inner-products (the innerproduct is as described in (1)).Using I, for each F s ( H p )with p 2 0, we
White Noise Analysis: Background and a Recent Application
33
have the corresponding space [HI, c L2(H’,p ) with the norm 11. 1, induced by the norm on the space F . ( H p ) c .From this the chain of spaces (2) can be transferred into a chain of function spaces:
[HI =
n
[HI, C ... C [HI2 C [HI1 C [HI0 = L 2 ( H ’ , p ) .
(5)
P>O
Observe that [HI is a nuclear space with topology induced by the norms {/I . 1, ; p = 0 , 1 , 2 , . . . }. Thus, starting with the pair H c HOone obtains a corresponding pair [HI C L2(H’, p ) . As before, the identification of HA with HO leads to a complete chain
H
H,
=
c . . . c H I c Ho N H-o c H P 1 c ... c
P20
u P
H-, = H’.
(6)
a
In the same way we have a chain for the ‘second quantized’ spaces F s ( H q ) cs1 [HIq. The unitary isomorphism I extends t o unitary isomorphisms
I : F s ( H - p ) c4
kf [HI; c [HI’,
(7)
for all p 2 0. In more detail, for a E F s ( H - p ) c the distribution I ( a ) is specified by
( I ( a ) ,(6) = (a,I-Y(6)),
(8)
for all (6 E [HI. On the right side here we have the pairing of F3(H-p)c and F s ( H P ) cinduced by the duality pairing of H-, and H p ; in particular, the pairings above are complex bilinear (not sesquilinear). 3.2. Properties of test functions
The following theorem summarizes the properties of [HI we need. The results here are standard (see, for instance, the monograph8 by Kuo), and we compile them here for ease of reference. Theorem 3.1. Every function in [HI is p-almost-everywhere equal to a
unique continuous function o n H’ . Moreover, working with these continuous versions, ( a ) [HI is a n algebra under pointwise operations; (b) pointwise addition and multiplication are continuous operations [HI x [HI [HI; +
34
J. Becnel and A . N . Sengupta
(c) for any
4 E H',
the evaluation map 64 : [HI 4 R : F
H
F(4)
is continuous; 1
2
( d ) the exponentials ei'-HlX'o, with x running over H , span a dense subspace of [HI '
A complete characterization of the space [HI was obtained by Y. J. Lee (see the account in Kuo [8, page 891). 3.3. The Segal-Bargmann transform The Segal-Bargmann transform takes a function F E L 2 ( H ' , p ) to the function S F on the complexified space H , given by
S F ( z )=
s,. +
e"z2/2Fdp,
with notation as follows: if z = a
Z(x)
def
=
z E H,
(9)
ib, with a , b E 3-t then
def
zz = ( % , a )+ i ( x , b ) ,
for x E H'
(10)
and z 2 = z z , where the product z u is specified through
zu if z
=a
def
=
( a ,s)
-
(b,t )
+ i ( ( a ,t ) + (b,s ) )
(11)
+ ib and u = s + it, where a , b, s , t E 'FI.
Let pc be the Gaussian measure HL specified by the requirement that
for every a , b E H . For convenience, let us introduce the renormalized exponential function c , = e8-w2/2 E L 2 ( H ' , p ) for all w E H,. It is readily checked that for any 20 E H ,
[Scw](z= ) ewZ,
for all z E H,.
(13)
Thus we may take S c , as a function on HL given by S c , = e8 where now 15 is a function on HL in the natural way. Then Sc, E L2(Hb,pc) and one has -
(Sew, s c u ) L 2 ( p c )= (cw,C u ) L 2 ( p c ) = ewu. This shows that S provides an isometry from the linear span of the exponentials c, in L 2 ( H ' ,p ) onto the linear span of the complex exponentials
White Noise Analysis: Background and a Recent Application
35
’e in L2(HL,p,). Passing to the closure one obtains the Segal-Bargmann unitary isomorphism
s : LZ(H’, p)
-+
HoP(H:, p,)
where Ho12(H,!,pc)is the closed linear span of the complex exponential functions ’e in L 2 ( ~ p; c, ) . An explicit expression for S F ( z ) is suggested by (9). For any F E [HI and z E HL, we have (SF)(z) = ( W X P ( Z ) ) 7 F)
(14)
where the right side is the evaluation of the distribution I(Exp(z)) on the test function F . Indeed it may be readily checked that if S F ( z ) is defined in this way then [Scw](z)= e w z . In view of (14), it natural to extend the Segal-Bargmann transform to distributions: for @ E [HI’, define S@to be the function on H , given by
ef(@,I(Exp(z)))
S@(z)
(15)
3.3.1. The S-transform over subspaces Recall from the construction of the measure p we have a p-almost everywhere defined Gaussian random variable 2 with mean 0 and variance 1x1;. For a subspace V of Ho, let ISV be the completed a-algebra generated by the functions .ir : H‘ -+ R as v runs over V . Note that the linear span of {ei’ : v E V } is dense in L2(H’,plav).For orthogonal subspaces V and V’ we have that for any v E V and u E V’ the random variables 2, i are independent. Moreover, there is an unitary isomorphism from L2(H’,plav)@ L2(H’,p1avl) onto L 2 ( H ’ , p ) given by f @ g + f g . This leads t o the following proposition about the S-tran~form.~
Prop 3.1. Let V be a subspace of Ho and V’ be the orthogonal complement. For 4 E L 2 ( H ’ , p l u v )and 1c, E L 2 ( H ’ , p l n v i ) we have S(41c,)= S(4)S(+). 4. Application to Quantum Computing Quantum Computation is the study of information processing tasks that can be carried out on a quantum mechanical system. The setting for a quantum mechanical system in continuous variables is a rigged Halbert space or Gel’fand triple. Such a triple has been the subject of discussion throughout much of this paper. It is a triple ( H ,H o ,H’) such that HOis a Hilbert space,
36
J . Becnel and A . N . Sengupta
H is a nuclear space imbedded in Ho and H‘ is the dual of H (in finite dimensions HO is simply C n ) . The most common triple is the one developed L2(R),S’(R)). throughout the beginning sections of this paper (S(R), In the following discussion we shall use some of the informal techniques and ideas prevalent in the physics literature. A (pure) state of a quantum mechanical system with one coordinate (in R) is represented in Dirac notation as Iz), with z E R.Typically, in the physics literature it is said that the rigged Hilbert space (S(R),L2(R),S’(R)) has orthonomnal basis { Iz); z E R} where by orthonorma1 it is meant that
M Y ) = S(z - Y) A function f in L2(R) or S’(R) can then be expressed as a formal integral
(In this sense, Iy) can be thought of as the delta function at y , S(z For a particular z o , we can define
-
y).)
which yields: (zolyo) = q.0
-
Yo).
The rigged Hilbert space described above is often denoted by
HR. We can develop analogous concepts for the rigged Hilbert space (S(Rn), L 2 ( R n )S’(Rn)). , We will denote this space by H p . Of course this can be generalized to any such rigged Hilbert space ( H ,H o ,H’). For such a space the “orthonormal basis” would be { lz); z E Ho} where we have the bracket product defined as (zly) = 6(z - y). We will use the notation ‘H for the rigged Hilbert space ( H ,Ho, H’).
4.1. Quantum algorithms Of the algorithms that have been developed for the hypothetical quantum computer, the best known is Shor’s Order Finding Algorithm. For a fixed integer N, the algorithm finds the order of an integer a with respect to N. That is, the output is an integer p such that u p = 1 mod N. This algorithm leads to an efficient algorithm for factoring an integer. As with all great ideas, Shor’s algorithm has been generalized to other settings. One such generalization is a continuous variable algorithm which
W h i t e Noise Analysis: Background and a Recent Application 37
given a periodic function f : R -+ C outputs the period o f f . This algorithm was developed by Lomonaco and Kauffman.” (Although the algorithm is not presented in a rigorous manner, it can be made rigorous through ordinary distribution theory.) More generally, if 4 is a function on a vector space E , the task is to determine the hidden subspace H of all ‘periods’ of #, i.e. all vectors h E E for which $(z h) = ~ ( I c for ) all IC E E. Lomonaco and Kauffmanll
+
developed an algorithm for finding the hidden subspace in such a setting, with an infinite-dimensional vector space V . The steps are done in the spirit of the Shor algorithm; however, as the authors admit, the algorithm is “highly speculative”. Below we present the algorithm first in this formal version and later we will make use of white noise analysis to present a rigorous account. To perform the algorithm we need to assume, as is the case for the ‘traditional’ hidden subgroup algorithm, that there exists a black box which can evaluate our function 4. Since 4 is arbitrary we cannot “hardwire” in the computation to perform 4, instead we must assume we can perform the operation in the setting of quantum computation. This black box is denoted by UdP
Remark 4.1. The algorithm also relies heavily on the following identity:
A rigorous formulation and proof of this is given in our work Ref. 12. Algorithm: Hidden Subspace Inputs: The elements (0) E ‘FI and (0) E H R ~a;black box U+t,which has C )I~z)ZIz) ~ ( I c ) ) ; the effect U + ~ , ~ I =
+
Output: A vector u E VL Procedure: 1. Start in the initial state: l0)lO) E 3-1 @ H p 2 . Apply the Fourier transform in 3-1, F-’@I to create superposition:
3. Apply the black box U+:
38
J . Becnel and A . N . Sengupta
4. Apply 318 I:
u
using the decomposition HO =
+
(u V')
WEV
by a change of variables
since $(x
where \ O ( U ) )=
svLDx
e-2ni(2iu)
+ u) = $(x)
I4(.>).
5. Measure the first register with respect to the observable
to produce a random vector u E
V1
Remark 4.2. In the original algorithm by Lomonaco and Kauffman," HO was taken to be the space Paths of all functions (paths) z : [0,1] -+ R" 1 which are L2 with respect to the inner-product z . y = z ( t ) y ( t d) t and the measure Dz is a formal 'Lebesgue measure'.
so
4.2. Hidden subspace algorithm
Here we use the concepts developed in white noise analysis to present a mathematically rigorous formulation of the hidden subspace algorithm. Suppose we have a functional $ : Ho -+ R" with a hidden closed subspace V c HO such that
$(x
+ u)= $(z)
for all w E V
White Noise Analysis: Background and a Recent Application
39
We would like t o determine V t o some extent. First we can extend q5 to the domain H O , ~as, follows: For z = z iy with z, y E Ho, we define
+
+ 4(Y>
q5(z) =
+
And of course we have $ ( z w) = $ ( z ) for all w E V,. In the original algorithm we used two rigged Hilbert spaces. The first we denoted by ‘H. In our version the space 3-1 becomes the white noise space ( [ H ] , L 2 ( H ’ , p )[HI’). , The original algorithm also made use of the rigged Hilbert space HRn. We now define HRn to be the space of all complex functions f on Wn such that f # 0 at only a countable number of points and CxERn lf(z)12< co.Equip HRn with the inner-product
This makes HRn a non-separable Hilbert space. For HRn we have an orthonormal basis given by Iz) = l{x}
(zly) = 6,,
with
We will again make use of a black box
U, : L2(H’,p ) 8 HRn
4
L2(H’,p) @ HRn
which performs the operation
U4@@ 12) = @ 8 I2
+U l ) )
where we use Wiener-It6 isomorphism to get I-’(@) = with each f n E H t F . We now outline each step of the algorithm in detail:
(fn)rE F,(Ho),
Step 1. In the original version we begin in the state l0)lO) E 7-1 @ HRn. Here we take a unit vector Q, in L2(H’,p) and begin in the state I$O)
= @ 10)
of L 2 ( H ‘ , p ) @ Hwn. For now the only restriction we place on @ is that for any subspace W of Ho we can decompose CP into Q, = @w@Wiwhere @W E L 2 ( H ’ , p l a w ) and Q,wi E L 2 ( H ’ , p l a W ~ )Then, . in particular, @ can be decomposed into Q, = Q , v @ ~where I, V is the hidden subspace of the functional d.
40 J . Becnel and A . N . Sengupta
Step 2. This step is the one that is altered the least from the original algorithm. Here we apply the black box U+ t o 1740) in order t o arrive at 1741) = U$l740)
= @ I4(fl))
where f l E H o , is ~ the element obtained through the Wiener-It6 isomorphism as previously described.
Step 3. In place of the Fourier transform in the original algorithm we use the S-transform (actually S 8 I) to get 1742) =
(S 8 I)1741)= S(@)l$(fl)).
Now we use that
to arrive a t
where Pvl is the orthogonal projection onto We can then write 1742) as 1742) =
VL.
S(@V)Ifl(PVlfl))
where I f l ( P V l f 1 ) ) = S ( @ V l ) I 4 ( P V l f l ) ) .
Step 4. Apply S-' 8 I to get 1743)
= (S-l 8 I)1"2)
= S-lS(@~)Ifl(P~lfi)) = @vIfl(Pvlf1)).
Step 5. We now need to measure @V in some way to obtain information about the subspace V . Up until this point we let @ be an arbitrary unit vector in L2(H',p). However, in order to complete the algorithm, we need t o be a bit more specific about our choice of initial state. We now take a random non-zero vector x E Ho and let @ be given by @ = e2-lzli/2, With this choice of @ it is easy to see that @ v and QVl work out to be QV
= ,+-lXVli/2
and
QVl
Recall that e2v-1zv1i/2 -+Exp(xv) = by the Wiener-It6 isomorphism. For n = 0,1,2,. onto the Hf" term.
= ekvl
-lxvl
li/2
$ x v where x v
. . , let P,
E Hf"
be the projection
White Noise Analysis: Background and a Recent Application 41 Therefore measuring with respect t o {Pn},",o produces i x v for some n. &om this we can extract xv E V. Thus we have a hidden subspace , runs through the hidden subspace V. And, of for 4, namely { R x ~ } which course, running the procedure k times will produce a k-dimensional hidden subspace (provided t h e dimension of the hidden subspace is greater t h a n equal t o k).
Acknowledgment Research supported by US NSF grants DMS-0201683 and DMS-0601141
References 1. 2. 3. 4. 5. 6. 7.
8. 9. 10. 11.
12.
C. Hermite, Comptes rendus d e 1'Academie des Sciences 14,93 (1864). B. C. Hall, The heat operator in infinite dimensions, In this volume, (2007). H.-H. Kuo, Introduction to Stochastic Integration (Springer, 2006). B. Simon, J . Mathematical Phys. 12 (1971). J. J. Becnel and A. N. Sengupta, The Schwartz space: A background to White Noise Analysis, http://www.math.lsu.edu/ sengupta/as20041.pdf. J. J. Becnel, Proceeding of the A M S 134,58l(February 2006). H.-H. Kuo, Gaussian measures on Banach spaces (Lecture Notes in Mathematics, Vol. 463, Springer-Verlag, Berlin, New York, 1975). H.-H. Kuo, White Noise Distribution Theory (CRC Press Inc, New York, New York, 1996). S. Albeverio, B. C. Hall and A. N. Sengupta, Infinite Dimensional Analysis, Quantum Probability and Related Topics 2, 27 (1999). S. J. Lomonaco and L. H. Kauffman, A continuous variable shor algorithm, http://xxx.lanl.gov/abs/quant-ph/O210141, 12 pages(October, 2002). S. J. Lomonaco and L. H. Kauffman, Continuous quantum hidden subgroup algorithms, in Proceedings of SPIE, Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium, (Washington, D.C., 2003). J. J. Becnel and A. N. Sengupta, Proceedings of the American Mathematical Society 9, 2995 (2007).
42
PROBABILITY MEASURES WITH SUB-ADDITIVE PRINCIPAL SZEGO-JACOBI PARAMETERS Aurel I. Stan
Department of Mathematics Ohio State University at Marion 1465 Mount Vernon Avenue Marion, OH 43302, U.S.A. E-mail: stan.
[email protected] It is known that for any normally distributed random variable X,any Bore1 measurable functions f and g, and any p and q positive numbers, such that l/p l / q = 1, if r(JjTZ)f(X) and r(&Z)g(X) are both square integrable, o g(X), of the random variables f (X)and g(X), then the Wick product f (X) is also square integrable, and the following inequality holds:
+
E[lf(X) 0 dX)l215 ~ ~ l ~ ( J j s ~ ) f ( X ) 1 2 1 E ~ I ~ ( ~ ~ ~ ~ ~ ~ where E denotes the expectation, Z the identity operator, and r(rZ) the second quantization operator of rZ, for any complex number r. We extend this inequality t o all random variables X,having finite moments of any order, whose SzegGJacobi omega parameters are sub-additive. Keywords: Szego-Jacobi parameters, Schwarz inequality
1. Introduction
It was proven in' that if X is a normally distributed random variable, f and g are measurable functions, and p and q are positive real numbers, such that (l/p) ( l / q ) = 1, and I?(@) f and r ( 4 I ) g are both square-integrable, then the Wick product fog is square integrable, and the following inequality holds:
+
E[If ( X )O d X )
121 I Ell %mf( X ) l21E[lW
m ( X )l21,
(1)
where I denotes the identity operator, and I?(cI) the second quantization operator of c I , for any complex number c. This inequality can be viewed as a Holder inequality for Gaussian Hilbert Spaces. The main result of this paper is the following. If X is a random variable, having finite moments of any order, whose Szego-Jacobi w-sequence
Probability Measures with Sub-Additive Szego-Jacobi Parameters
is sub-additive, i.e., wp+q 5 wp the inequality (1) holds.
43
+ wq, for all p and q natural numbers, then
In section 2 we provide a short background of the Szego-Jacobi parameters and in section 3 we define the Wick product. In section 4 we prove the main result announced above.
2. Background Let X be a random variable having finite moments of any order. This means E[lXlP]< 03, for all p > 0, where E denotes the expectation. Since X has finite moments of all orders, the random variables 1, X , X 2 , . . . are square integrable and thus, we can apply the Gram-Schmidt orthogonalization procedure to construct a new sequence of orthogonal polynomial random variables: f o ( X ) , f I ( X ) , f 2 ( X ) , . . . , where, for all n 2 0, if f n ( X ) is not almost surely equally to zero, then f n has degree n and is chosen in such a way that its leading coefficient is equal to 1. If p is the probability distribution of X I then we distinguish between two cases: 0
If p has a finite support of cardinality k, then for all n 2 k, fn = 0 (the
0
null polynomial). If p has an infinite support, then for all n 2 0, f n # 0, and thus, f n is a polynomial of degree n with a leading coefficient equal t o 1.
It is well-known that for all n 2 0, there exist two real numbers an and w,, such that
X f n ( X ) = f n + l ( X ) + a n f n ( X )+ W n f n - l ( X ) .
(1)
If n = 0, then f n - l = f - 1 := 0 and we may define wo however\we want. We define wo := 0. If p has finite support of cardinality k, and n = k - 1, then f,+l = f k := 0. The numbers a,, 0 5 n < N , and w,, 1 5 n < N , where N = 03 or N = k, are called the Szego-Jacobi parameters of X (or p). The numbers w,, 1 5 n < N , are called the principal Szego-Jacobi parameters of X (or p ) . It is also known that the square of the L2-norm of f n ( X ) is:
for all 1 5 n < N . Of course, since fo = 1, E [ f o ( X ) 2 ]= 1. It follows from ( 2 ) that W n = E [ f n ( X ) 2 ] / E [ f n - 1 ( X ) 2for ] l all 15 n < N . Thus w, > 0, for all 1 5 n < N .
44 A . Stan
Let 'H := @ o s , < ~ C f , ( X ) ,where N denotes the cardinality of the support of p. It follows from (2) that if {C,)O<,
11 f(x)112
C
:=
ICn12wn! < 00,
(3)
O
where
11 . 11
denotes the L2-norm, w,!
:=
w1w2...wn, for n 2 1, and
wo! := 1.
3. Wick product
We define the Wick product fm(X) o f n ( X ) of two orthogonal polynomial random variables fm(X) and f n ( X ) as:
fm(X)o f n ( X ) := fm+n(X),
+
(1)
for all m, n 2 0, such that m n < N . Formula (1) holds, in the finite support of cardinality k case, even for m n 2 k, since in that case we can define fm+, := 0.
+
We can use now formula (1)t o extend the definition of the Wick product, in a bilinear way. Formally, if cp(X) = Cn
c
cp(W o $ ( X ) :=
bnfn(X),
(2)
n
where
C
b, :=
cud,,
(3)
u+v=n
for all 0
I n < N . Of course yo$
E
'H if and only if C05n
4. Random variables with sub-additive w-parameters
In this section we will prove a Holder type inequality for random variables, having an infinite support and finite moments of all orders, whose sequence {w,},2l is sub-additive. This inequality involves the second quantization operator of a constant times the identity operator, whose definition we present below.
Probability Measures with Sub-Additive Srego-Jacobi Parameters
45
Let X be a random variable having finite moments of all orders, whose orthogonal polynomial random variables, obtained by applying the GramSchmidt orthogonalization procedure to the sequence of polynomial random . . . , are fo(X) = 1, fl(X), f2(X), . . . . Each f , has the variables 1, X ,X 2 , leading coefficient equal to 1. Let {a,},>O and {wn},21 be the SzegoAs before, N denotes Jacobi parameters of X . Let H := EB~<~
Definition 4.1. For any complex number c , we define the second quantization operator of c I , where I denotes the identity operator of the onedimensional Hilbert space C X ,spanned by X ,as a densely defined operator on H ,defined by:
where d, E C ,for all 0 5 n
A random variable f (X):= C05n
Lemma 4.1. If {w,},~lis a sub-additive sequence of positive real n u m bers, i.e., urn+, I w, w,, for all m, n 2 1, then for all k 2 r 2 0 ,
+
Proof. We will prove inequality (2) by induction on k . If k = 0 or k = 1, then = (f) = 1, for all 0 5 T I k . Let us assume now that (2) holds for k = n and all r E ( 0 , 1, . . . , n } .We want to prove that it also holds for k = n 1 and all T E (0, 1, . . . , n l}. Indeed, if r = 0 or r = n 1, then = (":l) = 1. If 1 5 r I n, then since the sequence {wp}p21 is sub-additive, by using the induction
(z:)
+
+
('' zf')
+
46
A . Stan
hypothesis, we conclude that:
We are now ready to present our main result.
Theorem 4.1. Let X be a random variable having finite moments of any order, and let {wn},21 be its principal Szego-Jacobi parameters. If the sequence {un},21 is sub-additive, then for all p and q positive numbers such that ( l / p ) ( l / q ) = 1, and all f ( X ) E 'H and g ( X ) E 'H, such that r ( f i 1 )f ( X ) E 'H and r ( f i I ) g ( X ) E 'H, we have f ( X ) o g ( X )E 'H and the following inequality holds:
+
E[lf( X )0 g(X)l21 5 E [ l ~ ( f i I () Xf ) I Z l ~ [ l ~ ( ~ ~ ) ~ ( X > I 2 1 ~
+
Proof. Let p > 0 and q > 0 be such that ( l / p ) ( l / q ) = 1. Let f ( X ) E 'H and g ( X ) E 'H, such that I'(,,$I)f(X) E 'H and r ( f i I ) g ( X ) E 'H. We analyze two cases:
Case 1. If the probability distribution p of X has an infinite support, then wn > 0, for all n 2 1. There exist two unique sequences of complex numbers { c n } , 2 ~ and {dn},LO, such that:
n=O
and M
n=O
Probability Measures with Sub-Additive Szego-Jacobi Parameters
47
where { f n ( X ) } n 2 ~is the orthogonal sequence of polynomial random variables of X , such that, for all n 2 0, fn has degree equal to n and a leading coefficient of 1. Since r ( J i r I ) f ( X )= C r = o p n / 2 c n f n ( X )E 'H, we have:
Since I'(&I)g(X)
=
Cr=oqn/2dnfn(X)E 'H, we have:
We have:
Applying the triangle inequality, we get:
2
Wk!
m+n
r
Using now the Cauchy-Bunyakovsky-Schwarz inequality, inequality (2) (which holds since {wn},>l - is assumed to be sub-additive), and Newton's
48
A . Stan
binomial formula, we obtain:
Ell f ( X )O d X ) ?I 5 k=O g w
[c
k ! u+w=k
1
1 wu!w,!p"q"
r
1
J
Lu+w=k
k
k=O
1 1
r
1
Lu+v=k
J
Lu+w=k
J
00
k=O u+v=k
=
00
00
u=o
w=o
E [ IJ3Jirl)f ( X )I7E [ I % m g ( X )17.
Case 2. If p has a finite support of cardinality k , then w, > 0, for all k E { 1 , 2 , . . . , k - l } , and w, = 0, for all k 2 T . Any random variable of the form $ ( X ) is equal almost surely with a random variable + ( X ) , where
Probability Measures with Sub-Additive Szego-Jacobi Parameters
49
q!~ is a polynomial of degree at most k
- 1. Thus we may assume that f and g are polynomial functions of degree at most k - 1. For all E > 0, let Y, be the random variable whose Szego-Jacobi parameters are I& = 0, for all n 2 0, and w; = w, nE, for all n 2 1. Since w; > 0, for all n 2 1, we conclude that the probability distribution pe of Y, has infinite support. Because the sequence {w;},?~ is the sum of the sub-additive sequence we conclude that {w;},>l is sub{w,},>1 - and additive sequence {n~},>1, additive. Thus according to the first part of our proof we have:
+
Since f and g are polynomial functions, only finitely many of their coefficients c j and d j , j 2 0, are different from zero, and thus passing to the limit, as E 4 0, in inequality (3), we conclude that
E[If(X)0 g(X)Il 5 ~ [ l ~ ( . J i T ~ ) f ( X ) l 2 l ~ [ l ~ ( ~ ~ ) S ( X ) l 2 1 0
Observe, that in the proof of Theorem 4.1 the fact that the sequence was sub-additive was needed to conclude that condition (2) holds. So, this theorem holds also under the weaker assumption that the sequence { w ~ } ~ satisfies > I the condition (2). We will show now that the condition (2) does not imply sub-additivity. {w,},>1 -
Counterexample: Let w1 := 1, w2 := 3/2, and
for all n
2 3.
Claim 1: For all n
5 )(:
2 k 2 0,);:(
Indeed, for n = 1, there is nothing to prove since: = 1.
(i)
For n = 2, ("') WO W ~ / W I= 3/2 < 2 =
=
=
(z:)
=
(,)1
=
(i) = 1 and (zz) = w ~ ! / ( w ~ ! w =I ! )
(zt)
=
(,)2
(",)
=
(z;) = )(:
=
(zt)
(;).
For n 2 3, we have
=
(z)
=
1, and for all 1 5 k 5
50
A . Stan
n - 1:
Claim 2. The sequence {wn},21 is not sub-additive.
+
To see this we will show that w3 > w1 w2. Let us compute first the value of w3. Using the recursive formula (4), for n = 3, we obtain:
= min{3,3}
= 3.
Thus
W I + ~2
=1
+ (3/2) = 5/2 < 3 =
~
3
.
Since { w , } , ~ l is a sequence of positive terms, by taking, for example, a, := 0, for all n 2 0, we know that there exists a random variable X , having finite moments of any order, whose Szego-Jacobi sequences are exactly {an}n>o and {Wn}n>l. This counterexample shows that inequality (1) does not imply that the sequence {wn),2l is sub-additive, since it holds for sequences {wn),>l which satisfy the weaker condition (2). We do not know whether (1) implies (2). The following examples illustrate the main result of this paper.
Example 1: If the principal Szego-Jacobi parameters of a random variable X , having finite moments of any order, are w, = (?inti, for all n 2 1, where I is a fixed natural number, and C1, C2, . . . , C I , t l , t 2 , . .. , t I are fixed real numbers, such that Ci > 0 and ti 5 1, for all 1 5 i 5 I , then since
xi=,
Probability Measures with Sub-Additive Szego-Jacobi Parameters
+
51
+
( m n)t 5 mt nt, for all t 5 1, and m, n 2 1, we conclude that { ~ ~ } ~ 2 l is sub-additive. Hence the inequality ( 1 ) holds for X . The Gaussian and Poisson random variables are in this category (I = 1 and t = 1 ) . Example 2: Since the principal Szego-Jacobi parameters of each of the following classic random variables: uniformly distributed over an interval, arcsine, and semi-circle are non-increasing, they are trivially sub-additive, and thus these random variables are enjoying inequality ( 1 ) . See,2,3and4 for explicit formulas of the principal Szego-Jacobi parameters of these random variables. Example 3: Let us consider the q-Gaussian random variable X,, where -1 < q 5 1. That means X, is the symmetric random variable whose principal Szego-Jacobi parameters are w, = 1 q q2 . . q"-' = (1 - q " ) / ( l - q ) , for all n 2 1. The sequence is non-decreasing if and only if q 2 0.
+ + +
+
Claim: For all q E (-1, 11, the sequence { ~ ~ }is~sub-additive. 2 l Indeed, for q = 1, w, = n, for all n 2 1 , and thus the sequence additive. In fact, X I is the standard Gaussian. On the other hand, for q E (-1, l ) ,we have: Wm+n
-W m
-Wn
1 - qm+"
1-q"
1-q
1-q
==
-
qm
_-
+ qn - qm+n
-
( ~ ~ }is ~ z l
1-q" 1-q
1
1-q ( 1 - q m ) P - 4") 1-q
< 0, for all m and n natural numbers, since 141 < 1. Thus {wn}">1 - is subadditive. Acknowledgement: It is a great honor for me to dedicate this paper t o Professor Kuo. Professor Kuo was for me an excellent Ph.D. adviser and has continuously influenced and encouraged me in my mathematical research and career. I would like to express my deep gratitude to Professor Kuo. I would also like to thank Professors Sengupta and Sundar, for including me among the people who were invited to speak during the session that they organized in honor of Professor Kuo, at the Joint A.M.S. Meetings in New Orleans, on January 7, 2007.
52
A . Stan
References 1. H.-H. Kuo, K. Sait6, and A.I. Stan, A Hausdorff-Young Inequality for White Noise Analysis, in Hida, T., Sait8, K. (eds.), Quantum Information IV, (World
Scientific Publishing Co., 2002). 2. N. Asai, I. Kubo, and H.-H. Kuo, Multiplicative renormalization and generating functions I, Taiwanese J. Math. 7,(2003). 3. N. Asai, I. Kubo, and H.-H. Kuo, Multiplicative renormalization and generating functions 11, Taiwanese J. Math. 8 , (2004). 4. N. Asai, I. Kubo, and H.-H. Kuo, Generating functions of orthogonal polynomials and SzegGJacobi parameters, Prob. Math. Stat. 23,(2003).
53
DONSKER’S FUNCTIONAL CALCULUS AND RELATED QUESTIONS Pao-Liu Chow
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA E-mail:
[email protected] Jiirgen Potthoff
Fakultat fur Mathematik und Informatik, Universitat Mannheim, 0-68131 Mannheim, Germany E-mail:
[email protected] The present paper is concerned with Donsker’s functional calculus in his derivation of the Feynman-Kac formula. In attempting t o validate his calculations in a white noise analysis setting, several related mathematical questions arise and are investigated. Based on the results thus obtained, we introduce an alternative approach via the S-transform in which all computational steps can be justified.
Keywords: Functional calculus, White noise analysis, Partial differential equation, Feynman-Kac formula.
1. I n t r o d u c t i o n In a paper which appeared in 1962,l M. Donsker showed a certain connection between functional integrals and functional differential calculus in the classical Wiener space. In particular he considered the initial-value problem for the heat equation with a “potential” V :
d dt
1 a2 u ( z , t ) V(z) u(z,t ) , 2 dx2
+
- u ( z , t )= --
4x7
(1)
0) = S(z),
u ( z , t )= 0,
t > 0,5 E R,
as z
-+ f
m,
where V(z) is assumed to be bounded and continuous on R,and 6(z) is the Dirac-delta function. Let v denote the Wiener measure on the space Co
54
P.-L. Chow and J . Potthoff
of continuous functions p ( t ) , 0 5 t < co with p ( 0 ) = 0. By the well-known Feynman-Kac formulaI2 the solution of (1) can be expressed as a Wiener integral:
4x74 =
11 w
co
1 t
d ( P ( t ) - z) exp{
=E P ( W
V ( p ( s ) ) d s }d v
0
-4
(2)
t
exp{J’ 0 V ( p ( s ) ) d s ) ) ,
where Ep denotes the expectation with respect to the Wiener measure for the process p ( . ) and d(pt - x) is now known as the Donsker’s delta function (see, e.g., Ref. 3 and several other references to follow). Instead of Eq. (2), Donsker parameterized Eq. (1) by introducing an imaginary drift term with a white-noise q(t)
a
1 a2 -u(t,x) 2 ax2
- u(t,z) = -
at
a u ( t ,z), + V(z) u(t,x) + i q ( t ) 0
ax
(3)
u(x, 0) = d b ) , where q ( t ) is another Brownian motion independent of p ( t ) . Let F E L 1 ( v ) . Define the Fourier-Wiener transform
JqQ) = Ep{F(P + 4 ) = G(4),
(4)
and the inverse Fourier-Wiener transform
G) = E,{G(q
-
iP)) = F ( P ) .
(5)
As is shown in Section 2, after applying the Fourier-Wiener transform to the parametric equation ( 3 ) , the transformed problem can be solved easily. Then the solution u ( t , z ; p ) can be obtained by an inverse transform. As a special case, u(t,z; 0) yields the Feynman-Kac formula. The informal calculations involved will be called Donsker ’s functional calculus or simply Donsker’s calculus, and for the problem just described we give a detailed account of it in Section 2. Though informal, the calculations do lead t o the correct answer. The natural question arises whether they can be justified in some sense, and this question is the main motivation of the present paper. In an attempt to find a mathematically rigorous version of Donsker’s calculus, we shall consider the problem in the framework of white noise analysis or the theory of white noise distributions developed by Hida,4 Kuo5 and many other authors. In the process we need to overcome three main technical problems, namely the extension of Fourier-Wiener transforms to a space of white-noise distributions, the measurability and independence of generalized stochastic processes and the It8 formula for generalized functionals
Donsker’s Functional Calculus 55
of Brownian motion. These problems will be studied in the subsequent sections. In Section 3 we will give an overall review of white noise analysis, such as chaos decomposition, S-transform, spaces of white noise distributions as well as the characterization theorem. The key issue of Fourier-Wiener transforms is treated in Section 4. It will be shown in Theorem 4.1 that such a transform can be defined as a bijection from the space R of test functionals into itself. However we have not been able to extend the Fourier-Wiener transforms to the space R* of generalized functionals, which is the main obstacle to justifying Donsker’s calculus rigorously. In Section 5 we take up some questions regarding generalized stochastic processes. In particular the questions of measurability and independence are addressed, and It8’s formulas for some generalized Brownian functionals will be provided. As an alternative approach given in Section 6, we modify Donsker’s parametric Eq. (3)’ where the imaginary white noise i G(t)is replaced by a smooth function h ( t ) (see Eq. (1)). By the S-transform method, similar to Donsker’s derivation of the Feynman-Kac formula, the calculations can be justified by the results obtained in Section 5. The chaos expansion of Donsker’s delta function is reviewed in Appendix A. The somewhat technical proofs of Theorems 5.1, 6.1, and 6.2 are deferred to Appendices B and C, resp. 2. Donsker’s Calculus The informal calculations presented in this section are partly based on the original lectures of M. Donsker from 1971.6 Consider the initial value problem for the stochastic PDE (partial differential equation):
av at
- 1 d2v
2 8x2
+ iq(t)
0
dV -
dX’
t > 0,x E R,
which should be interpreted as the stochastic differential equation:
v(0,). = q.), and o denotes the Stratonovich product convention. By the It8 formula, it is easy to check that the solution is given by
4 4 x; 4 ) = 9 (6 x +
m) I
(3)
56
P.-L. Chow and J . Potthoff
where g is the standard one dimensional heat kernel
The solution w given by (3) is the fundamental solution of the following initial-value problem
au
1 d2U -- --
+ V ( x )u + i q(t) 2 ax2
at
au
0
-
ax’
(5)
4 2 , O ) = b(z),
where V is a bounded continuous function on R. From the fundamental solution of (5) we find the associated Green’s function given by
S ( t , .( - Y) for t
+ i ( q ( t )- q ( 4 ) )
> s, in terms of which Eq. (5) can be written as
u(t,x;Q) = g ( t , 2 + i q ( t ) )
+
i”
J’g
(t
-
s, .(
- Y)
+ i ( q ( t ) - 4.)))V(Y) 4%Y;4 ) dY d s , (6)
where 6, ( p ( t ) ) = b ( p ( t ) - x) is Donsker’s delta function which is known to be generalized Brownian functional or Hida distribution. Assume that
Hypothesis 2.1. The Fourier- Wiener transforms can be generalized to the space of white-noise distribution^.^ Then we can calculate the inverse Fourier-Wiener transform of Donsker’s delta function
-
6, ( q ( t ) ) = g(t’ 32 + W ) ,)
(7)
which implies the Fourier-Wiener transform
G(t’x;d t ) ) = d x
(m).
(8)
By taking the Fourier-Wiener transform of Eq. (6) and making use of (8) and the independent-increment property of a Wiener process, we get
.^(t, P ) = 6, (&)>
+ J’0
t
J’ hX-Y ( d t )
- P b ) ) V(Y) .^(%Y; P )
ds.
(9)
We note that, in order for the inverse Fourier-Wiener transform of equation (9) to yield (6), we have assumed that
Donsker ’s Functional Calculus
57
Hypothesis 2.2. The generalized stochastic process S,-,(p(t) - p ( s ) ) is independent of any Fs-adapted Brownian functional @ ( s ; p )in a suitable sense. Furthermore we suppose that
Hypothesis 2.3. The following It6 formula holds for a generalized functional of Brownian motion
a
d 6, ( P ( t ) ) = -a x 6, ( P W )O dP(t).
(10)
Then, by using the It6 formula and exchanging the order of differentiation and integration, we obtain from (9) that
or
B ( t ,x;p)= 6(x). By the method of characteristic^,^ the above initial-valued problem can be readily solved to give t
G(t, x;PI
= 6, ( P W ) exp
(/V ( P ( S ) )d s ) .
(12)
0
The fact that it is a solution can be easily verified by the It6 formula. We apply the inverse Fourier-Wiener transform t o B, and find U ( t , x;4 ) = E p
{ 6,( d t )- i q ( t ) ) exp
where we set
6, ( p ( t ) - i q ( t ) ) =
1
/
w
(I
t
V ( P ( 4- i q ( s ) ) d.)},
exp{iA(p(t) - i q ( t ) -
(13)
1
d ~ .
In particular, by setting q = 0, we get the Feynman-Kac formula (2) as expected. The above calculations reveal that, t o justify them, it is necessary t o deal with the questions raised in the Hypotheses 2.1-2.3. These issues will be treated in the subsequent sections within the framework of white noise analysis.
58
P.-L. Chow and J . Potthoff
3. Tools from White Noise Analysis and Malliavin Calclus In this section we set up our framework, and collect some tools from white noise analysis and Malliavin calculus that we shall use in the sequel. Since we are only dealing with functionals of Brownian motion, it is natural to consider R+ = [ O , + c o ) as the underlying time parameter domain, while in the standard white noise framework (e.g., Refs. 5,8,9) it is taken t o be R. Therefore we choose the white noise analysis framework as presented in Refs. 10-12. Let (0,A, P ) be a complete probability space on which a standard Brownian motion B = ( B t ,t 0) is defined. The expectation with respect to P will be denoted by E ( . ). Throughout this paper we make the following assumption:
>
Hypothesis 3.1. The P-completion of a(Bt, t 2 0) is equal to A. 3.1. Chaos Decomposition Let C 2 ( P ) denote the pre-Hilbert space of square integrable real-valued random variables on (0,A, P ) , and let L 2 ( P ) be the corresponding Hilbert space, consisting of P-equivalence classes of random variables in C2( P ) . Loosely speaking, Hypothesis 2.1 says that every random variable on (0,A, P ) is a “functional of Brownian motion”, and indeed it is not hard to show” that we have for L 2 ( P ) the usual chaos decomposition: Every 2 E L 2 ( P ) is in one-to-one correspondence with a sequence (fn, n E No), where fn E L2(Rn+)is (Lebesgue-a.e.) symmetric, and
n=O
with 00
n=O
where In(fn),n E N,denotes the standard n-fold Wiener-It6 integral of f n , and I O ( f 0 ) = fo E R. 3.2. S-Transform Let C r ( R + )denote the space of Cw functions on R+ which have compact support. For h E CT(R+) set 00
Xh =
h ( s )d B ( s ) .
(2)
Donsker's Functional Calculus
59
Clearly, ( X h , h E Cr(IR+))is a centered Gaussian family with covariance given by the inner product of L2(R+) (with Lebesgue measure A). Therefore, the mapping X : C?(R+) -+ L 2 ( P ) which , maps h E CF into X h has a unique continuous extension to L2(R+) as an isomorphism from L2(R+) into L 2 ( P ) ,and this extension will be denoted by X ,too. Let h E Cy(R+),and consider the stochastic integral equation
€t(h)= 1
+
t
h ( s )& ( h )dB,,
whose solution is the well-known martingale (&,(h),t
(3)
2 0) given by
Et(h) = exp Xht - -
(
(4)
where we have set ht = l p t ] h , and I . 12 is the norm of L2(R+).Note that the (P-a.s.) limit of this martingale as t -+ +m,
& ( h )= exp X h (5) ( 2 belongs to all LP(P),p 2 1. Moreover, Hypothesis 3.1 entails that the set & = {E(h), h E CF(IR+)}is total in L 2 ( P )(e.g., Ref. 10). For Z E L 2 ( P )define its S-transform S Z as the family of L2(P)-inner products of Z with E :
S Z ( h )= E ( Z & ( h ) ) ,
h E Cy(R+).
(6)
Note that S is normalized in the sense that S l ( h ) = 1, h E C r ( R + ) . Moreover, since & is total in L 2 ( P )it follows that S is an injective mapping from L 2 ( P )into the space of real valued functions on Cy(R+). The most elementary example is the S-transform of & ( g ) , g E L2(R+):
S&(g)(h)= e('lh),
h E C?(R+),
(7)
where ( 9 ,h) is the inner product of g and h in L2(R+). Among the remarkable properties of the S-transform we mention here only the following two. For 2 E L 2 ( P ) with chaos decomposition as in equation (l),and h E CT(R+)we have
Moreover, if then
Y = (Y,,t 2 0) is an adapted stochastic process in L 2 ( P @ A ) ,
60
P.-L. Chow and J . Potthoff
Here is a short proof of (9). Let h E CF(R+). Since the stochastic integral is measurable with respect to .Ft = a(B,, s 5 t ) ,we can compute as follows
where we used equation (3). Now we can apply the It6 isometry and get
=
I
t
(.Y,(h)) h ( s )d s ,
and in the second step we used that Y, is Fs-measurable. We conclude this subsection with the following remark. Assume that we have realized our setup on a suitable white noise probability space, i.e., informally & ( w ) = w ( t ) , w E 0, t 2 0. Then it follows from the translation formula for Gaussian integrals13 that S Z ( h ) = E ( Z ( .+ h ) ) ,and an analogous formula can be proved for a realization on Wiener space.
3.3. Smooth and Generalized Random Variables In this subsection we quickly review in the present framework the construction of spaces of smooth and generalized random variables. The first dual pair of such spaces we consider here is the analogue of the Hida spaces ( S ) ,( S * ) (e.g., Refs. 8,14) in the current setting where we deal with white noise analysis over the half line R+ as time parameter domain. We give a construction which is similar to the one introduced originally into white noise analysis by Kubo and Takenaka.14 For h E C r ( R + ) , consider the following differential operator A:
I d d Ah(t) = -- - t - h(t) 2 dt dt
+ (1 + -14 t ) h ( t ) ,
t E R+.
(10)
This operator has been studied extensively in Refs. 10,12. A has a unique self-adjoint extension in L2(IR+). Moreover, the Laguerre functions are
Donsker’s Functional Calculus
61
+
eigenfunctions of A , its spectrum is equal t o { n 3/2, n E N}, and each eigenvalue has multiplicity one. Denote by S,(R+), p E No = ( 0 ) U N, the domain of A,, and equip this space with the norm IAP . l2 so that it becomes a Hilbert space. Clearly, for q 2 p we have S,(R+) c S,(R+), and for q > p the associated injection is Hilbert-Schmidt. Denote by S(R+) the projective limit of the chain (S,(R+), p E NO), and call it Schwartz space on R+. Obviously, S(R+) is a nuclear space. It can be proved that h E S(R+) (has a representative which) is the restriction of a function in -the usual Schwartz space S(R) to R+. We denote by S’(R+) the dual of
S(R+)Denote the set of all 2 E L 2 ( P )with chaos decomposition (1) having CF(Rn+),n E N, and only finitely many non-zero terms by ’DO.Then for 2 E ’DO,p E NO,we can define fn
n=O
For every p E N the operator I’(A)P has a unique self-adjoint extension in L 2 ( P )with domain denoted by R,. The natural Hilbertian norm Ilr(A)”.112 of R, will be denoted by 11 . II2,,. Obviously, we have R, c R,, whenever q 2 p , and we denote by R the projective limit of the chain (R,, n E NO). It is a space of smooth random variables. It follows from the fact that the spectrum of A is strictly larger than 1 that the embedding of R,into R,, q > p , is Hilbert-Schmidt (e.g., Ref. 15, or Ref. 8, Chap. 3.B). Therefore, R is a nuclear space. For the present paper there are two properties of R which have to be mentioned: Firstly it is an algebra under pointwise multiplication (which is continuous) , and secondly the exponential random variables {exp(Xh), h E S(R+)} belong to R (in fact form a total subset). The dual R*of R is a space of generalized random variables, and upon identification of L 2 ( P )with its dual we obtain the following Gel’fand triple
R c L 2 ( P )c R*.
(12)
The dual pairing between an element E R*and 2 E R will be denoted by ( @ I 2 ) . Since for h E CF(R+) we have that exp(Xh) E R, we can extend the S-transform to R*: S Q , ( ~=) e-lh1z/2 ( Q , , e X h ) ,
Q, E
R*, h E c~(R+).
(13)
We shall also make use of the Wiener-Sobolev space^'^-^' which we quickly recall now. Let N denote the number operator, i.e., N acts on the
62
P.-L. Chow and J . Potthoff
n-th chaos, n E No,by multiplication with n. Let p > 1, k E No,and denote by Dp,k the LP(P)-domain of N k l 2 ,equipped with the norm
The dual space of V p , k , p > 1, k E No,will be denoted by V p J , - k , where p' is conjugate to p . The Meyer- Watanabe space V of smooth random variables is defined as the projective limit of the family (Dp,k,p > 1,k E No),and V* denotes its dual. Clearly, D* is the union of the spaces DP,)-k,p > 1, k E No. In view of the applications in the present paper, it will be convenient to introduce still another pair of dual spaces G, G* defined as follows.2o Let L;(P) denote the subspace of L 2 ( P )consisting of those 2 E L 2 ( P )for which the chaos decomposition Eq. (1) has only finitely many non-trivial terms. On L i ( P ) define the following family of norms
The completion of L i ( P ) with respect to 11 . 1 1 2 , ~is denoted by GA. G is the projective limit of the chain (GA, X E R), G* its dual (which - as a set coincides with the inductive limit of this chain). It is not hard to see that20
R c G c D c P ( P ) c D* c G* c R*.
(16)
Moreover, it has been proved in Ref. 20 that Q is an algebra under pointwise multiplication. We shall use for all dual pairings the notation introduced above. Furthermore, since the constant random variables belong to R, we can define the expectation of a generalized random variable @ in R*, G* or D* by E ( @ )= (@, 1).
3.4. Differential Operators In this subsection we only sketch the minimum of the theory of differential operators which is necessary for the present paper. Let h E L2(R+). On the white noise or on the Wiener space the derivative operator Dh can straightforwardly be defined as the usual Giiteaux derivative in direction h (on the white noise space) or in the direction of the anti-derivative of h (on the Wiener space). In our general setup, where no natural notion of translation is available, one can do the following construction.12 Consider Y E L 2 ( P )of the form y = f(X,,,
. . . ,Xg,),
Donsker’s Functional Calculus
63
where f is a smooth function on R”, 91, . . . , gn E L2(R+).We assume that the following expression also belongs to L2( P )
2
W ( X g , ,..,XgJ . ( h ,g i b ( I R + ) r
ai is the usual i-th
partial derivative. Then we set
a= 1
and
n
i=l
It has been proved in Ref. 12 that DhY is P-a.s. well-defined, and that on the white noise space it coincides with the usual differential operators of white noise analysis. In particular, Dh maps V 2 , 1 continuously into L 2 ( P ) . D i denotes the adjoint of Dh in the sense that for all Y , Z E V 2 , 1 ,
(GAy )p ( p ) =
D h Y ) LZ(P)’
(21
It is well known (and easy to see) that D i = Xh - Dh. Assume that { e k , k E N} is an orthonormal basis of L2(R+).Then we have the following useful formula for the number operator 00
N
=CD:,D~~.
(17)
k=l
Let ? denote i the Hilbert space L2 8 L 2 ( P ) .For Y E V 2 , 1 we denote by VY the element in Fl given by the sequence (D,,Y, k E N). Moreover, if Z E e2 m V2)1, we write
k=l
and find
N = v*v. 3.5. Characterization Theorem and Wick Product
We first define U-functionals which play an important role in the theory of the S-transform.
Definition 3.1. Assume that F is a real-valued function on Cy(R+) so that (a) F has a m y entire extension, i.e., for all h, k E Cr(W+),the mapping X I+ F ( k Ah) from R t o W has an entire analytic extension t o C,and
+
64
P.-L. Chow and J . Potthoff
(b) there exists CF 2 0, and a continuous norm all z E C l h E C r ( R + ) ,
I . IF
on
S(R+)so that for
( ~ ( z h )Il C F exp(Iz12 IN$).
(19)
Then F is called a U-functional. The space of all U-functionals is denoted by U. The following theorem has been proved in Refs. 21,22:
Theorem 3.1. The S-transform is a bijection from R* onto U . We remark that more results of this type can be found in numerous papers. In Refs. 23,24 "characterization theorems" are proved to handle Ufunctionals which depend on parameters, e.g., to control derivatives with respect t o space or time variables. Next we define the Wick product Q, o 9 of two generalized random variables Q,, 9 E R*. To this end, observe that the space U of U-functionals is an algebra with respect t o pointwise multiplication. It follows then from Theorem 3.1 that there is a bilinear mapping from R* x R* into itself, written as . o ., so that for all Q 1 Q E R*,
S ( @0 ") = (SQ,)(se)
(20)
holds true. It turns out that the Wick product is actually continuous on R* (cf., e.g., Ref. 8, chapter 4). A description of the Wick product in terms of chaos expansion is as follows. Assume that Q,, 9 E R*have a (generalized) chaos expansions given by (Fn, n E No), (Gn1n E No) with Fn, Gn being symmetric elements in S'(Rn+), n E NO. Then Q, o 9 has chaos expansion given by ( I f n ,n E NO), with n
Hn =
C Fn-mGGm,
(21 1
m=O
where 6 denotes the symmetric tensor product. Thus we get the following formula which resembles the Cauchy product for two absolutely convergent series:
n=O m=O It has been proved in Ref. 20 that product.
B and B* are algebras under the Wick
Donsker's h n c t i o n a l Calculus
65
4. Fourier-Wiener Transform
In this section we give a rigorous definition of the Fourier-Wiener transforms, Eqs. (4), (5), and discuss some of their properties. For the definition, we shall essentially follow the article of Hiaa, Kuo and Obata.25We deviate here from our convention that all random variables are real valued, and consider complex valued (generalized) random variables. Everything developed before has a natural extension to this case, but we want to emphasize that we extend the S-transform in such a way, that it becomes complex-linear (i.e.l not anti-linear). Consider a generalized random variable 2 E R*, and let h E C r ( R + ) . Theorem 3.1 entails that h ++ SZ(ih) defines a new element in R*. Moreover, in view of formula (8) (and its obvious generalization to R*) it follows that this element is given by a chaos expansion which is the chaos expansion of 2, and the n-th kernel is multiplied by i n . That is, we can write SZ(ih) = S o i N Z ( h ) . Obviously the same is true if we restrict 2 to L 2 ( P )or to R,and in particular, iN is unitary on L 2 ( P ) .(YanZ6calls this mapping Fourier-Wiener transform - we reserve this name for a different transformation defined below.) Consider a smooth random variable 2 E R with chaos expansion as in Eq. (l),and note that the kernels f, are symmetric functions in S(Rn+), n E N.Therefore its S-transform is given for h E Cy(R+) by the formula
and this function of h E CF(R+) obviously extends t o a function on S'(Rn+): we just have t o interpret the integrals on the right hand side as dual pairings between S(Rn+)and S'(Rn+) N S'(R+)Bn. (In the last isomorphism we used the nuclear theorem, which holds true because S(R%n+) is a nuclear space as can be read off from the construction for the case n = 1 in Subsection 3.3, i.e., from the spectrum of the operator A introduced there.) We would like to argue now, that SZ defines an element in R. Informally speaking, we want to replace the integration against the tensor products of h by multiple Wiener-It6 integrals against a Brownian motion - in a sense, the integrals on the right hand side of Eq. (1) are multiple Stratonovich integrals of the kernels. Let us introduce the following notation: for k 5 [n/2J ( [rnJ denotes the largest integer less than or equal t o rn) we set
66
P.-L. Chow and J . Potthoff
In terms of the truce operator Tr E S’(Rt)
Tr : S(IW?)-+
R
we can write this also more compactly as (recall that f n is symmetric) fn,k =
(n’”, f n ) .
(3)
Then we can use the well-known formula (e.g., Ref. 8, Chapter 5.D) which re-expresses the multiple Stratonovich integral of f n with respect t o Brownian motion B (which is well-defined because f n E S(Rn+))by a sum of multiple Wiener-It6 integrals, viz.,
Thus we would like to define a random variable
2 by
Now one can use well-known estimation techniques (e.g., Refs. 8,25) to prove that 2 defines indeed an element in R. Moreover, the mapping Z H 2 is continuous from R into itself. For simplicity, we continue to denote this element by SZ,and in this sense consider the S-transform as acting on R. It is clear that we can now compose the two mappings defined on R: Definition 4.1. The Fourier-Wiener transforms 3* are the transformations from R into itself defined by F*Z = S O i k N Z ,
Remark 4.1. 3;- is denoted in Ref. 25 by
z E R.
(6)
T.
The following theorem is almost trivial, but nevertheless it seems to be quite useful. Theorem 4.1. O n R,3+and 3- act as inverses of each other.
Proof. Due t o hypothesis (3.1) and by the construction of R, the set I (cf. Subsection 3.2) is total in R, and therefore it suffices to show the statement
Donsker’s Functional Calculus
Under the identification of a function of h with an element in above, we obtain
F+E(g)
= e--1gI2/2E ( i g ) .
67
R discussed (7 )
Similarly,
and therefore equation (6) yields
so that we find
The converse equation is proved in the same manner.
0
We conclude this section with the remark that in Ref. 25 it has been proved that the adjoint of F+ acting on R* is Kuo’s Fourier transform Ic, which can be defined as
where € 0 is the element in R* whose S-transform a t h E CF(R+) is equal to exp(-(h, h ) / 2 ) , the so-called valeur en zero.26 5. Independence and It6 Calculus
As shown in Section 2 , in order to justify Donsker’s informal derivation of the Feynman-Kac formula, it is necessary to verify in our framework Hypotheses 2.2, 2.3 which are concerned with the independence and the It6 formula for generalized random processes, respectively. These two technical problems will be considered in this section separately.
68
P.-L. Chow and J . Potthoff
5.1. Independence of Generalized R a n d o m Variables First we bring in some new notions. Let I be an interval in R+,and denote by 31the a-algebra generated by the random variables Bt - B,, s, t E I , s
5 t.
Definition 5.1. Q, E G* is called .F-measurable, if for all h E
SQ,(h)= S Q , ( l I h )
S(R+), (1)
holds true. R e m a r k 5.1. Note that the S-transform of an element in G* has a continuous extension from S(R+)to L2(R+), so that there is no ambiguity in the evaluation of SQ, at I l h on the right hand side of Eq. (1). Consider 11 as an operator on L2(R+),and define its second quantization I'(1I) in the same way as r ( A ) in Subsection 3.3. The following lemma, some parts of which are the generalization of Proposition 4.7 in Ref. 27, is almost trivial. L e m m a 5.1. Q, E G* is TI-measurable i f and only if one of the following equivalent conditions is satisfied:
(4 I ' ( l I ) Q , = Q,, (b) SQ,(h)= S Q , ( l I h ) , (c) for every n E N, the n-th kernel of the chaos expansion of Q, has its essential support in I", ( d ) for every n E N, the n-th homogeneous chaos In(Fn) of Q, is 31measurable in the usual sense, (e) for some X L 0 , exp(-XN)@ is an 31-measurable random variable in L2(P). If I = [ s i t ]s, , t E
W+,s 5 t , we shall also write Fs,tfor T I .
Definition 5.2. Two generalized random variables Q,, E G* are called strongly independent if there exist intervals I , J C R+ whose intersection have Lebesgue measure zero, and such that Q, is 31-measurable, and Q is FJ-measurable. Remark 5.2. Strong independence of generalized random variables has also been discussed in Ref. 28. One of the interesting aspects of strong independence is the following simple result which is proved in Ref. 29.
Donsker's Functional Calculus
69
Lemma 5.2. T h e pointwise product . of r a n d o m variables h a s a welldefined e x t e n s i o n t o pairs @, @ of strongly independent generalized r a n d o m variables in G* so that @ . '3 E G*. Moreover, t h e f o r m u l a
holds.
If @, 9 E G* are strongly independent we shall write for a . @ also @'3and call it their pointwise product. In the following corollary two direct consequences of Lemma 5.2 are formulated. Corollary 5.1. A s s u m e t h a t @, @ E
G* are strongly independent. T h e n
S ( @ @= ) (S@)(SQ). Moreover, for all
(3)
X E R,
Proof. Eq. (3) follows directly from equations (20) and (2). For the proof of formula (4) let ( F n ,n E NO),( G n ,n E NO)be the sequences of kernels of the chaos expansions of @, 9 respectively. Then by Eqs. (22) and (2) the chaos expansion of @@ is given by c o n
n = O m=O
Therefore we have c o n
n=O m=O c o n
n=O m=O =
( e X N @o) ( e X N Q )
=
( e X N @() e X N 9 ) ,
where we used again Eq. (2), and the fact that exp(XN)@,exp(XN)@ are also strongly independent.
70
P.-L. Chow and J . Potthoff
5. 2. It6 Calculus f o r Generalized Stochastic Processes
By a generalized stochastic process we mean a mapping from an interval c R+, the t i m e parameter d o m a i n , into G*. A generalized stochastic process Q, = (at,t E I ) is called t a m e d , if there exists X 2 0, so that for all t E I , Ckt E s-x, i.e., if for all t E I , exp(-XN)Qt E L ~ ( P > . A tamed generalized stochastic process with values in G-x is called measurable, if the mapping
I
exp(-XiV)Q, : I x w
-+
R
(t,w>++ exp(-XN)Q,t(w)
(6)
is measurable. It is not hard to see that such a Q, is measurable, if and only if every term in its chaos decomposition is a measurable stochastic process in the usual sense. Theorem 5.1. F o r every T E S’(R), Q, = (T o Bt, t measurable generalized stochastic process.
> 0 ) is a t a m e d
For the choice T = 6,, z E R, the statement of the theorem follows rather directly from the chaos expansion of Donsker’s delta function with an estimation similar to the one in KUO’Spaper.3 This will be shown in Lemma 5.3 below. The proof of the general case is deferred to Appendix B. In order to simplify the language we shall from now on mean by a generalized stochastic process a measurable tamed generalized stochastic process. A generalized stochastic process Q, = (at,t E I ) , I c R+,will be called adapted, if for every t E I , Qt is .Fo,t-measurable. Now we can construct an It6 integral of adapted generalized processes as in Ref. 29, and this construction follows essentially the same steps as in the case of ordinary adapted processes. Assume that I is a finite interval, and for a simple adapted generalized stochastic process of the form M
i= 1
where
t1
< t 2 < - .. < t M + 1
is a partition of I , set
which is well defined in G* because for every i = 1,.. . ,M , Q,i and Bti+,-Bti are strongly independent. With the help of Corollary 5.1 one can show the
Donsker’s Functional Calculus
following generalization of the It6 enough so that Q,, E S-X for all t E I , then
71
Assume that X 2 0 is large
By an application of the generalized It6 isometry (9) we can now extend the stochastic integral from simple adapted processes to the general case as in the classical case, cf. Ref. 29. Assume that Q, = (Q,)t, t E I ) is an adapted stochastic process as before, and let X >_ 0 be such that for all t E I we have at E S-X. Assume furthermore that
l
llQ,tll;,-A
< +co.
(10)
Then Q, has an approximation in L 2 ( I ;$A) by a sequence of simple adapted stochastic processes, whose sequence of generalized It6 integrals is Cauchy in 8-A. Its limit is defined t o be the stochastic integral SIQ,tdBtof Q,. Moreover, also for this stochastic integral the It6 isometry (9) holds true. It is important to observe that this stochastic integral can be written as the image of a classical It6 integral under exp(XN):
JI ( e P X N Q t )dBt .
Qt dBt =
(11)
Moreover, its S-transform is given by
S ( 1 at dBt) ( h )=
l
SQt(h)h(t)dt,
h E C:(R+).
(12)
By Theorem 5.1 all this holds true for Q,t = T o Bt, t E [a,b],0 < a 5 b < +co, T E S’(R). For at of this form we shall occasionally write T ( B t )in order to avoid conflicts with the notation for the Stratonovich integral.
5.3. Donsker’s Delta Function
Donsker’s delta function S, o Bt, i.e., the composition of the Dirac distribution S, concentrated a t x E IR with Brownian motion Bt, t > 0, has been investigated in many papers. Its first constructions as a generalized random variable were by Hida,4 K ~ b o and , ~ K~u o , ~and in the same volume edited by Kallianpur31 appeared the article of Watanabe’? with a theorem that offers another construction in Sobolev-Wiener spaces. Kuo3 was the first t o derive the chaos expansion of Donsker’s delta function, and we shall review this expansion below and in Appendix A.
72
P.-L. Chow and J . Potthoff
In Ref. 11 it has been proved that for given t > 0, Donsker’s delta function is the unique weakly continuous mapping z H @ t ( z from ) lR into R*so that for all f E S(R),
holds, where the integral on the right hand side is a Pettis integral in R*. It is straightforward t o calculate the S-transform of Donsker’s delta function, and the result is t
S(6X o B t ) ( h )= g ( t , z -
hods),
h E C?(R+),
(14)
where g is the heat kernel Eq. (4). Now we present the chaos decomposition of Donsker’s delta function in an especially convenient form, the details can be found in Appendix A. First we have to introduce some notation. For t 2 0, n E No,we set
Because
. for n E No, the set {e,(Bt), n E NO}is orthonormal in L 2 ( P ) Furthermore, t > 0, we define the following function on the real line
where g ( t , z) is the heat kernel (4), and Hn is the n-th Hermite polynomial. Then the chaos expansion of 6,(Bt) takes the form oc)
6 x oBt
= Cvn(t,z)en(Bt)*
(17)
n=O
Following now Kallianpur and Kuo,~’for T E S’(R) we write T = T and obtain the chaos expansion of T o Bt:
* 6,
03
T OBt
=
C(T,vn(t,.)) en(Bt).
(18)
n=O
In the sequel we shall make use of the following notation: for f E Ck(R) or T E S’(R) we write for their Ic-th derivatives f ( k ) , T(’”)resp. Lemma 5.3. Donsker’s delta function has the following properties:
Donsker’s Functional Calculus
73
(a) For x E R, t > 0, k E No, hik) o Bt belongs to 2 ) 2 , - ( k + l ) . I n particular, o Bt, t > 0 ) is a tamed measurable generalf o r all x E R, k E NO, ized stochastic process in G*, and the mapping t ++ G* from (0, CQ) to S* is continuous. (b) Let s > 0. Then ( 6 , o (Bt - B s ) ,t > s), is an (.Fs,t, t > s)-adapted generalized stochastic process in G * , which i s strongly independent of any .Fo,s-measurable generalized random variable in G * .
(6ik)
Proof. The first statement in (a) has been proved by Watanabe in Ref. 33 with methods from Malliavin calculus. Here we give an elementary argument based on Lemma B.3: A glance at Eq. (18) shows that
n=O Hence we get w
and the estimate (B.16) in Lemma B.3, Appendix B , shows that the last sum is convergent. In particular, for every X > 0, the series M
n=O
converges for all Ic E No,2 E R,t > 0 , in L 2 ( P ) .Therefore, the processes t ++ hik) 0 Bt are tamed, measurable generalized stochastic processes. The last statement in (a) follows from the continuity o f t H q i k ) ( t , x )on (0, +m), and the preceding convergence argument. With Definition 5.2 and the Lemmas 5.1, 5.2, property (b) is easily verified. 0 The following lemma provides the important It6 formula for a generalized function of Brownian motion.
Lemma 5.4. For any x E
R and 0 < s < t , the following It6 formula holds
which can be written in the Stratonovich f o r m Jz(Bt)
- Jz(Bs) =
I”
Jk(Br)O d B r .
74
P.-L. Chow and J . Potthoff
More generally, for every T E S’(R), the following equations hold true: T ( B t )- T(B,) =
I’
T’(B,) dB,
:I’
+-
T”(B,) dr
(21)
Proof. We give a proof which is quite in the spirit of Donsker’s calculus. Let 0 < s < t , x E R, and consider the S-transform of &(Bt - B,) at h E C F ( R + ) , Similarly to Eq. (14), we can derive p h ( s ,t;). = Sdz(Bt - & ) ( h )
which is the Green’s function of the parabolic equation
d
1
a2
- u ( t ,x) = - -u ( t ,). at 2 ax2
d + h(t)u ( t ,x), ax
t > s,
(24)
(25) Now, by noting that ph(O, t;x) = g(t,2;h ) and by taking Eqs. (12), (14) into account, we interchange the order of differentiation and integration with the S-transform (which can be justified, e.g., as in Ref. 24), and obtain from Eq. (25), with the injectivity of the S-transform 6z(Bt) - &(a,)=
I”
Sk(BT) dB,
+ 5l I ’ K(B?-)dr,
(26)
which is Itti’s formula Eq. (19). We want t o remark here that It6’s formula has been derived in several articles for generalized stochastic processes with different methods. Next we can combine equation Eq. (26) with the method of Kallianpur and K u o , ~ viz., ~ to write for a general T E S’(R), T = T * 6, where * . denotes the convolution of distributions. Then we obtain ItG’s formula in the form of Eq. (21):
T ( & ) - T(B,) = The formulas (20), (22) follow from Eqs. (19), (21) by rewriting their righthand sides as Stratonovich integrals. 0
Donsker’s Functional Calculus
75
6. Towards Donsker’s Calculus
In Section 2 we presented Donsker’s derivation of the Feynman-Kac formula by means of the method of imaginary parametric function combined with the Fourier-Wiener transforms. To justify his approach in the white noise analysis setting, the main obstacle, as mentioned before, is the extension of such transforms from the space R (or G), as done in Theorem 4.1, t o the space R* (G*, resp.). Instead we shall introduce an alternative approach based on the method of real parametric function and the S-transform, which is well-defined on Q* (see Theorem 3.1). As t o be seen, this new approach can be fully justified. In lieu of Eq. (l),consider the initial-value problem
a
1 dZu(t,z)
at u(t,x) = 2
4 0 ,).
8x2
%g +
+ h(t)
V ( x )u ( t ,z),
t > 0, (1)
= S(z),
u ( t , x ) = 0,
as x
--+
kco,
where h E CF(R+) is the parametric function, and V is a given function on R. Throughout this section we shall make the following assumptions on the potential V: Hypothesis 6.1. V belongs to C4(R), is bounded from above, and the derivatives V ( k ) k, = 1, . . . , 4 , grow at most polynomially at infinity. Then, by the standard PDE theory,34 the initial-value problem has a unique solution denoted by uh(t,x) which satisfies Eq. (1) for t > 0 and the initial condition u(0,x) = S(x) in the sense that, for any 4 E S(R), l i m l uh(t,x)q5(x)dx= +(O). tl0
w
In terms of the Green’s function p h ( s , t ;x) given by formula (23), Eq. (1) can be converted into the integral equation
We will solve the initial-value problem (1) or, equivalently, the integral equation (3) by the method of S-transform in G*. To this end, consider the initial-value problem for the stochastic PDE:
d d U ( t , x )= -- U ( t ,X) 0 dBt ax U(O1x) = +),
+ V(Z) U ( t ,X) d t ,
t > 0, (4)
76
P.-L. Chow and J . Potthofl
which can be written as the It6 equation (t > 0) 1 d2 d U ( t ,Z) = 5 a22 U ( t ,Z) dt
+ V ( X U) ( t ,
Z)
d dt - - U ( t ,X) d B t ,
dx
(5)
U ( 0 , x )= 6(x). Similar to the initial condition (2) for the problem (l),the initial condition U ( 0 , x ) = 6(x) should be interpreted as lim!
tl0
w
a.s., for 4 E S(R).
U ( t , x )+(x) dx = 4(0),
(6)
Notice that Eq. (5) resembles the model equation studied in the paper Ref. 35 for the turbulent transport problem. It is a degenerate (noncoercive) parabolic It6 equation (see Ref. 7) which has only a generalized solution U ( t ,x). By the same method of proof, it can be shown that U ( t ,z) belongs to R*,and is weakly continuous for t > 0, x E W. In fact, as to be seen below, the solution is also in B*. Now we claim that the solution of the problem (4) is given by t
U ( t , x )= &(&) exp{
V ( B s d) s } .
(7)
In order to prove our claim, we first state the following two results, whose proofs are sketched in Appendix C. Let us denote t
V(Bs)ds}.
Kt = exp{
(8)
Theorem 6.1. Under Hypothesis 6.1, for all t > 0 , x E W, the following products exist in E*:
@(&)
Kt,
6,(Bt) V ( & )Kt,
IC = 0,1,2.
Furthermore, all expressions define generalized stochastic processes in 6' with time parameter domain (0, +cm),which are measurable and tamed. Theorem 6.2. Under Hypothesis 6.1, for all t
d & ( & ) Kt = d:(Bt) K t d &
+ (51
> 0, x
E
R, the It6 formula
+ 6,(Bt) V ( B t ) )Kt dt
(9)
and its Stratonovich f o r m d 6 z ( B t ) Kt hold true in 6'.
= S;(Bt) Kt
0
dBt
+ S,(Bt) V ( B t )Kt dt
(10)
Donsker’s Functional Calculus
77
By applying the It6 formula Eq. (19) t o Eq. (7) in differential form, and noting that
63%) we get for t
a
= ---6,(Bt),
ax
> 0,
d U ( t , x ) = --
ax a(
6,(Bt) K t ) 0 dBt
+ 6,(Bt) V ( B t )Kt dt
d U ( t ,Z)o dBt + V(Z) U ( t ,X) d t , (11) ax which gives the stochastic PDE in Eq. (4). For 4 E S(R), we have with Eq. (13) = --
lim tl0
/
6,(Bt) Kt 4(x)dx = lim 4(Bt)Kt
p
tL0
= 4(0),
as.
This shows that U ( t , x ) given by Eq. (7) is the solution as claimed. To connect U ( t ,x) with uh(t,x) via the S-transform, we rewrite Eq. (4) as the integr a1 equation:
U ( t ,Z) = S,(Bt)
+
I’
F ( s ,5 - Bt
+ B,) ds,
(12)
where we put F ( s ,y) = V(y) U ( s ,y). This can be verified easily by applying the It6 formulas in Lemma 5.4 t o Eq. (12) t o get for t > 0,
d U ( t , x ) = -% a (6,(Bt)+/
t
F(s,x-Bt+B,)ds)odB,+F(t,x)dt.(13)
0
Clearly the initial condition is satisfied. Since we have
F ( s ,x - Bt
+ B,)= F ( s , .) * a,-.(&
- B,),
Eq. (12) can also be written as
U ( t ,).
= 6,(Bt)
+
I’ s,Ly
(Bt - B,) F ( s ,Y) dy ds
or
Notice that, by Lemma 5.3(b), 6,-y (Bt - B,) and U ( s , y ) are strongly independent. Now we take of the last relation the S-transform at h E C r ( R ) . Then with Corollary 5.1 and Eq. (23), we obtain from Eq. (14)
P.-L. Chow and J. Potthoff
78
which coincides with the integral equation (3), corresponding to the initialvalue problem (1).Therefore, in view of Eq. (7), we can conclude that
u h ( t ,z)
= SU(t,z ) ( h ) =E
where for t E
1 t
+
(hx(Bt ht) ~ X P {
+
V ( B , h,) d s } ) ,
(16)
R+ we used the notation
1 t
ht =
h(r)dr.
Finally, by setting h = 0 we obtain the Feynman-Kac formula ( 2 ) .
Appendix A. Chaos Decomposition of Donsker's Delta Function We recall that for n E by
No,the standard Hermite polynomial Hn is defined
d" e - x 2 , XER. (A.1) dxn In Appendix B we shall make use of the following well-known relations:
H,(z)
=
(-1)"e"
2
-
H L ( z ) = 2nHn-1(z),
x E R,
(A.2)
and
Hn+l(z) - 2xH,(z)
+ 2n H,-l(z)
= 0,
z E
R.
(-4.3)
The n-th Hermite function is defined by
{Cn, n E No} forms an orthonormal basis of L2(R,B(R),A). The Hemite polynomials with parameter o2 > 0 are defined by
Their relation to the standard Hermite polynomials is given by
Let n E N and fn be a symmetric, square integrable function on R".Its n-fold Wiener-It6 integral is defined by r
Donsker’s Functional Calculus
79
whereS,isthesimplex{(sl, . . . , s n ) E R n , s l < s z < . . . < s , < + o o } . A s a particular case we consider f = l p t l , and obtain
In(l;3
= hn(Bt;t).
(A.8)
In Ref. 3 , H.-H. Kuo showed tha,t Donsker’s delta function has the following chaos expansion
which can be re-expressed, with the help of Eq. (A.4) as follows
Observe that for
vn, n E No,defined in Eq. (16) we have (A.ll)
Thus we can write M
(A.12) n=O
Appendix B. Proof of Theorem 5.1 In this appendix we shall prove certain properties of the functions qn introduced in Eq. (16), and some of these will in turn provide a proof of Theorem 5.1. We begin with the remark that it has been proved in Ref. 36, p. 141 f., that the standard family of seminorms on the Schwartz space S(R) is equivalent to the family 11 . ( ( a , ~ , 2ar , p E No,defined by
We fix an arbitrary t
> 0, and define the following operators on S(E%): A t = Z1( - t D 2 + - x12 ) , t at
= - (“+&D),
Jz&
a* =-(“-&D). 1
Jz&
80
P.-L. Chow and J . Potthoff
Note that
Furthermore we have
Define for n E No
&(s;t)= t-
5 E
1/4~n(%)1
R
(B.3)
where J n is the usual n-th Hermite function (cf. (A.4)). Since A, is the unitary transform of the standard Hamiltonian of the harmonic oscillator, At has spectrum { n 1/2, n E NO} (each of multiplicity one), with eigenfunctions tn(* ; t ) ,n E No. We define another family of norms 11 . I ~ Z , p~ ,E No, by
+
Ilfll2,P
f E S(W
= IIA;fll2,
(B.4)
Lemma B.l. For all a , ,D E No there exists a constant K a , p > 0 so that f o r all f E S(R), IlfIla,P,2
5 Ka,Pt(a-p)'2l l f / 1 2 , P ,
(B.5)
+
+
with p = [ ( a ,D)/21 (smallest integer greater or equal to ( a ,D)/2). Moreover, the families of n o m s 11 . I l u , p , ~ a, , ,Ll E NO,and I/ . / / z , ~p , E No, are equivalent. Based on the commutation relations of the operators A t , a,, and a;, the proof can be carried out along the lines as indicated in Cha'pter V of Ref. 36. Therefore the proof is omitted here. Now consider for n E No the following function f n E S(R): f n ( x )= H,
( 5e-x2/2t1 ) 0
x E R.
A comparison with Eq. (16) yields
X
n! ( n - 2m)!
J .-
Donsker’s Functional Calculus
81
Sketch of the proof. It will be convenient to make the convention fk = 0 for k E Z, k < 0. With the relations (A.2) and (A.3), it is straightforward to compute for x E R, n E No,
Thus we get
Using this equation recursively we find the estimate P
( n - 2m)!
m=O
llfn-2m112.
(B.12)
With the well-known normalization of the Hermite functions in L2(R) it is easy to estimate as follows
Now we insert the last estimate into inequality (B.12), and obtain the bound (B.8). 0 Recall the family (q,, n E Eq. (16).
No) of
functions on ( O , + c o ) x
IW defined in
Lemma B.3. (a)
For all p , n E No,t > 0, (B.13)
(b) For all a , /3 E No there exists a constant Ca,p > 0, so that for all n E No,t > 0, llrln(t1 .)IIa,P,2
- ca ,P t(a-P)/2-1/4 <
(B.14)
+
where p = [ ( a /3)/2]. (c) For all n E No, t > 0, x E
R, (B.15)
82
P.-L. Chow and J . Potthofl
( d ) For every k E t > 0,
No there is a constant
c k
> 0 so that for all n
SUP(VLk)(t,2)I< - ckt - ( k + + ' ) / 2 nk/2--1/12
E
No,
(B.16)
XER
Proof. To prove statement (a) we use Eq. (B.7) and Lemma B.2 so that we get
and the proof of (a) is finished. Statement (b) follows now from (a) and Lemma B . l . The formula (B.15) follows by an elementary calculation from Eqs. ( A . l l ) ,(A.4) together with Eqs. (A.2),(A.3). Finally, we prove statement (d) of the lemma. To this end, we iterate Eq. (B.15), and obtain
v L k ) ( t , z )= (-1)ktt-k/2J ( n + l ) ( n + 2 ) . . . ( n + k ) v ~ + k ( t , 5. ) Using the well-known bound sup lqn(t,x)l I C t p 1 I 2n - l / l 2 X
for some numerical constant C (e.g., (21.3.3),p. 571, in Ref. 37), we estimate now in the following way
5 Ct-112 ( n .+ k p 1 2 - Ct-1/2 n-1/12 <
and possibly the constant C had t o be increased in the second step. Thus we get the following estimate
.)I
suplvLk)(t,
I c k t-(k+1)/2 ?l,k/2-1/12,
XER
where
c k =C
J m , and the proof of the lemma is complete.
0
Donsker’s Functional Calculus
83
Finally, part (b) of Lemma B.3 entails the following corollary, with which we also conclude the proof of Theorem 5.1.
Corollary B . l . Assume that T E S’(R), and that t > 0. Then there exists + N)-q(T o Bt) E L 2 ( P ) .I n particular, for every X > 0 , T 0 Bt E 6-A.
q E No so that (1
Proof. Since T E S’(R), we may assume that there exist constant KT > 0, so that for all f E S(R) l(T,f)l I KT
Q,
,O E No,and a
Ilf Ila,P.2
holds. Thus we get for n E No,
I(T,vn(t,.))I 5 KT Ka,p t(a-P)/2-1/4 - +n
+
(;
where p = [ ( a p)/21. Choose q sion (18). Then we have
=p
) p l
+ 1, and consider the chaos expan-
n=O M
which converges.
0
Appendix C. Proofs of Theorem 6.1 and Theorem 6.2 First we prove a result that - under certain conditions on the potential V the product S,(Bt) Kt of Donsker’s delta function with the FeynmanKac weight Kt (cf. Eq. ( 8 ) ) belongs t o D* c G*. Throughout we suppose that the time parameter t belongs t o some interval [O,T],T > 0.
-
Lemma C.1. Assume that
(a) v E C1(W, (b) V is bounded from above, (c) for every t E [0,TI, the random variable
belongs to L 2 ( P ) .
84 P.-L. Chow and J . Potthoff
Then there exist p > 1 so that f o r all t E (O,T],6,(Bt) Kt E Dp9-1. I n particular, there exists a X 2 0 so that f o r all t E (O,T],S,(Bt) Kt belongs to L A . Before we enter the proof, we remark that we immediately have the following
Corollary (3.1. Assume that V is as in the hypothesis (a) and (b) of Lemma C.l, and such that V’ grows at most polynomially at infinity, i.e., there are constants K > 0 , and m E N,so that [V‘(x)l 5 K (1
+ Ixlm),
for all x E
EX.
Then the conclusion of Lemma C.l holds true.
Proof of Corollary C . l . From Holder’s inequality we get for all t E [0,TI,
Proof of Lemma C . l . Without loss of generality we may assume that V is negative, so that for all t E [O,T],IKtJ 5 1. The product of 6,(&) with Kt will be defined for 2 E D by
(&(&) Kt, 2) = (~z(Bt),Kt z), and we have to prove that the right hand side makes sense. We know from Lemma 5.3 that 6,(Bt) E D z , - ~ .Therefore with
1(6z(Bt),Kt z)I L l l l ~ ~ ~ ~ t ~ l IlIIK l ztZ,1-1l12,1 we have to prove that the under the hypothesis of the proposition the last norm can be bounded by a constant times l l Z l l p , k for some p > 1, k E N.
Donsker’s h n c t a o n a l calculus
We fix an orthonormal basis { e k , k E of Eq. (18), and it follows that
85
N} of L2(R+).Then we can make use
Ill~tzlll;,l= IlKtZll; + IlvKtzII;. The first term on the right hand side can be bounded by 1 1 2 1 1 ; . The second term can be estimated in the following way:
where we used again that IKt I 5 1. For last expression we have by Holder’s inequality
Therefore it remains to prove that the first of the last two norms is finite. We use De,B, = ( e k , l p , s ~ ) L ~ ( to R +compute )
so that
By hypothesis the L2(P)-norm of the last expression is finite. For the last statement in Lemma C.l, we only have to remark that it has been proved in Ref. 20, that for all p > 1, k E No,there exists a X 2 0 0 SO that Dp,-k C G-x.
86
P.-L. Chow and J . Potthoff
Theorem C. 1. Assume that
v
( 4 E C3(R), (b) V is bounded from above, (c) V ( k )k, = 1, 2, 3, are of at most polynomial growth at infinity. Then there is a p > 1 so that for all t E (O,T], 6 $ k ) ( B t )Kt E v p , - ( k + l ) ,
k
= 0,1,2,
d z ( B t ) V ( & )Kt E
Vp,-l.
I n particular, there exists a X 2 0 so that for all t E (O,T], all expressions belong to 8-x. The proof of Theorem C.l is a - somewhat tedious - elaboration of the preceding proofs, which however does not use any new idea. Therefore we will only sketch it. Obviously, Theorem 6.1 follows from Theorem C . l .
Sketch of the Proof of Theorem C . l . Consider the expressions 6(')(Bt)Kt, k = 1, 2. In Lemma 5.3 we have already proved that S(2)(Bt)E v z , - ( k + l ) , k = 1, 2. Following the pattern of proof of Lemma c . 1 , we then have to show that for all Z E V ,we can bound the norms
IllKm
, 2 ,
llWt 21112,3
by norms of some spaces v p , k , p > 1, k E No. Using the commutation relation of N and V, one can express the squares of the above norms as before by sums of expressions involving only terms of the form V k K t Zwith k = 1, 2, 3. In each case we use the product and chain rules for V, and the resulting expressions involve the derivatives of V up t o order 3. Finally one uses the triangle and Holder inequalities in order to separate the terms and estimate them as above. The product 6,(Bt) V ( B t )Kt is handled similarly. 0 For a sketch of the proof of Theorem 6.2 we first show the following result.
Lemma C.2. Let p be a positive COO-function with support in [-1/2,1/2]. For x E R, m E N set 6,,m(Y) = mp(m(x- Y ) ) ,
Y E R.
Let I be any closed subinterval of (O,T]. Then for all k = 0 , 1, 2, t E I , the sequence (6z,m(&), (k) m E N) converges in v 2 , - ( k + 2 ) to 6,( k )( B t ) , the convergence being uniform in t E I .
Donsker’s Functional Calculus
87
Proof. Let n E No,m E N,t E I , then with the mean value theorem it is straightforward to show that we have the following bound
Now we use inequality (B.16), and obtain the estimate
If we apply this estimate to the convergence of the chaos expansions (17), (18), we get the statement of the lemma. 0 We combine the last lemma with the estimation techniques in the proofs of Lemma (2.1, Corollary C.l and Theorem C.l, and obtain the following
Corollary C.2. Let I be a closed subinterval of (0,Tl. Consider the sequence (6x,m, m E N) in Lemma C.2. Under hypothesis 6.1, there exists p > 1, so that f o r all t E I , the expressions
S g L B t ) Kt,
6z,m(&)
V ( & )Kt,
IC = 0 , 1 , 2
converge in Dp,-(k+2) to
6ik’(&) Kt,
SX(Bt) V ( & )Kt,
IC = 0,1,2,
resp., in every case the convergence being uniform in t E I Now we can finish our sketch of the proof of Theorem 6.2. We just have to notice that the usual It6 formula holds for the approximating terms constructed above. Since for some p > 1 every term in the It6 formula converges in Dp,-4 to the corresponding term of the integrated version of formula (9), uniformly in t on every closed subinterval of (0, TI, this formula is proved to hold in V p , - 4 .Hence with the same arguments as above, the formula holds in G*. Thereby also the Stratonovich form of It6’s formula in G*, Eq. (lo), is proved.
References 1. M. Donsker, Analysis in function space (MIT Press, Cambridge, MA, 1964), Cambridge, MA, ch. On function space integrals, pp. 17-30. 2. M. Kac, R u n s . American Math. Sac. 6 5 , 1 (1949). 3. H.-H. Kuo, Donsker’s delta function as a generalized Brownian functional, in Theory and Applications of Random Fields, ed. G . Kallianpur (SpringerVerlag, Berlin, Heidelberg, New York, 1983).
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4. T. Hida, Generalized Brownian functionals, in Theory and Applications of Random Fields, ed. G. Kallianpur (Springer-Verlag, Berlin, Heidelberg, New York, 1983). 5. H.-H. Kuo, White noise distribution theory (Chapman and Hall / CRC, Boca Raton, London, New York, 1996). 6. M. Donsker, Integration in function space, Lecture Notes, Courant Institute, New York University, New York, (1971). 7. P.-L. Chow, Stochastic Partial Diflerential Equations (Chapman and Hall / CRC, Boca Raton, London, New York, 2007). 8. T. Hida, H. Kuo, J. Potthoff and L. Streit, White Noise - A n Infinite Dimensional Calculus (Kluwer Academic Publishers, Dordrecht, 1993). 9. N. Obata, White noise calculus and Fock space (Springer-Verlag, Berlin, Heidelberg, New York, 1994). 10. T. Deck, J. Potthoff and G . VBge, Acta Appl. Math. 48, 91 (1997). 11. J. Potthoff and E. SmajloviC, On Donsker's delta function in white noise analysis, in Mathematical Physics and Stochastic Analysis - Essays in Honour of Ludwig Streit, eds. S . Albeverio et al. (World Scientific, Singapore, 2000). 12. J. Potthoff, Acta Appl. Math. 6 3 , 333 (2000). 13. H.-H. Kuo, Gaussian Measures in Banach Spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1975). 14. I. Kubo and S. Takenaka, Proc. Japan Acad. 5 6 , 411 (1980). 15. I. Kubo and S. Takenaka, Proc. Japan Acad. 5 6 , 376 (1980). 16. P. A. Meyer, Quelques rhsultats analytiques sur le semi-groupe d' OrnsteinUhlenbeck en dimension infinie, in Theory and Applications of Random Fields, ed. G . Kallianpur (Springer-Verlag, Berlin, Heidelberg, New York, 1983). 17. S. Watanabe, Malliavin's calculus in terms of generalized Wiener Functionals, in Theory and Applications of Random Fields, ed. G. Kallianpur (SpringerVerlag, Berlin, Heidelberg, New York, 1983). 18. H. Sugita, J . Math. Kyoto Univ. 2 5 , 31 (1985). 19. H. Sugita, J . Math. Kyoto Univ. 2 5 , 717 (1985). 20. J. Potthoff and M. Timpel, Potential Analysis 4, 637 (1995). 21. J. Potthof and L. Streit, J . Funct. Anal. 101,212 (1991). 22. Y. Kondratiev, P. Leukert, J. Potthoff, L. Streit and W. Westerkamp, J . Funct. Anal. 141,301 (1996). 23. J. Potthoff, White noise methods for stochastic partial differential equations, in Stochastic Partial Dioerential Equations and Their Applications, eds. B. Rozovskii and R. Sowers (Springer-Verlag, Berlin, Heidelberg, New York, 1992). 24. T. Deck, J. Potthoff, G. VBge and H. Watanabe, Appl. Math. Optim. 40,393 (1999). 25. T. Hida, H.-H. Kuo and N. Obata, J. Funct. Anal. 111,259 (1993). 26. J.-A. Yan, Sur la transformee de Fourier de H. H. Kuo, in Se'minaire d e Probabilite's X X I I I , eds. J. AzBma, P. Meyer and M. Yor (Springer-Verlag, Berlin, Heidelberg, New York, 1989)
Donsker’s Functional c a k u h s
89
27. T. Hida, Brownian Motion (Springer-Verlag, Berlin, Heidelberg, New York, 1980). 28. H. Gjessing, H. Holden, T. Lindstrom, B. Bksendal, J. Ubme and T. Zhang, The Wick product, in Frontiers in Pure and Applied Probability, eds. H. Niemi et al. (TVP Publishers, Moscow, 1993). 29. F. Benth and J. Potthoff, Stochastics and Stochastics Reports 58, 349 (1996). 30. I. Kubo, It6 formula for generalized Brownian functionals, in Theory and Applications of R a n d o m Fields, ed. G. Kallianpur (Springer-Verlag, Berlin, Heidelberg, New York, 1983). 31. G. Kallianpur (ed.), Theory and Applications of R a n d o m Fields (SpringerVerlag, Berlin, Heidelberg, New York, 1983). 32. G. Kallianpur and H.-H. Kuo, Appl. Math. Optimization 12, 89 (1984). 33. S. Watanabe, Some refinements of Donsker’s delta functions, in Stochastic Analysis o n Infinite Dimensional Spaces, eds. 11. Kunita and H.-H. Kuo (Longman Scientific & Technical, 1994). 34. A. Friedman, Partial differential equations of parabolic type (Prentice-Hall, Englewood Cliffs, N.J., 1962). 35. P.-L. Chow, Appl. Math. Optim. 20, 393 (1999). 36. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, London, 1972). 37. E. Hille and R. Phillips, Functional Analysis and Semigroups, Amer. Math. SOC.Colloq. Publ., Vol. XXXI, revised edn. (Amer. Math. SOC.,Providence, Rhode Island, 1957).
90
STOCHASTIC ANALYSIS OF TIDAL DYNAMICS EQUATION U. Manna Department of Mathematics, University of Wyoming, Laramie, W Y 82071, USA E-mail:
[email protected]
J. L. Menaldi Department of Mathematics, Wayne State University, Detroit, MI 48202, USA E-mail: jlmQmath.wayne. edu
S. S. Sritharan Department of Mathematics, University of Wyoming, Laramie, W Y 82071, U S A E-mail: sriQuwyo.edu In this paper the deterministic and stochastic tidal equation in 2-D have been studied. T h e main results of this work are t h e existence and uniqueness theorem for strong solutions. These results are obtained by utilizing a global monotonicity property of the sum of the linear and nonlinear operators and exploiting the generalized Minty-Browder technique.
Keywords: Tidal dynamics equations, Maximal monotone operator.
1. Introduction The ocean tides have long been of interest to humans. A full account of the general theory and history of the tidal waves can be found from Lamb' and summarized as follows. First, Newton2 established the foundations for the mathematical explanation of tides, which motivated Maclaurin3 t o investigate the effect on tidal dynamics due t o Earth's rotation. Although Euler realized that the horizontal component of the tidal force has more effect than the vertical component to drive the tide, the first major theoretical formulation for water tides on a rotating globe was made by L a p l a ~ ewho ,~ formulated a system of partial differential equations relating the horizontal
Stochastic Analysis of Tidal Dynamics
91
flow to the surface height of the ocean. The Laplace theory was developed further by Thomson and Tait,5 and Poincar6.6 In the last few decades many rapid progress in theoretical and experimental studies of ocean tides can be observed. These days both the experimental and the theoretical information on ocean tides are being used t o study important problems not only in oceanography but also in atmospheric sciences, geophysics as well as in electronics and telecommunications. For extensive study on the recent progress in this field we refer the readers Marchuk and K a g a r ~ ,and ~ * ~P e d l o ~ k y . ~ Marchuk and Kagan7 constructed the tidal dynamics model from the 3-dimensional Navier-Stokes equations by integrating along the z-axis and then considering on a rotating sphere, which is a slight generalization of the Laplace model. The existence and uniqueness of the deterministic tide equation by using the classical compactness method have been proved in Ipatova,l0 and Marchuk and Kagan.7 In this paper we have considered the model described in Refs. 7,lO and proved the existence and uniqueness of strong solutions for the stochastic tide equation in bounded domains. A brief description of the model has been given in Section 2. In Section 3 we have defined few standard well known function spaces and then proved the global monotonicity property of the nonlinear operator of the tidal equation. Then we establish certain new a priori estimates which play a fundamental role in the proof of existence and uniqueness of weak and strong solution proved in the second half of Section 3 and in the Section 4 for the deterministic and stochastic cases respectively. The monotonicity argument used here is the generalization of the classical Minty-Browder method for dealing with global monotonicity. Here the use of global monotonicity avoids the classical method based on compactness and thus the results apply to unbounded domains and hence the existence and the uniqueness results are new even in the deterministic case. The Minty-Browder technique for dealing with local monotonicity was first used by Menaldi and Sritharan'l for the stochastic Navier-Stokes equation and by Manna, Menaldi and Sritharan12 for the stochatic Navier-Stokes equation with artificial density. Similar ideas were used by Sritharan and Sundar13 for the stochastic Navier-Stokes equation with multiplicative noise and also in Barbu and Sritharan.l41l5 2. Tidal Dynamics: The Model Under the following assumptions: (1) the Earth is a perfectly solid body, (2) tides in the ocean do not change the Earth's gravitational field, and (3) there is no energy exchange between the mid-ocean and the shelf zone,
92
U. Manna, J . L. Menaldi, and S. S. Sritharan
Marchuk and Kagan7 obtain the following mathematical model
r + A l w - KhAW + -1WIW h 6& + Div(hw) = 0, &W
4-
= f,
(1) (2)
in 0 x [O,T],where 0 is a bounded 2-D domain (horizontal ocean basin) with coordinates x = ( ~ 1 ~ x and 2 ) t represents the time. Here & denotes the time-derivative, A, V and Div are the Laplacian, gradient and the divergence operators respectively. The unknown (dependent) variables (w, E ) , functions of (x,t), represent the total transport 2-D vector (i.e., the vertical integral of the ve1ocity)and the deviation of the free surface with respect t o the ocean bottom. The coefficients A1 = [ai j ] is a 2-dimensional antisymmetric square matrix with constant coefficients a l l = a 2 2 = 0 and --a12 = -a21 = 2w,, the Coriolis parameter (i.e., wz = w cos(cp), w is the angular velocity of the Earth rotation and cp the latitude), fch > 0 the constant horizontal macro turbulent viscosity coefficient, r > 0 the constant bottom friction coefficient equal to a numerical constant, g the Earth gravitational constant, h = h(x) is the (vertical) depth at x in the region 0 and f = yLgVE+ is the known tide-generating force with y~ the Love factor. The domain contour 6 0 consists of two parts, a solid part I'l coinciding with the shelf edge and the open boundary r 2 . The vector wo of full flow is considered a known function of the horizontal coordinates and time. Thus a non-homogeneous Dirichlet boundary condition are added t o the above PDE, namely
w = WO
on
8 0 x [O,T].
(3)
The full flow satisfies w0=0
on
rl
(4)
and
where the time integral is extended to one time-period and n is a normal to the contour r 2 . Here (4)represents the fact that the no-slip boundary condition is fulfilled on the contour rl while (5) means that the water transfer via the open boundary r2, integrated during the tidal cycle turns t o zero.
Stochastic Analysis of Tidal Dynamics
93
To take into consideration the redistribution of water masses, one should add an integro-differential term of the form
where K ( x ,y) = G(X', ' p l , X, p) with X the longitude and p the latitude has the form n
rnln
where k,, hn are the Love factors of order n, an := (0.18)(3/2n+ l),gno := (2n 1 ) / ( 4 ~ )grim , := [(2n 1 ) ( ~ ~ ! ) ~ ] /-[ m)!(n 2 ~ ( n m ) ! ]and , Pnm are the associated Legendre functions. Note that the n-term correspond to the expansion in spherical harmonics. Denote by A the following matrix operator
+
+
+
)
-p p -aA
A : = ( -aA and the nonlinear vector operator
where a := nh and 3!, := 2w, are positive constants, y(z) := r / h ( z ) is strictly positive smooth function. In this model we assume the depth h ( z ) to be continuously differentiable function of 2 , nowhere becoming zero, so that minh(z) = E XEU
> 0, m a x h ( s ) = /I, and XEU
where M is a some positive constant which equals t o zero at a constant ocean depth. To reduce to homogeneous Dirichlet boundary conditions the natural change of unknown functions u(5,t ) := w(5, t ) - wO(z, t ) ,
(11)
1
(12)
and z(z,t ) := ( ( 5 ,t )
+
t
Div(hwo(z, s))ds,
which are referred to as the tidal flow and the elevation. The full flow wo, which is given a priori on the boundary aU, has been extended t o the whole domain U x ] O , T ]as a smooth function and still denoted by wo.
94
U. Manna, J . L. Menaldi, and S. S. Sritharan
Then the tidal dynamic equation can be written as
+ Au + ylu + wol(u + w") + gVz = f' &z + Div(hu) = 0 in 0 x [O,T], &u
u=O
in
0 x [O,T],
on d 0 x [O,T], Z=ZO
U=UO,
in
(13)
Ox{t=O},
where f'=f--
at
+ gV /tDiv(hwo)dt - Awo, 0
So the nonlinear and linear partial differential equations in (13) are coupled with an integro-differential equation (14). An integro-differential term of the form
may be added, where the kernel is a symmetric function K(x, y) = K(y,x) for any x, y in 0. Then the first equation in (13) will be replaced by
~ t ~ + A u + y ~ u + w o ~ [ ~ + ~ o ] + g V=[ fz' + K in z 0 ] x [O,T],(18) where K denotes the integral operator (17). 3. Deterministic Setting: Global Monotonicity and
Solvability In this paper the standard spaces used are as follows: MIA(0)with the norm
and IL2(0) with the norm
c W - l ( 0 ) we may consider A or Using the Gelfand triple WA(0) c IL2(0) V as a linear map from HA (0)or L2( 0 )into the dual of MI; ( 0 )respectively.
Stochastic Analysis of Tidal Dynamics 95
The inner product in the iL2 or L2 is denoted by (., .) and the induced duality by (., .). So clearly (ulV)LZ =
s,
U(X)
. v(x)dx
for any u and v in L2(0). Notice that by the divergence theorem, (gVz, hv)L2 = -(gz,Div(hv))Lz,
Vz E L 2 ( 0 ) , v E iL2(0). (3)
Lemma 3.1. For any real-valued smooth functions cp and 11, with compact support in R2, the following hold: 2
llP~llL2 I llcp~lcpll,l l l @ ~ 2 @ l l L,l
(4)
Proof. The results stated above are classical and well known.16
0
Notice that by means of the Gelfand triple we may consider A, given by (8), as mapping WA(0)into its dual W-'(O). Define the non-symmetric bilinear form a(u,V ) := a [ ( d i u i ,&w)
+ ( 8 2 ~ 2~, Y Q )+] P[(ui,
-
~ 2 ) ( ~ 2vi)] ,
(6)
on the (vector-valued) Sobolev space Wh(O) := HA(0,JR2), where (.;) denotes the inner product in the (vector-valued) Lebesgue space IL2(0):= L 2 ( 0 , R 2 ) ,e.g., see Adams17 for detail on Sobolev spaces. Thus if u has a smooth second derivative then
a ( u , v) = (Au, v) for every v in W h ( 0 ) . Moreover, the bilinear form a(.,.) is continuous and coercive in W;(O), i.e., la(u,v)l
I C1IIullw; IIVIIW~, vu,v E W i W L
(Au, U)
=~
( uU), = c ~ J J u JVU J ~E; ,W;(O),
(7)
(8)
+
for some positive constants C1 = a P. Now we state an useful result as a lemma from Kesavan." Lemma 3.2. Let X be a normed linear space and let R c X be open. Let J : R -+ JR be twice differentiable in 52. Let K c R be convex. Then J is convex i f and only iJ f o r all u , v E K , d2
J (v;u,u) = -J ( v + 8u + au) dB da If
U. Manna, J . L. Menaldi, and S. S. Sritharan
96
Let us denote the nonlinear operator B(.) by
v
++
B(v) := ylv
+ w"/[v+ w"].
(9)
Then we have the following lemma: Lemma 3.3. Let u and v be in IL4(0,R2). Then the following estimate
holds: (B(u) - B(v),u - V) 2 0.
(10)
Proof. Suppose J ( v ) denotes the functional
Then simple calculations give, d - J ( ~+ eu + aw)= dcx and
s,
yw(v
+ eu + a W ) l v + eu + c x W p 5 ,
d2 J ( v + 8u + QW) = 2 de d a
YUW(V
+ 6u + awldz.
Hence
since , , y(z) is a positive function. for any u, v E ~ ~ R(~ )0 Thus in view of Lemma 3.2, J ( v ) is a convex functional and so
+
~ ( e u (1- e)v) I e q U )+ (1- e)J(v) which yields J(V
+ e(U
-
v) - ~ ( v )5) e[J(U) -J ( ~ ) I .
Dividing both sides by 9 and letting 8 -+0 we deduce
(J'(v), u - v) 5 J(u) - J(v) and then
(J'(u) - J'(v), u - v) L (J'(u),u - v) - [J(u) - J(v)]
L -J(v)
+ J(u>- [J(u)- J(v)]
= 0,
for every u and v where the above integrations in 0 make sense.
(11)
Stochastic Analysis of Tidal Dynamics
97
Then with the help of (11) we can conclude that
(B(u)- B(v), u - V)
=
(J'(u
+ w") - J'(v+ w"), [U + w"] -
[V
- w"])
2 0, i.e., inequality (10). Notice that the nonlinear operator B ( . ) is a continuous operator from
IL4 ( 0 )into IL2 ( 0 ) We . can check IIB(v)lllLz 5 C2IIVIIJL4, V V , W 0 E IL4(0) (12) IIB(U) - B(V)IILZ 5 c 2 [IIUI~IL~ I I V ~ ~ ((u L ~] v ( I ~ 4 , VV, w0 E L 4 ( 0 ) , (13) where the constant C2 is the sup-norm of the function y. With the above notation, the tidal dynamics equation can be written in a weak sense as
+
(~,v)Lz
(2
+ a(u,v) + (B(u),v)Lz+ ( g V z , v ) p = (f,v)Lz,
+ Div(hu), C)LZ = 0,
u(0) = U",
V[ E L 2 ( 0 ) ,
z ( 0 ) = 20,
b'v E WA(0) (14)
(15) (16)
where f is given by the integro-differential equation (14).
Proposition 3.1 (energy estimate). Under the above mathematical setting let W" E
L 2 ( O 1 T ; W ~ ( O ) )E, fL2(O1T;lL2(0)),uo E lL2(0), 20 E L 2 ( 0 ) . (17)
Let u in L2(0,T;Wi(0)) and z in L 2 ( 0 , T ; L 2 ( 0 ) ) be the solution of deterministic parabolic variational equality (14)-(16) such that u belongs to L 2 ( 0 , T ; W 1 ( 0 ) ) and i belongs to L2(0,T;L2(0)), with W1(0)the dual space of W i ( 0 ) . Then we have the energy equality
which yields the following a priori estimates
98
CJ. Manna, J . L. Menaldi, and S. S. Sritharan
where the constant C depends on the coeficients and the norms Ilfll L2(0,T;W-') 7 llwOIlL2(o,T;w;) 7 IIU(O>l l L Z and Ilz(0)Il LZ.
Since in L2-norm divergence is bounded by gradient,
Also
Stochastic Analysis of Tidal Dynamics 99
Hence from (24) we have,
Integrating the above equation in t we have,
Now equation (22) yields
I d --JJz(t)lJ$ 2 dt = -(Div(hu(t)), z ( ~ ) ) L z . Notice that
Now using the assumption on h from
(lo), we have
(28)
100
U. Manna, J . L. Menaldi, and S. S. Sritharan
Integrating in t we have,
Let
K = max{l+ M
2g2 2p2 + -,r 7 + (y + M } , &
then the above equation becomes
Now by virtue of equation ( 5 ) in Lemma 3.1 and by the assumption on wo in the proposition we have, Ilw0(t)llL2(0,T;L4)I Cllw0(~)IIL2(o,T;W~) F K’I where C is any positive constant. Hence using the Gronwall’s inequality we have the desired a priori estimates (19) and (20). 0
Remark 3.1. Note that since h=h(x) is a bounded function, we can get similar energy estimate by taking the inner product of the tidal dynamics equation with hu. If we denote
F(u) := AU + B(u) - f ,
Stochastic Analysis of Tidal Dynamics
101
where the operators A and B are defined by (8) and (9) respectively, then the tidal dynamics equation can be written in a weak sense as
{
( U P ) ,~
+
+ ( g V z ( t ) ,hu(t))x,Z = 0,
u ( ~ ) ) I L z (J'(u(t)),hU(t))gz
V u E WA(O) (31)
+
( i ( t ) Div(hu(t)),2(t))Lz = 0 , V.? E L 2 ( 0 ) .
Now by divergence theorem, .sd ( g V z ( t ) ,h4t))ILZ = -(Sz(t),Div(hu(t)))Lz= ( g z ( t ) .i.(t))LZ , = --llz(t)ll[z. 2 dt Thus from (31) we have the energy equality, d
d t [ I I ~ U ( t ) l l 2+ 2 g l l Z ( t ) 1 1 2 2 ] + 2 ( F ( u ( t ) )hU(t))n.Z , = 0.
(32)
Note that the energy equality (32) also yields a priori estimates similar to (19)- (20).
Proposition 3.2 (Uniqueness). Let (u,z ) be a solution of the deterministic tide equation (14)-( 16) with regularity u E Co(O,T;lL2(0)) fl L2(0, T ;WA(O)), z E L2(0, T; L 2 ( 0 ) ) ,
and let the data f , uo and zo satisfy the condition f E L2(0, T;W-l(O)), uo E IL2(O),
20
E L2(O).
If (v, Z) is another solution of the deterministic tide equation (14)-(16), such that v E Co(0,T;lL2(0))ilL2(0,T;WA(0)) and 2 E L 2 ( 0 , T ; L 2 ( 0 ) ) , then [IIU(t)- v ( t ) 1 1 2 z + I I Z ( t ) - w l l $ l e - K t I ltu(0)- vco>tlEz + IlZ(0) - Z(O)Il&, for any 0 5 t
IT,
(33)
where K is a positive constant.
Proof. Notice that if u and v are two solutions then w = u - v solves the deterministic equation d -w(t)+Aw(t)+gV(z(t)-6(t))
= B(v(t))-B(u(t))in IL2(0,T ;W-'(O)). dt Multiplying the above equation by w ( t ) , taking the inner product and applying the result from Lemma 3.3, we get
I d 2 dt
102
U. Manna, J. L. Menaldi, and S. S. Sritharan
Thus
Again notice that
d - Z ( t ) ) Div(hw(t)) = 0. dt Taking inner product with t ( t )- Z ( t ) we have as (29),
+
-(z(t)
d dt
-Ilz(t)
- Z(t)1122
I Mllw(t)11;2
2P2 + (cy + M)Ilz(t) -
W
l
2
2
a +,Ilw(t)ll&
(35)
Let us denote 292
2p2
K = - +a- + M .a
Then adding (34) and (35) and rearranging we have d -[llw(t)1122 dt
+ Il4t) - WIIE~II K[llw(t)1122 + Ilz(t) -
@,ll22].
(36)
Hence using Gronwall's inequality we have the desired estimate (33) for any 0 OltiT. A finite-dimensional Galerkin approximation of the deterministic tide equation can be defined as follows. Let { e l ,e2, . . .} be a complete orthonorma1 system (i.e., a basis) in the Hilbert space L2(0)belonging to the space IHIA ( 0 )(and IL4 (0)). Denote by ILK ( 0 )the n-dimensional subspace of L2(0) and Wi(0)of all linear combinations of the first n elements { e l ,e2, . . . ,e n } . Let us consider the following ODE in Rn
{
d(u"(t), V(t))L2
+
+ a(u"(t),v(t))dt + (B(u"(t)), V(t))L2dt + (gVz"(t), V(t))L2dt = (f(t), V(t))L2dt,
(37)
(.in(t) Div(hu"(t)), <(t))L2= 0,
in (0, T), with the initial conditions (U(O),V)L2= (UO,V)L2, and
(Z(O),C)L2
= (ZO,OL2,
(38)
Stochastic Analysis of Tidal Dynamics
103
for any v in the space IL;(O) and C in L i ( 0 ) . The coefficients involved are locally Lipschitz and we use the a priori estimates (19) and (20) to show global existence of a solution u"(t) in the space Co(O,T ,lLz(0)). P r o p o s i t i o n 3.3 (2-D existence). Let f , uo and zo be such that
i
f E L2(0, T ;w-l(o)),
uo E IL2(0),zo E L 2 ( 0 ) .
(39)
Then there exists a solution (u(t,x),z(t, x)) with the regularity
i
uE
c0(o, T ; L ~ ( on ) )~
z , i. E
~ (T;w;(o)), 0 ,
L~(o,T;L~(o))
(40)
satisfying the deterministic tide equation (14)-(16) and the a priori bounds (19)-(20). Proof. Let us denote F ( u ) := A U
+ B(u) - f ,
where the operators A and B are defined by (8) and (9) respectively. Then du"(t)
+ F(u"(t))dt + gVz"(t)dt = 0.
Using the a priori estimates (19)-(20), it follows from the Banach-Alaoglu theorem that along a subsequence, the Galerkin approximations {un} have the following limits: U"
-
Z" -+
F(un)
u weakly star in Lw(0,T;IL2(0))nL2(0,T;W;(0)),
-
z weakly in L2(0, T ;L2(0)),
FO weakly in L2(0,T;IHI-l(0)),
where u has the differential form du(t)
+ Fo(t)dt + gVz(t)dt = 0,
in L2(0,T ;W-l(O))
and the energy equality similar to (32) holds, i.e., d[IIhu(t)ll22
+ gllz(t)11;2] + 2(Fo(t>,hu(t))L2dt = 0.
Since the Galerkin approximations {u"} satisfy the energy equality d [ l l h u n ( t ) l & +g11zn(t)ll&]
+ 2 ( F ( ~ " ( t ) ) , h u ~ ( t ) ) ~=, d0,t
104
U.Manna, J . L. Menaldi, and S. S. Sritharan
integrating in 0 5 t 5 T we have, IIJilu"(T)lI;2
+ 911zn(T)I122+ 2
1
T
0
(F(un(t)),hun(t)),2dt
= IIJilu"(~)11:2
+ 911zn(0)11~2.
Hence -2
LT
( F ( un( t ) ),hu"(t)),,dt
=
IIJilU"(T)IIE2
- IIfiu"(o)11:2
f gllzn(T)1122
- 911Z"(O)1122.
(41)
Considering the fact that the initial conditions u"(0) and z"(0) converge to u(0) = u g and z(0) = z0 respectively in L2, and the lower-semi-continuity of the IL2-norm, we deduce
2 IIJilu(T)Il& +gllz(T)1122
1
- I I ~ u ( o ~ l l : 2-
g11z(0,1122
T
= -2
(Fo(t),hu(t)),,dt.
(42)
Here notice that from the equation (8) and the monotonicity property of the nonlinear operator B(.),i.e. from Lemma 3.3, we have (F(un(t)) - F(v(t)),hu"(t) - hV(t)),2 L 0.
(43) Multiplying both sides of (43) by 2, integrating in 0 5 t 5 T and rearranging the terms we have
lT
(2F(v(t)), hv(t) - h W ) ) , Z d t (2F(un(t)), hv(t) - hun(t))L2dt
Stochastic Analysis of Tidal Dynamics
Let us consider v := u L2(0, T ;H i p ) ) . Then we have
+ Xw with X
>0
105
and w E L"(0,T;JL2(0))
n
Dividing by X on both sides of the inequality above, and letting X go to 0, one obtains
1''
(F(u(t))- Fo(% hw(t)),,dt
2 0.
Since w is arbitrary and h = h(x) is a positive, bounded and continuously differentiable function, we conclude that Fo(t) = F ( u ( t ) ) .Thus the existence of a solution of the deterministic tide equation (14)-(16) has been proved. 0 4. Stochastic Tide E q u a t i o n
Let us consider the tide equation subject to a random (Gaussian) term i.e., the forcing field f has a mean value still denoted by f and a noise denoted by G. We can write (to simplify notation we use time-invariant forces) f(t) = f ( z , t ) and the noise process G ( t ) = G ( x , t ) as a series dGk = xkgk(x,t)dwk(t), where g = ( g l , g z , . . . ) and w = ( ~ 1 ~ ~..)2 , . are regarded as 12-valued functions in x and t respectively. The stochastic noise process represented by g(t)dw(t) = gk(z, t ) d w k ( t , ~ )is normal distributed in W with a self-adjoint trace-class co-variance operator denoted by g2 = g2(t) and given by
We interpret the stochastic tide equations as an It6 stochastic equations in variational form
( (i+ Div(hu), C)LZ = 0, in (0, T ) ,with the initial condition
106
U.Manna, J . L. Menaldi, and S. S. Srithamn
for any v in the space
G(0)and any < in L 2 ( 0 ) .
Proposition 4.1 (energy estimate). Let
for any 0 5 t 5 T and K is a positive constant and the constant C depends on the coefficients and the norms [If IILZ(O,T;W-I), ~ ~ w O I I L Z ( ~ ; L Z ( O , T ; W ~ ) ) , IIU(0)llL~and 1140)llL2. Proof. It is straightforward to see that (2) and (8) yield the energy estimate (5). Next we take the first equation of (2) by replacing v by u and proceed in the same way as in Proposition 3.4 to get the estimate similar t o (27)
Stochastic Analysis of Tidal Dynamics
Similarly we consider the second equation of (2) by replacing the estimate like (30)
107
by z to get
Now let us set
r 2g2
212
K = max(1 + M + -,++M}. & a a Then summing up (7) and (8) and rearranging the terms we have
Taking the sup norm in [0,TI and then taking the mathematical expectation we have
108
U.Manna, J . L. Menaldi, and S. S. Sritharan
Now by means of martingale inequality, we deduce
Notice that, by the assumption on wo in the proposition, we have
[llwo
( t )/ l L Z ( 0 , T ; L 4 ) ]5 CZE
[llwo(t)
llLZ(O,T;B~)]5 c3,
where C2 and C3 are positive constants. Applying (11) in (lo), rearranging the terms and finally using the Gronwall's inequality we get the desired estimates (6). 0 Now we deal with the existence and uniqueness of the SPDE and its finite-dimensional approximation.
Proposition 4.2 (uniqueness). Let u be a solution of the stochastic tide equation (2) with the regularity
{
u E L~(R;CO(O,T;L~(O))
n~~(o,T;@(u))),
z E L2(R x 0 x (O,T)),
(12)
and let the data f , g, uo and zo satisfy the condition
{
f E L2(0,T;W-1(0)), g E L2(0,T;C,(lL2(0))), uo E L2(0), zo E L 2 ( 0 ) .
(13)
If v in L2(R;Co(O,T ,IL2(0)) n L2(0,T ,IHIA(0)))is another solution of the stochastic tide equation (2), then [IIU(t) - v(t)112z+IIZ(t)
-
w122]e-Kt
I IlU(0) - v(O)llEz + IlZ(0) - ~(0)1122,
(14)
Stochastic Analysis of Tidal Dynamics
109
with probability 1, for any 0 5 t 5 T and K is a positive constant.
Proof. Indeed if u and v are two solutions then w = v - u solves the deterministic equation d -w(t)+Aw(t)+gV(.z(t)-Z(t)) = B(v(t))-B(u(t)) in IL2(0,T; IK1(O)). dt Thus the proof of uniqueness follows directly from Proposition 3.6, with probability 1. 0 If agiven adapted process u in L2(R;L"(0,T;IL2(0))nL2(0,T;IHIA(0))) satisfies d(u(t),V)L2
Wt),V b d t + (g(t),V)ILZdW(t),
(15) for any function v in HA(0) and some functions F in L2(0, T; W-'(O)) and g in L2(0,T;C2(L2(0))),then we can find a version of u (which is still denoted by u) in L2(R;Co(O,T; L2(0)))satisfying the energy equality =
d l l ~ ( t , l l 2= ~ [2(F(t),U(t))LZ
+ n.(g2(t))]dt + 2(g(t), U(t))LZdW(t)
(16)
see e.g. Gyongy and Krylov,lg and Pardoux.20
Definition 4.1. (Strong Solution) A strong solution u is defined on a given filtered probability space (R, F,Ft,P ) as a L2(R;L"(0, T ;lL2(0)) n L2(0,T; IHIA(0))n Co(O,T; L2(0))) valued function which satisfies the stochastic tide equation (2) in the weak sense and also the energy inequality (6). Proposition 4.3 (2-D existence). Let f , g, uo and zo be such that
i
f E L2(0,T;H-1(0)), g E L2(0,T;C2(IL2(0))), uo E P ( O ) , 20 E L 2 ( 0 ) .
(17)
Then there exist adapted processes u(t, x, w ) and z(t, x,w ) with the regularity
{
u E L2(0;Co(O,T;IL2(0)) n L2(0,T ;W;(O))), (18)
z , t E L2(R;L2(0,T; L2(0)))
satisfying the stochastic tide equation (2) and the a priori bound (6).
Proof. Consider a finite dimensional Galerkin approximation of the stochastic tide equation. Let us denote F(u) := A U
+ B(u)
-
f,
110
U. Manna, J . L. Menaldi, and S. S. Sritharan
where the operators A and B are defined by (8) and (9) respectively. Then dun(t)
+ F ( u n ( t ) ) d t+ gVzn(t)dt = g(t)dw(t).
Then using the a priori estimates ( 6 ) , it follows from the Banach-Alaoglu theorem that along a subsequence, the Galerkin approximations { un} have the following limits: un
zn
--
u weakly star in L ~ ( Q L ~ ( O , T ; I L ~ (n O ~ ~) )( o , T ; i @ ( O ) ) ) ,
L2(R;L2(0,T;IL2(0))), Fo weakly in L2(fl;L2(0,T;M[-1(0))),
z weakly in
F(u")
where u has the It6 differential du(t)
+ Fo(t)dt + gVz(t)dt = g(t)dw(t)
in L 2 ( QL2(0,T ;M[-'(O))),
and the energy equality similar t o stochastic version of (32) holds, i.e.,
Since the Galerkin approximations {u"} satisfy the energy equality
Integrating between 0 5 t 5 T and taking the mathematical expectation we have
Considering the fact that the initial conditions u"(0) and z n (0 ) converge to u(0) and z ( 0 ) respectively in IL2, and the lower-semi-continuity of the
Stochastic Analysis of Tidal Dynamics
L2-norm, we deduce
Rearranging the terms we have
111
112
U.Manna, J . L. Menaldi, and S. S. Sritharan
Dividing by X on both sides of t h e inequality above, and letting X go t o 0, we obtain
Since w is arbitrary and h = h(x) is a positive, bounded and continuously differentiable function, we conclude t h a t Fo(t) = F ( u ( t ) ) .Hence the existence of a strong solution of the stochastic tide equation (2) has been proved.
0
Acknowledgments First and third authors gratefully acknowledge the support from Army Research Office, Probability and Statistics Program (grant number DODARMY1736). References 1. 2. 3. 4.
5. 6. 7.
8. 9. 10. 11.
12.
13.
14.
H. Lamb, Hydrodynamics, Dover Publications, New York, 1932. I. Newton, Philosophiae Naturalis Principia Mathematica, 1687. C. Maclaurin, De Causci Physic6 Fluxus et Refluxus Maris, 1740. P. S. Laplace, Recherches sur quelques points du systkme du monde, Mem. Acad. Roy. Sci, Paris, 88 (1775), 75-182. W. Thomson and P. G. Tait, Treatise of Natural Philosophy, Vol. I, 1883. H. Poincark, Lecons de Me'canique Celeste. 3. The'roe des mare'es, Cauthier Villars, Paris, 1910. G. I. Marchuk and B. A. Kagan, Ocean tides. Mathematical models and numerical experiments, Pergamon Press, Elmsford, NY, 1984. G. I. Marchuk and B. A. Kagan, Dynamics of Ocean Tides, Kluwer Academic Publishers, Dordrecht / Boston / London, 1989. J. Pedlosky, Geophysical Fluid Dyanmics I, 11, Springer, Heidelberg, 1981. V. M. Ipatova, Solvability of a tide dynamics model in adjacent seas, R U M . J . Numer. Anal. Math. Modelling, 20, No. 1 (2005), 67-79. J. L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes Equation, Appl. Math. Optim., 46(2002), 31-53. U. Manna, J. L. Menaldi and S. S. Sritharan, Stochastic 2-D NavierStokes Equation with Artificial Compressibility, Communications on Stochastic Analysis, 1,No. 1 (2007), 123-139. S. S. Sritharan and P. Sundar, Large deviations for two dimensional NavierStokes equations with multiplicative noise, Stochastic Processes & Their Applications, 116, No. 11(2006), 1636-1659. V. Barbu and S. S. Sritharan, m-accretive quantization of the vorticity equation, Semigroups of operators: theory and applications (Newport Beach, CA, 1998), 296-303, Progr. Nonlinear Differential Equations Appl., 42, Birkhuser, Basel, 2000.
Stochastic Analysis of Tidal Dynamics
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15. V. Barbu and S. S. Sritharan, Flow invariance preserving feedback controllers for the Navier-Stokes equation, J. Math. Anal. Appl., 255(2001), no. 1, 281307. 16. 0. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. 17. R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. 18. S. Kesavan, Nonlinear Functional Analysis. A First Course, Hindustan Book Agency, New Delhi, 2004. 19. I. Gyongy and N. V. Krylov, On stochastic equations with respect to semimartingales It6 formula in Banach spaces, Stochastics, 6 (1982), 153-173. 20. E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 6 (1979), 127-167.
114
ADAPTED SOLUTIONS TO THE BACKWARD STOCHASTIC NAVIER-STOKES EQUATIONS IN 3D P. Sundar Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803, USA E-mail: sundarOmath.1su. edu
H. Yin Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA E-mail: h y i n a m t u . edu The existence of adapted solutions to the backward stochastic Navier-Stokes equations in three-dimensional bounded domains is shown by using a monotonicity argument. The terminal condition is assumed t o be bounded. Uniqueness is proved by using finite-dimensional projections, linearized equations and suitable truncations.
Keywords: Backward stochastic Navier-Stokes equations; the It6 formula; Galerkin approximation; adapted solutions.
1. Introduction Backward stochastic differential equations (BSDEs, for short) are terminal value problems of stochastic differential equations that involve the It6 stochastic integral. The adapted solution of a BSDE is a pair of adapted processes wherein the second process specifies the noise coefficient of the equation, and the first process is the solution of the resulting backward equation. The following simple example explains the meaning of BSDEs. Let (0,F,{Ft},P ) be a complete stochastic basis on which a standard one-dimensional Wiener process {Wt} is defined, such that {Ft} is the natural filtration of {Wt},augmented by all the P-null sets in F. Let T denote the terminal time, and let [ be a square integrable .&-measurable random variable known as the terminal value or terminal random variable. Consider the following terminal value problem for
Adapted Solutions to Backward Stochastic Navier-Stokes
115
all t E [O,T]:
We are interested in finding an 3t-adapted solution u of (1).Such a problem is not solvable since u ( t ) = E, V t E [O,T]is the only solution and it is not &-adapted. Therefore, a reformulation of the problem is called for. A natural modification of the problem is to set u(t)as E((lFt)so that u is an adapted process and u ( T ) = E. However, such a process u doesn't satisfy the differential equation (1).Hence, we proceed to modify the equation (1) by taking advantage of the fact that u by the present definition is a square integrable .Ft-martingale. In fact, by invoking the martingale representation theorem, there exists an adapted square integrable process Z such that
E(ElFt)= 4 0 ) + That is,
u ( t )= u(0)+
1
t
Z(s)dWs.
t
Z ( s ) d W s and u ( T )= E .
Therefore in integral form,
~ ( t=) -
4'
i i ( ~ ) d W S'Jt E [O,T].
The reformulation of the terminal value problem (1) is thus given for t E [O,TI by
The solution of equation (2) consists in finding an adapted pair ( u ( . )Z, ( . ) )of processes subject to certain integrability conditions. The pair ( u ( . )Z(.)) , is known as the solution of the BSDE (2). Our aim in this paper is to solve the backward stochastic Navier-Stokes system in a bounded three dimensional domain with the strong assumption that the terminal value is bounded almost surely. The three-dimensional stochastic Navier-Stokes system in a bounded domain G with no-slip condition is given by du
+ {-vAu +
(U
. V)u}dt= {-Vp
v.u=o
+ f(t)}dt + ~ ( U) t ,dW(t)
116
P. Sundar and H. Yin
with u(t,z) = 0 for z on the boundary of G, and u ( 0 , z ) = uo(z) for x E G. As explained in the next section, the above system can be written as an abstract evolution equation: du(t)
+ { ~ A u ( t+) B(u(t))}dt = f(t)dt + a ( t ,~ ( t )dW(t) )
(3)
with u(0) = UO.The backward stochastic Navier-Stokes equation corresponding to the equation (3) is given by the following terminal value problem: for 0 5 t 5 T,
{
du(t) = -VAu(t)dt
u(T) =
-
B(u(t))dt
<
+ f(t)dt + Z ( t ) d W ( t )
which should be understood in its integral form:
u(t) = E
+
T
{vAu(s)
+ B(u(s))
-
f(s)}ds -
LT
Z(s)dW(s)
The problem consists in finding an adapted solution {u(t),Z ( t ) } for 0 5 t 5 T in a suitable function space. A backward stochastic Navier-Stokes system arises as an inverse problem wherein the velocity at time T is observed and given, and the noise coefficient has to be determined from the given terminal data. A systematic study of backward stochastic differential equations was initiated by Pardoux and Peng, Ma, Protter, Yong, Zhou, and several other authors in a finite - dimensional setting. Ma and Yong' have studied linear degenerate backward stochastic differential equations motivated by stochastic control theory. Later, Hu, Ma and Yong2 considered the semi-linear equations as well. Backward stochastic partial differential equations were shown to arise naturally in stochastic versions of the Black-Scholes formula by Ma and Yong . The usual method of proving existence and uniqueness of solutions by fixed point arguments do not apply to the stochastic system on hand since the drift coefficient in the backward stochastic Navier-Stokes equation (BSNSE) is nonlinear, non-Lipschitz and unbounded. In the previous works cited above, the equations are driven by finite-dimensional Wiener processes whereas the driving process is a Hilbert-valued Wiener process in this paper. Much of the paper owes itself to a surprising a priori estimate of the L"([O,T] : H ) norm of u(t) that holds almost surely. The proof of the uniqueness of solutions is wrought by establishing the closeness of (a) solutions of finite-dimensional projections of the BSNSE, (b) solutions of a linearized projected BSNSE, and (c) finite-dimensional projections of solutions of the BSNSE. The present paper is a partial extension of our work
Adapted Solutions to Backward Stochastic Navier-Stokes
117
on the two-dimensional backward Navier-Stokes equations wherein the estimates on the coefficients are much nicer.3 For the stochastic Lorenz model, the solvability of the backward system is known.4 Though such a system is finite-dimensional, the coefficients in the model share certain properties of the terms in the three-dimensional Navier-Stokes equations. In section 2, the setup of the problem, the function spaces, and the background results are presented. The a priori estimates for the solutions are proved in section 3. The existence of solutions to the backward stochastic Navier-Stokes equations is shown by the Minty-Browder monotonicity argument in section 4 for bounded terminal data. Pathwise uniqueness of solutions is established in section 5.
2. Preliminaries Let V be the space of three-dimensional vector functions u on G which are infinitely differentiable with compact support strictly contained in G, satisfying V . u = 0. Let V, denote the closure of V in W",' for all real a. In particular, let H = Vo and V = V1. The notation L 2 ( G ) ,W;12(G)etc. would mean vector functions with each coordinate in L 2 ( G ) ,W;l2(G)etc. The following characterizations of the spaces H and V are well-known:
H = {u E L 2 ( G ): V . u = 0 and u . n = 0) and V
=
{u E W i ' 2 ( G :) V . u
=0
and u
=0
on d G }
where W,'"(G) = {u E L2(G) : Vu E L 2 ( G ) , u J a c = 0}, and n is the outward normal. Let V' be the dual of V. We have the dense, continuous and compact embedding (see Ref. 5): V C, H = H' C, V'. Let D(A) = W 2 ) 2 ( Gn) V . Define the linear operator A : D(A) -+ H by Au = -Au. Since V = D(A1/2),we can endow V with the norm JIuIJv= J J A 1 / 2 ~The J J ~V-norm . is equivalent to the W1i2-norm by the Poincare inequality. The operator A is known as the Stokes operator and is positive, self-adjoint with compact resolvent. The eigenvalues of A will be denoted by 0 < A1 < A2 5 . . . , and the corresponding eigenfunctions by e l , e2 , . . . . The eigenfunctions form a complete orthonormal system for H . Define b(., ., .): V x V x V R by --f
118 P. S u n d a r a n d H. Yin
using which we can define B : V x V -+ V' as the continuous bilinear operator such that < B(u,v), w >= b(u,v,w) for all u, v, w E V . B(u) will be used to denote B(u, u). The external body force f(t) is assumed to be V'-valued for all t. Let { W ( t ): 0 5 t 5 T } be an H-valued {Ft}-adapted Wiener process with covariance given by Q , a positive, symmetric, trace cIass operator on H . Let LQ denote the space of linear operators 2 such that ZQ112is a HilbertSchmidt operator from H to H . Define the norm on the space LQ by
121iQ= tr (ZQZ*). The noise coefficient 2 takes values in LQ such that E I Z ( S ) d(s~< o~o . The stochastic Navier-Stokes equation can be written in the abstract evolution equation setup (see Ref. 5 ) for bounded domains. The theory of Wiener processes on infinite-dimensional spaces can be found in the book by Kuo.' Let P be the orthogonal Leray projector:
P : ( L 2 ( G ) ) 3+ H so that ( L 2 ( G ) ) = 3 H
+H I
where H I can be characterized by
H I = {g E ( L 2 ( G ) ) 3: g
= V h where
h E W1)2(G)}.
(1)
We can write the Navier-Stokes system as
{
+
du(t)= - ~ A ~ ( t )-d B t ( ~ ( t ) ) d tf(t)dt u(0) = uo(z).
+ Z(t)dW(t)
If the terminal value is given as u(T) = E, then
{
+
d u ( t ) = - ~ A ~ ( t )dB t ( ~ ( t ) ) d tf(t)dt
+ Z(t)dW(t)
u ( T )= E
(2)
is known as the backward stochastic Navier-Stokes equation.
Remark 2.1. In this paper, we consider the particular case when
Qek
= q k e k for all
k, and { b k ( t ) } be a sequence of iid Brownian motions in R.The Wiener 00
process { W ( t ) }is taken as W ( t ) = x& b k ( t ) e k k=l
00
with
qk
< 03.
k=l
For any Banach space K, let Lc(R; L p ( 0 , T ; K ) )be the set of all {F}t>oadapted K-valued processes X(.) such that E IlX(t)Ilgdt < 03.
Adapted Solutions to Backward Stochastic Navier-Stokes
119
Definition 2.1. A pair of Ft-adapted processes (u(t),Z ( t ) ) is called a solution of the backward stochastic differential equation (2) if the following holds:
[+
(1) ~ ( t=) in V',
4"
(vAu(s)+B(u(s))-f(s))ds-
LT
Z ( t ) d W ( t )holds P-as.
(2) u(.) E L$(R; L 2 ( 0 , T ;V))n Lm(O,T;L$(R; H ) ) . (3) E L$(R; L2(0,T;L Q ) ) .
z(')
The following estimates are well-known, and the reader is referred to Temam.5
Lemma 2.1. Let G c R3 be a bounded domain of class C2. If u E V n H 2 ( G ) , then B ( u ) E H c L 2 ( G ) and llB(u)llH
5 cGllul\hAUIIL
Lemma 2.2. Under the hypothesis of the above Lemma, for any u, v E V , there exists a positive constant CG such that
11B(u)11 V' 5
1
3
CGIIUIIgIIUllt,
Remark: For any u, v E V , it follows that
3. A Priori Estimates
Let PN : H + H N be the projection where H N =span{el,e2,... , e ~ } . The space H N being finite-dimensional, it follows that Notice the fact that VN = H N = Vh for all N . First we restrict the domain of A and B to H N and still denote it by A and B. This is an abuse of notation, However, it simplifies the notation in this section. Define
AN := PNA and B N := P N B The projected backward Navier-Stokes equation is defined by
+
d u N ( t ) = --vANuN(t)dt- B N ( U N ( t ) ) d + t fN(t)dt ZN(t)dWN(t) U N ( T ) ="
(1)
120
P. Sundar and H. Yin
N
for 0 5 t
5 T , where f N
= P N f , W N ( t )=
&$‘@)el,,
EN
=P
N ~and ,
k=l
Z N ( t ): [O,T]x R * L ( H N , H N ) . Proposition 3.1. Let the external body force f E L2(0,T ;V’), and the terminal value satisfy the bound Il
<
Proof. The proof follows along the same steps as in the two-dimensional case given in Ref. 3. We outline it here. An application of the multidimensional Itti’s formula to IIuN(t)ll$yields T IIUN(t)ll$
=
1l”IlL - 2
s (-A
N N
u (S),UN(S))V’,VdS-
t
T
lT
IIZN(s)ll?,,ds
where (ZN)* is the adjoint of Z N , and the duality pairing (., .)vt,v is just the H-norm. For 0 < r 5 t 5 T , we take the conditional expectation to obtain
+
6’
EFrlluN(t)l/&EFT
IIZN(s)lliQds
T
= E F r ~ ~+~2NE~ F r~l & (.ANuN(s),uN(s))V,,Vds - 2E3.. l T ( f N ( s )u, N ( s ) ) V / , v d s
By the Gronwall inequality, one can deduce that
Omitting the first term on the left hand side of the above inequality and taking expectation on both sides, one gets
Adapted Solutions t o Backward Stochastic Navier-Stokes
121
which is bounded by a constant, say, K‘. Taking r to be t and omitting the last two terms on the left hand side of inequality ( 3 ) ,one obtains IIuN(t)JJ& 5 K’ Hence if KO= 3K’, then
If the external body force f has higher order of integrability, one would expect better bounds on the solution u N .One such result is given below which can be proved along a similar line of arguments used in the proof of the above proposition. Corollary 3.1. Let the conditions in Proposition 3.1 hold. Additionally, we assume that f E L4(0,T ;V’). Then there exists a constant K1, independent of N , such that
H ) n L$(R;L 2 ( 0 , T ;V)). i.e. { u N ( t ) }is bounded in L”(s1 x [O,T]; Likewise, if the V-norm of E is bounded a s . , better bounds on u N accrue asin the following corollary.
Corollary 3.2. Let the conditions in Proposition 3.1 hold. Additionally, let ll[ll$ 5 K for almost all w E and some constant K . Then there exzsts a constant K2, independent of N , such that
For every M E
N,let LM be a smooth function on R1satisfying
LM(2) =
i‘ 0 0
I L M ( x )5 1
if 1x1 < M if 1x1 > M
+1
otherwise
Proposition 3.2. The operator B N truncated by using L M is Lipschitz
continuous. That is, IILM(IIxllv)BN(x)
for
- LM(IIYIIV)BN(y)llHI C N , M l l x
any x,y E H N and M E N.
-
YllV
122
P. Sundar and H. Yin
Adapted Solutions to Backward Stochastic Navier-Stokes
123
Proposition 3.3. Under the conditions of Proposition 9.1,the projected system (1) admits a unique adapted solution (uN(t),ZN(t))for each N and
Proof. First, we introduce some notation. For 1 _< i 5 N , suppose that i$'(t)
:= (uN(t),ei)H and let UN(t):=
P. Sundar and H. Yin
124
Define Z N ( t )as
and
l&"(t)
=
( "r)
where {bj(t) : 1 5 j 5 N } are N independent
b"t)
standard 1-dimensional Brownian motions. Thus for any N E N,the projected system 1 is equivalent to
i
dUN(t) =
-vA%N(t)dt
UN(T)=
r^"
- BN(UN(t))dt
+ZN(t)dt + P ( t ) d W ( t ) (5)
Define the associated truncated system as follows:
i
dUNJqt) =
-vANUN,M(t)dt - L M ( 11 UNIM ( t )11 V ) B N (UN,M( t ) ) d t
UNiM(T)=
iN
+I.N(t)dt
+2 N y t ) d f i y t )
(6)
+
Let h N , M ( t , z )= - V A N S - L M ( I I ~ ~ ~ ~ ) B N Z N( ( ~t ) ). Clearly, h N i M ( t , z is ) Lipschitz on [O,T]x RN.Thus we know that (6) admits a unique adapted solution (UN>M(t), Z N ) M ( t )E) M[O,T](see Yong and Zhou ~ . 3 5 5 ~where ) , M[O,TI equipped with the norm
and here 1 2 1 '= tr(ZZT). Exactly as in the proof of Proposition 3.1, it can be shown that
sup
0ltST
I I ~ ~ ) ~ ( t )5l KO l & for a constant KOindependent of N , and M . Therefore, sup IluNsM(t)ll$ 5 KO,N for a constant K ~ , N independent of M. Thus
0StST
for M > K ~ , NL, ~ ( I I u ~ y ~ ( t ) l=l v1, ) and the solution of (6) is also the solution of (5). This allows us to complete the proof. 0 4. Existence of Solutions First of all, let us prove some simple results.
Adapted Solutions t o Backward Stochastic Navier-Stokes
125
The next corollary immediately follows for u, v which are functions of time.
$ Jot
Corollary 4.1. Let r l ( t ) = Ilu(s)11$ds and r2(t) = all u,v E L$(R; L 4 ( 0 , T ;V ) ) .Then
% Jot
\Iv(s)I~$~s
for
+
( ~ A w B(u) - B(v)
1 + -i'i(t)W,W)Vf,V 20 2
i
=1,2
where w = u - v. Lemma 4.3. For any u, v, and w E V , we have 1
l(B(u)-B(v), W ) V ) , V l 5
3
~~ll~ll~ll~ll~+ll~ll~ll~ll~~ll~-
126
P. Sundar and If. Yin
Proof.
Thus
The existence of solutions to the backward Navier-Stokes equation (2) is proved in the next theorem by using the estimates obtained above.
Theorem 4.1. Assume that ll(/l$ < K for some constant K, P-a.s., and f E L2(0,T;V’).Then the Navier-Stokes equation (2) admits an adapted solution (u(t),Z(t)) E L”(R x [O,T]; V ) x L%(R;L2(0,T ;L Q ) ) . Proof. We will prove the theorem in the following steps. Step 1: First, let us find some bounds for the projected system. By Corollary 3.2, we know that {uN(t)}F=l is bounded in L“(R x [0,TI;V). Hence {uN(t)}TT1 is bounded in L’$(R; L 2 ( 0 , T ;V’)). There exists a constant C , such that IIAuIIv, 5 C((ullv for all u E V . Thus from (a), we know that {ANuN(t)}F=l is bounded in L’$(R; L2(0,T;V’)). By Lemma 2.2 and Proposition 3.1, IIBN(UN(t))llV/I CGK$IIUN(t)ll~. Since {uN(t)}F=l is bounded in Lm(R x [O,T];V), {BN(uN(t))} is bounded in Lg(R; L2(0,T;V’)). By Proposition 3.1 it follows that {ZN(t)} is bounded in
L%(R;L2(0,T ;L Q ) ) . Step 2: Clearly we have the following strong convergence:
IN
4
E
and f N ( t ) + f ( t )
in Lg(R; L2(0,T ;V’)).
Adapted Solutions to Backward Stochastic Navier-Stokes
127
Since L$(R; L2(0,T ;V ’ ) )and L%(R;L2(0,T ;L Q ) ) are Hilbert spaces, and from the results in Step 1, there exist u(t), Y ( t ) ,G ( t ) , Z ( t ) , and { N ~ } ~ such ! l , that
u N k ( t.% ) u(t), v A N k u N k (% t ) Y ( t ) ,and B N k ( u N k ( t% ) ) G(t) in L$(R; L2(0,T ;V ’ ) ) ,and
For every t , we define
Lt : L$(R; L 2 ( 0 ,7‘; L Q ) )+ L$(R; L2(0,T ;V’))
M(t)H
lT
M(s)dW(s).
The operator Ct is bounded and hence maps the weakly convergent sequence {ZNk(t)}glto a weakly convergent sequence Z N k ( s ) d W N k ( s ) } g lwith limit s,’Z(s)dW(s).
{L’
Here we have used the fact that letting ZN(t)(ei)=Ofor i > N . Likewise,
JTZ “ ( s ) d W ( s ) = Z~ N~ ( s ) d W N ( s by)
l
T
( v A N k u N k +BNk(uNk(s)))ds (s) -5
( Y ( s +) G ( s ) ) d s
in L$(R; L 2 ( 0 , ~ V’)). ; Let
J”‘(t)
=tNk +
I’
{ v A N k u N k+ ( sB) N k ( u N k ( s-fNk(s)}ds ))
Then uNk( t )= F N k( t )P-a.s. for every k and they both weakly convergent in L$(R; L2(0,T ;V’)).Hence the weak limits agree P-as, i.e.,
u(t) = 6
+
l
T
+
( Y ( s ) G(s) - f(s))ds -
(1)
Step 3: Now let us prove the existence. From now on, we will denote the index of those convergent subsequences by N again, instead of Nk.
128
P. Sundar and H. Yin
% Jot
Let r ( t ) = Ilv(s)ll$ds for any v(t) E Lm(R x [O,T]; V ) .Apply It6's formula to e-r(t)IIuN(t)I)$,we get
Now by taking expectation, we get
= - 2E
I
T
1 e-'(t)(vANuN(t) BN(uN(t)) T+(t)uN(t), uN(t))v/,vdt
+
+
Likewise, equation (1)and the It6 formula applied to e-T(t)llu(t)ll$yield
Taking the limit, (2) becomes
Adapted Solutions t o Backward Stochastic Navier-Stokes
Combining
(a), (3) and
129
(4), we get
T
+1
lim { 2 E l e-'(t){(vANuN(t) + BN(uN(t)) 2+(t)uN(t), uN(t))v),v
N+CC
Hence
LT
e-'(t){(vAuN(t)
-EL
1 + B(uN(t))+ 5+(t)UN(t),V(t) - uN(t))v',v
T
lim N-CC
e-T(t)llZN(t)lliQdt5 -E
I'
e-T(t)IIZ(t)lliQdt
(7)
Taking the limit and by ( 5 ) , and ( 7 ) , one obtains
1 e-T(t)(Y(t) G ( t ) 2+(t)u(t),v(t)- u(t))vt,vdt
ElT
+
+
+ B(v(t)) + ~1+ ( t ) v ( tv(t) ) , - u(t))v),vdt (8) Now we take v(t) = u(t) + Aw(t) for any w(t) E L"(R x [0,TI;V ) and e-'(t)(vAv(t)
x > 0.
130 P. Sundar and H. Yin
Therefore
and (8) becomes
E
FE
e-'(t)(Y(t)
+ G(t)- vAu(t) - B(u(t) + Xw(t)), Xw(t))vr,vdt
lT
e-'(t)(XvAw(t)
X + --+(t)w(t), Xw(t))vj,vdt 2
It is easy to obtain
ElT
+
e-T(t)(Y(t) G(t)- vAu(t) - B(u(t)),w(t))vl,vdt T
e-'(t)(vAw(t) Letting X we get
E
4
1 + B(w(t),u(t)) + s+(t)w(t), w(t))vr,vdt
0, since the right hand side of the last inequality is finite,
IT
e-'(t)(Y(t)
+ G ( t )- vAu(t)
-
B(u(t)),w(t))v,,vdt5 0
for all w(t) E Lm(R x [O,T]; V). Hence Y(t) G ( t ) = vAu(t) B(u(t)) P-as. and this completes the proof of the existence of the solution. 0
+
+
5 . Uniqueness of Solutions
Theorem 5.1. Under the conditions of Theorem 4.2, the adapted solution of (2) is unique in the space L"(R x [O,T];V) x L'$(0;L2(O,T;L~)).
Proof. Let (u(t),Z ( t ) ) , and (v(t),a(t)) be in L"(R L'$(R; L2(0,T;LQ)) be two solutions of (2).
x [O,T]; V)x
Adapted Solutions t o Backward Stochastic Navier-Stokes
131
Step 1: For every N E N,let p N ( t ) = PNV(~), and we define the following finite dimensional system
dx ( t )= - v A N X N ( t ) d t
+ f N ( t ) d t+ Y N ( t ) d W N ( t )
- BN(pN(t))dt
X y T ) = EN N (1) Since AN is Lipschitz, it is easy to see that (1) admits a unique adapted solution ( x N ( t )y, N ( t ) ) . Since llEllv is uniformly bounded, similar to the Corollary 3.2 , one can obtain {
for some constant K , independent of N . Step 2: It is clear that p N satisfies
+
i
d p N ( t ) = - v A N p N ( t ) d t - BN(v(t))dt f N ( t ) d t
+ PNo(t)dW(t)
P N m = EN (3)
Let wN(t)= x N ( t )- p N ( t ) , then
i
dwN(t)= - v A N W N ( t ) d t - (BN(pN(t)) - BN(v(t>))dt
+ ( Y N ( t-) PNa(t))dW(t)
WN(T) = 0
where, for convenience, we set Y N ( t ) ( e k=) 0 for all k > N . The It6 formula yields T
First,
P. Sundar and H . Yin
132
Applying the above inequality to (4), and by using a Gronwall argument, one obtains
upon letting T = t . Since V(S) E L"(Q x [0,TI;V ) ,we apply the Lebesgue dominated convergence theorem to get that as N -+ 00,
(6)
Step 3: Let W N ( t )= u N ( t ) x N ( t )and the difference of (1) and (l),we get
Z N ( t )= Z N ( t )- Y N ( t )Take .
i
d W N ( t ) = -vANWN(t)dt- ( B N ( u N ( t ) ) - B N ( p N ( t ) ) ) dZtN+( t ) d W ( t )
W N ( T ) = 0.
(7)
An application of the It6 formula to llWN(t)lltyields
dllWN(t)II$ = - 2(vANWN(t),A Nw - N (t))~dt
- 2(BN(uN(t))- B N ( pN ( t ) ) , A N W N ( t ) ) ~ d t
+ 2 (ZN(t)dWN( t ),ANW N( t ) )+~tr[ANZN(t)Q(Z N( t ) *) ] d t and
Adapted Solutions to Backward Stochastic Navier-Stokes
Thus for 0 5
T
133
5 t 5 T , (8) becomes
Step 4: It is clear that we have the integrability to apply Gronwall's inequality. Thus one gets
134
P. Sundar and H. Yin
Letting r = t and applying Lebesgue dominated convergence theorem, we
get
Since u N ( t )- p N ( i )
u ( t ) - v(t), we have
=O Thus we have shown that ( u ( t )Z(t))=(v(t), , a@)) P-as. for all t E [0,TI. T h e proof is thus completed. 0
References J. Ma and J. Yong, Stoch. Proc. and Appl. 7 0 , 59 (1997). J. M. Y. Hu and J. Yong, Probab. Theory Relat. Fields 123,381 (2002). P. Sundar and H. Yin (2007), preprint. P. Sundar and H. Yin, Comm. on Stoch. Anal. 1,473 (2007). R. Temam, Navier-Stokes Equations (North-Holland, New York, 1979). H.-H. Kuo, Gaussian Measures in Banach Spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1975). 7. J. Yong and X. Y. Zhou, Stochastic Controls (Springer-Verlag, New York, 1999).
1. 2. 3. 4. 5. 6.
135
SPACES OF TEST AND GENERALIZED FUNCTIONS OF ARCSINE WHITE NOISE FORMULAS Abdessatar Barhoumi Department of Mathematics Higher School of Sci. and Tech. of Hammam-Sousse Sousse University, Sousse, Tunisia E-mail: abdessatar. barhoumiOipein.rnu. tn Anis Riahi and Habib Ouerdiane Department of Mathematics Faculty of sciences of Tunis University of Tunis El-Manar 1060 Tunas, Tunisia E-mail: habib.ouerdianeOfst.rnu.tn E-mail: alriahiQyahoo. f r In this paper white noise calculus for the arcsine process is investigated as an example of a non-Uvy white noise. Namely, we consider, on an appropriate space of distributions, X ’ , an arcsine white noise measure, A, and construct of test and generalized funca Gel’fand triple F ( X ) c L 2 ( X ’ , A ) c F*(X) tions. By using the S-transform, we prove a general characterization theorems for arcsine white noise distributions and white noise test functions. Our main results are theorems 4.1 and 4.2. Keywords: Arcsine white noise, Chaos decomposition, Characterization theor em, S-transform .
1. Introduction The main purpose of the present paper is to present the construction and description of a new Gel’fand triple of test and generalized functions with respect to the so-called arcsine white noise measure. This is intended as a continuation of our previous paper’l where a study on white noise functionals of general non-LQvyclass is started. In fact, our development aims at realizing Hida’s idea of white noise at a non-LQvy level and at investigating applications to infinite dimensional harmonic analysis and quantum physics.
136
A . Barhoumi, A . Riahi, and H. Ouerdiane
The white noise theory for Brownian motion was first introduced by T. Hida, in the Gaussian case, in his celebrated lecture notes Ref. 2. Later, Kubo and Takenaka reformulated Hida’s theory by taking different test functions space and using the S-transform as m a ~ h i n e r yFor . ~ non-Gaussian white noise analysis, Y. ItG4 constructed a Poissonian counterpart of Hida’s theory and Kondratiev et al.5-7 established a purely non-Gaussian distribution theory in infinite dimensional analysis by means of a normalized Laplace transform. (See also Ref. 8). In Refs. 9,10, a theory of the Lkvy white noise analysis for the general case of L6vy processes is developed. In view of these different developments, it is natural to ask if a white noise theory can also be developed for other processes. In particular, since Gegenbauer processes are becoming increasingly important in many applications, it would be of interest to have a white noise theory for such processes. In fact, Gegenbauer processes are long memory processes and are characterized by an unbounded power spectral density at zero. From this last singularity property, one can observe that a natural tool to analyze such processes appears to be a generalization of the wavelet transform in quite a different way from the L6vy processes. Our aim is to derive a general structure of Gegenbauer white noise analysis as a counterpart class of the non-Lkvy white noise. In our present paper, we focus on the particular case of the arcsine process. The contents of the present paper are as follows. Section 2 is devoted to a quick review of arcsine white noise functionals. In Section 3, we use the second quantization procedure to construct a nuclear triple of test and generalized functions. Finally, in Section 4, we define the S-transform as a main tool in working in these spaces. The S-transform serves to prove characterization theorems for test and generalized functions in terms of analytical functionals with an appropriate growth condition.
2. Arcsine White Noise Space 2.1. Arcsine space i n one dimension Let pc be the arcsine distribution with parameter c E EX given by
where, 60 is the Dirac measure concentrated on the point 0 and 1, is the indicator function of a subset A of R. For c = 1, we have the standard
Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
137
arcsine distribution :
From Ref. 11, we recall the following useful background. Apply the Gram-Schmidt orthogonalization process to the sequence { 1,IC, I C ~ , . . ,xn , . . . } to get a sequence {Tn; n = 0 , 1 , 2 , . . } of orthogonal polynomials in L 2 ( p ) .Here To(z)= 1 and Tn is the Chebyshev polynomials of the first kind given by +
where for y E
R,p E N*, we have
(p’)
- 7(7-1).**(7-P+1)
P!
It is well-known that these polynomials Tn satisfy the recursion formula x T n ( z )= Tn+l(z)+ w n T n - l ( ~ ) , with 1,
ifn=O
1/2,
ifn=1
1/4,
ifn22
Furthermore, the generating function of the arcsine distribution is given by
The Bessel function of the first kind of order a
>
-+
can be defined by
where r(.)is the Gamma function. Moreover, we have the following PoissonMehler integral representation
Using Eqs. ( 2 ) and (3), the Fourier transform of the arcsine distribution, in Eq. ( l ) , is given by
138 A . Barhoumi, A . Riahi, and H. Ouerdiane
2.2. Construction of the arcsine white noise space For simplicity, put I = ] - 1 , 1 [ . From the Favard theorem,15 one can easily obtain ~ ~ T n- ~ 2-2nf1. ~ ~Therefore, ~ ~ ~we, define p ~the corresponding Chebyshev functions X n ( x ) by
This gives an orthonormal basis { X n ; n = 0 , 1 , 2 , . . . } for H := L2(1,dx). Define the operator A, acting on an appropriate dense subset of H , by
d2 A = ( x 2- 1)dx2
+ 22-dxd - 4 ( ~2 22 -1) + 2.
Then the Chebyshev functions 'Hn are eigenvectors of A: 3 A X n = XnXn with X - - + n 2 , n = 0 , 1 , 2 , . . n-2 and, for any p > A - p is of Hilbert-Schmidt type. Now, for each p E R,define
i,
where 1 . 1 0 and ( . , . ) are , respectively, the norm and the inner product of H . For p 2 0, let X , be the Hilbert space consisting of all f E H with lflp < cm,and X - , be the completion of H with respect to I . I--p. Since A-l is of Hilbert- Schmidt type, identifying H with its dual space we come to the real Gel'fand triple12
x := n x p c H P20
U X - ~:= x'. P20
Being compatible to the inner product of H , the canonical bilinear form on X ' x X is denoted by ( . , . ) again. An application of the Bochner-Minlos t h e ~ r e m leads l ~ ~ us ~ ~t o the following.
Definition 2.1. The probability measure A on X ' of which characteristic function is given by
ei("+"PdA(w)= J,,((cp)), cp E X ,
sI
(4)
where (cp) = cp(x)dx,is called the arcsine white noise measure. The probability space ( X I ,B ( X ' ) ,A), B ( X ' ) being the cylinder a-algebra on X ' , is called the arcsine white noise space.
Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
139
Remark. For t E R', setting in (4) 'p = X l I t - , twhere + l , t- = min(0, t ) and t+ = max(0, t ) ,we obtain
L,
ei("+')dA(w) =
{
J,(Altl) if It1 < 1 if It1 2 1
J,(X)
This function coincides with the Fourier transform of the measure pltl if It1 < 1 and with the measure p if It1 2 1. On the probability space (XI, B(X'), A), the random variable X t := ( . , l ( t - , t + has l )an arcsine distribution. Moreover, taking the time derivative formally, we get X t ( x ) = x ( t ) . Thus the elements of X' are regarded as the sample paths of arcsine white noise and members of L2(X', B(X') are called quadratic integrable arcsine white noise functionals. So the family of random variables
x
=
xo=o
{Xt, t E R } ,
(5)
is called a white noise arcsine process. The image of the arcsine white noise measure A under the random variable Xt(.) is the arcsine distribution p if It1 2 1 and the arcsine distribution pltl if It1 < 1. Then, for t > s > 0, the characteristic function lE [exp{iX(Xt - X,)}] of Xt - X, has the following form :
1
J,(X(t - s)) if s, t E (0,1)
IE [exp{iX(Xt - X,)}] =
Jo(X(l
-
s)) if 0
J o ( 0 )= 1
< s < 15 t
if s, t
>1
We conclude that for s, t E ( 0 , l ) the random variable X t - X, has the arcsine distribution with parameter t - s. Furthermore, similarly as in the classical case, one can verify that the process X in (5) is a non-Lkvy process. This was our motivation for defining the arcsine L2-space as a space of nonLkvy white noise functionals. Let ( L 2 ) := L2(X',B(X'),A) be the real Hilbert space of square Aintegrable functions with norm denoted by 11 . 110. For n E N we denote by X@""the n-fold symmetric tensor product of X equipped with the Ttopology and by X$" the n-fold symmetric Hilbertian tensor product of X,. We will preserve the notation I . , 1 and I . I-, for the norms on X F n and X?;, respectively. In order to give the chaos decomposition of the white noise arcsine space ( L 2 ) ,we introduce the following definition of wick product.
140 A . Barhoumi, A . Riahi, and H. Ouerdiane
Definition 2.2.l For w E X' and n = 0 , 1 , 2 , . . . , we define the wick tensor product : w @ : ~as the continuous linear functional on X g n characterized by
and for any orthogonal vectors <1 , . . .,
+. +
(: wBn
:,
@
= (: ~
x and nonnegative integers n j ' s . . (: W B n k
@ : , <~ P n 1 ) .l
:,
The Fock space r ( H ) over H is defined as the weighted direct sum of the n-th symmetric tensor powers IT@'", n E N, +m
r(H) := @ 2 - 2 n + ' ~ g n . n=O --f
Thus r ( H ) consists of sequences f = (f"), E H S n and n E N, f
f ( l ) ,. . . )
such that, for any
+m
n=O 4
Theorem 2.1.l For each F E ( L 2 ) , there exists a unique sequences f = M (f(")),=, E r ( H ) such that +m
n=O --f
in the L2-sense. Conversely, for any f = (f(")),"=,
E
r ( H ) , ( 6 ) defines
a function on ( L 2 ) satisfying the isometry property
n=O
3. Arcsine Test and Generalized Functions Spaces Consider the space
P(X')of
c (:
continuous polynomials on X' :
n
'p,
p(w) =
k=O
w m k :,'p(k)),
'p(k) E
x Q w~ E, XI,n E N
Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
141
This space may endow with a topology such that P(X’) becomes a nuclear space. Therefore, if we denote by P*(X’) the dual space of P(X’) with respect to ( L 2 ) we get the following nuclear triple
P(X’) c ( L 2 ) c P*(X’). The bilinear dual pairing (( ., .)) between P*(X’) and P(X’) is then related to the inner product on ( L 2 )by ((PI
cp)) = ( K ’ ? j j ) ( L Z ) I
F
E
(L2)
I
cp E P(X’),
where ’?jj denotes the complex conjugate function of cp. Now, we consider the following family of norms M
n=O Cm
where cp(w)
= n=O
(:w@’
P(X’). For p , q E N,let F q ( X p ) P(X’)with respect t o ]I . Ilp,q. Then, we E
stands for the completion of define the space F ( X ) of test functions, from the projective system { F q ( x p ); PI 4? E W I by
F(X)= proj limFq(X,). P,P E
We recall from the general duality theorems12 that the strong dual F * ( X ) of the nuclear space F ( X ) is given by
F ( X ) = ind lim F-q(X-p), P,qE F4
where F - q ( X - p ) is the dual, with respect to ( L 2 ) ,of F q ( X p ) .The canonical bilinear form on F*(X) x F(X)will be denoted (( . , . )). The space F*(X) is the space of generalized functions and we obtain the Gel’fand triple
F(X)c (L2) c F*(X) . The chaos decomposition gives a natural expansion of @ E 3 * ( X ) into generalized kernels @ ( n ) E X ’ B n . Let E X’Bn be given. Then there exist a distribution (: wBn :, dn)) in F * ( X )acting on cp E F ( X ) as
Any @ E F ( X ) then has a unique decomposition +m n=O
142
A . Barhoumi, A . Riahi, and H. Ouerdiane
where the sum converges in 3 * ( X ) and we have +Do
((@,p)) = -p2nfl(@(n),p(")),
pE
F(X).
n=O
Finally, notice that F-q(X-,) is the Hilbert space with norm
n=O
4. Characterization Theorems 4.1. The S-transform
+
+
Let E = X i X and Ep = X , i X p , p E N,be the complexifications of X and X,,respectively. For E E X, we define the exponential function 4t by +Do
(br(w) =
C2"(:wBn : , < B y ,
w E X'.
n=O
Calculating its ( p ,q)-norm we find +m
n=O
if and only if 2qIEl; < 1. In contrast to usual white noise analysis, the exponential function are not test functions, they are only in those F p ( X q ) for which 2'Jl[15 < 1. Let @ E F * ( X ) ,then there exist p , q E N such that @ E F-q(X-,). For all E E X with 2qlEl; < 1,we can define the S-transform of @ as
S@((E) := ((a,$d). Using the duality form (l),we conclude that the S-transform of generalized Do
function
CP =
C (. *
.Bn
., *
is given by
n=O
n=O
This definition extends immediately to complex vectors f E E with 2 q l f l ; 1
c
<
Do
S q f ) := ((a,(bf)) =
n=O
2-"f1(@(4,
f@'").
(1)
Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
143
Hence, for E F q ( X P p ) (1) ] defines the S-transform for all f from the following open neighborhood of zero up,,
=
{f E E , qfl;< 1)
Therefore, it is noteworthy that { q b f ; f E Up,q } is a dense subset of F ' J ( X p ) . Then the 5'-transform is well defined on F-q(X-,).
-
4 . 2 . Characterization of test and generalized functions Let U c E be open. A function F : U @ is said to be Giiteauxholomorphic if for all Jo E U and for all J E E the mapping from @ to @ z ---t &'(to zJ) is holomorphic in some neighborhood of zero in @. By the general theory, the n-th Giiteaux derivative
+
1 F n ( t l , . . . , t n ) = -n! -<,
...DtnF(0)
(2)
becomes a continuous n-linear form on E . If F is Giiteaux-holomorphic, then there exists for every q E U a sequence of homogeneous polynomials 4n. D;F(O) such that
for all J from some open set V c U . The function F is said to be holoniorphic on U , if for all 77 in U there exists an open set V c U such that " 1 n! D ; F ( J ) converges uniformly on v to a continuous function. We say
c
n=O
that F is holomorphic at t o if there is an open set U containing JO such that F is holomorphic on U . We consider germs of holomorphic functions, i.e. we identify F and G if there exists an open neighborhood of zero U such that F ( 6 ) = G(J) for all J E U . Thus, we define Holo(E) as the algebra of germs of functions holomorphic at zero equipped with the inductive topology given by the following family of norms
With these notations and definition, we have the following characterization theorem for generalized functions.
Theorem 4.1. The S-transform defines a topological isomorphism of the space of generalized function P ( X ) onto the space Hob(E).
144
A . Barhoumi, A . Riahi, and H. Ouerdiane +w
Proof. Let Q,
=
(:wBn
E
F ( X ) and set F
n=O
there exist p , q E N such that Q, E F - q ( X P p ) .For inequality, we have
= SQ,. Then
< E U p , q ,by Cauchy's
4-m
n=O
This shows uniform convergence of (1) on U p , q .Hence, F E Holo(E) and we obtain
This implies that S is injective and continuous from F * ( X )to Holo(E). Conversely, let F E Holo(E) be given. There exist p , q E N such that +w
+m
n=O
n=O
where the right hand side of (3) converges uniformly and is bounded on E E with l
U p , q .For
<
The polarization identity gives
Spaces of Test and Generalized Functions of Arcsine W h i t e Noise Formulas
145
For s 2 0, we have M
Di7
We put @(w)=
2"-'
(: wBn :, Fn). For s , q/ >_ 0, the formula (4) yields
n=O
The last series converges if we chcose q / E N such that e224-4'+1 IIA-sll$s < 1. This shows that @ E . P ( X ) . hhrtbermore, it is clear that Sap([) = +m
C(Fn,CBn) F(<). This proves that S is sJrjectke m d bicontinuous. =
n=O
0
To characterize the space F(X)we start by tile definition of the Stransform o€ test functions. First of all, note that for any z E X I , we can define an exponential q5z in P ( X ) . Taking p E N such that z E X-, and using the decomposition coo
n=O
we have, for q E N,
n=O
if and only if 2-41212, < 1. Then, for all z E XI with 2-41z12_, define the S-transform of 'p E F(X)as
S d z ) := ( ( 4 z , ' p ) ) .
(5)
Hence, for 'p E Fq(X,), (5) defines the S-transform of following open neighborhood of zero
Up,,
-4
= { z E E-, ;
< 1, we can
'p
< l}.
2-4).~)2_,
for all z from the
146 A . Barhoumi, A . Riahi, and H. Ouerdiane
Let ‘FI(E’)stands for the space of all entire functions on E’. By definition, any cp E ‘H(E’)is an entire function on any E-p, p E N. For p , q E N, we define the Banach space
W - q (E-p) = {f E WE-,) ; lllf Ill-p,
-q
<)..
1
where
IIIf III--P,
:=
--P
sup
{If (.)I
ZE u-p,-q
(f
- 2-P-’,21Lp)
”’)
We denote by W (E’) the subspace of X ( E ’ ) of all entire functions on E’ with minimal type, i.e.
w(E’) =
n w-~E-,).
P9420
With the above definition, we have the following characterization theorem for test functions.
Theorem 4.2. The S-transform defines a topological isomorphism of the space of test functions 3 ( X ) onto the space W ( E ’ ) . Proof. Let ‘p E F(X)be given. Then, for p , q E N and z E L L P ,--P,by Cauchy’s inequality, we have +oo n=O
This gives Sq E W ( E ’ ) .It follows that S is injective and continuous from F(X)to W ( E ’ ) . Conversely, let u E W(E’)be given. There exist p , q E N such that +oo
u(2) =
c
C(un,
+oo
n=O
n=O
28’”) =
DlZ”U(0)’
where the right hand side of (6) converges uniformly and is bounded on E C with (XI < 24/’, as u is bounded by a constant c on
U-p,--P.For X
Spaces of Test and Generalized Functions of Arcsine White Noise Formulas
147
U-p,-q, Cauchy's integral formula yields
The polarization identity gives
-
For given p' E N, we choose p E N such that the embedding operator i,,,, : E-,! E-, is Hilbert-Schmidt. Let ( e j ) j c N be a complete orthonormal system in E+. Then (7) gives 00
Iunl:,
C
=
2 1 ( u n , e j 1@'.'ejn)1
j ~ , . ,j,=O ..
00
2n-1 (: w g n :, un).Then, the formula (8) yields
Put y(w)= n=O
If we choose q / E
N such that
e22q'-Q+111i,,,,11~s
< 1, the last series con+m
verges and then
'p E
F ( X ) . Furthermore, we get S'p(z) =
C(U,,,Z@~ =
n=O
Thus, we have found 'p E F ( X ) such that Sp = u.This proves that S is surjective and bicontinuous. 0
~(2).
References 1. A. Barhoumi, H. Ouerdiane and A. Riahi, Infinite dimensional Gegenbauer functionals, Banach Center Publ., Vol. 78, pp. 1-11, Polish Acad. Sci., Warsaw, 2007. 2. T. Hida, Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes 13, 1975. 3. I. Kubo and S. Takenaka, Calculus on Gaussian white noise I, 11, 111, I V , Proc. Japan Acad. Ser. A Math. Sci. Vol. 56 (1980), 376-380; Vol. 56 (1980), 411-416; Vol. 57 (1981), 433-437; Vol. 58 (1981), 186-189.
148 A . Barhoumi, A . Riahi, and H. Ouerdiane
4. Y. It6, Generalized Poisson Functionals, Probab. Theory Related Fields 77 (1988), 1-28. 5. S. Albeverio, Yu.L. Daletsky, Yu.G. Kondratiev and L. Streit, NonGaussian infinite dimensional analysis, J. Funct. Anal. 138 (1996), 311-350. 6. Yu.G. Kondratiev, L. Streit, W. Westerkamp and J.A. Yan, Generalized functions in infinite dimensional analysis, Hiroshima Math. J. 28 (1998) , 213-260. 7. J.L. Silva, Studies in non-Gaussian Analysis, Ph.D. Dissertation, University of Maderira, 1998. 8. R. Gannoun, R. Hchaichi, H. Ouerdiane and A. Rezgui, U n the'ordme de dualite' entre espaces de fonctions holomorphes ri croissance exponentielle J. Funct. Anal. Vol. 171, No.1 (2000), 1-14. 9. G. Di Nunno, B. ksendal and F. Proske, White noise analysis f o r Lkvy processes, J. Funct. Anal. 206, No. 1 (2004), 109-148. 10. Y.-J. Lee and H.-H. Shih, Analysis of generalized Lkvy white noise functionals, J. Funct. Anal. 211 (2004), 1-70. 11. N. Asai, I. Kubo and H.-H. Kuo, Multiplicative Renormalization and Generating Functions II., Taiwanese Journal of Mathematics, Vol. 8, No. 4 (2004), 583-628. 12. I.M. Gelfand and N.Ya. Vilenkin, Generalized Functions, Vol. 4, Academic press New York and London 1964. 13. H.-H. Kuo, White noise distrubition theory, CRC press, Boca Raton 1996. 14. N. Obata, White noise calculus and Fock space, Lecture Notes in Math. Vol. 1577, Springer-Verlag, 1994. 15. T.S. Chihara, A n Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. 16. W. Schoutens, Stochastic processes and orthogonal polynomials, Lecture Notes in Statist., Vol. 146, Springer Berlin 1999.
149
AN INFINITE DIMENSIONAL FOURIER-MEHLER TRANSFORM AND THE LEVY LAPLACIAN Kimiaki SaitB, Kazuyoshi Sakabe
Department of Mathematics, Meijo University Shiogamaguchi 1-501, Tenpaku, Nagoya 468-8508, Japan In this paper we present recent results on t h e role of the LBvy Laplacian in t h e infinite dimensional stochastic analysis. We discuss a stochastic process associated with t h e Laplacian and consider the operator that transfers from regular white noise functionals into functionals of exponential white noise. T h e operator gives relationships between the Beltrami Laplacian and the Ldvy Laplacian. An infinite dimensional Fourier-Mehler transform introduced by Kuo is also connected with the Ldvy Laplacian by the operator. Moreover t h e operator implies a Gauss-Poisson correspondence if we consider the LBvy Laplacian acting on generalized multiple Wiener integrals by a LBvy process.
1. Introduction
An infinite dimensional Laplacian was introduced by P. L6vy in his famous book.' This Laplacian has been studied by many authors from various aspects in Refs. 1-4,6,7,11,12,16,17,19,21,23 and 27, and references cited therein. On the other hand an infinite dimensional Fourier-Mehler transform was introduced by Kuo in Ref.14. This transform has been also studied by many authors (see Refs.5,9,15,26 and others). The purpose of this paper is to give a relationship between the L6vy Laplacian AL and KUO'SFourier-Mehler transform FO, 0 E R, by an operator K which plays the role of a Gauss-Poisson correspondence given as a homeomorphism from a space of Gaussian noise functionals onto a space of Poisson noise functionals (see Section 6). The relationship is given by
K[FO'p]=
e-zlTIO(A~-l)
K'p,
for all 'p in some domain of Gaussian noise functionals, where IT1 is the length of an interval T . (See Theorem 6.2 and Corollary 6.1) Since we can also give a stochastic process generated by the L6vy Laplacian on the
150 K. Saito and K. Sakabe
domain as a stochastic expression of a semigroup associated with the Laplacian, it implies a stochastic expression of K[F0'p]. This paper is organized as follows. In Section 2, we summarize basic elements of an infinite dimensional analysis based on a stochastic process given as a difference of two independent LBvy processes. In Section 3, extending recent works in Refs.11 and 28, we formulate the Levy Laplacian acting on a Hilbert space consisting of some LBvy distributions and give an equi-continuous Co-semigroup generated by the Laplacian. This situation is further generalized in Section 4 by means of a direct integral of Hilbert spaces. In Section 5, we give an infinite dimensional stochastic process generated by the extended Lkvy Laplacian through the semigroup. Moreover in Section 6, we give a relationship between KUO'SFourier-Mehler transform and the unitary group generated by the LBvy Laplacian through an operator transferring from Gaussian regular functionals into functionals of exponential white noise. 2. A compensated LQvyprocess and the LQvy distributions
Let {Lk,,,,(t)}tER and { L 2 , a , A ( t ) } tbe E ~independent LBvy processes of which the characteristic functions are given by ~ [ e i z ' ; , a , ~ ( t )= ] eth(z),
t
E
R, j
= 1,2,
where m E R, 0 2 0 , a 2 0, and X > 0. Set R,,,,,x(t) = Li,a,A(t) - L:,,,A(t) for all t E R. This form (2.1) comes from a necessary and sufficient condition for eigenfunctionals of the L6vy Laplacian in Ref.ll. Define N j ( t ,G ) , j = 172, by ~ j ( t ,= ~ )
C
lG(L:,a,A(S) - ~ t , u , A ( s - ) )
s:o<s
for t E R and G E have
%(R),and set N ( t ,G) = N 1 ( t ,G) - N 2 ( t ,G). Then we
~ [ e i z " c , a , ~ (= t)e ] t(h(z)+h(-z))
+
and R,,a,~(t)= a B ( t ) J R u N ( t , d u ) , t nian motion.
2 0, where
{B(t)}tER
is a Brow-
An Infinite Dimensional Fourier-Mehler Transform 151
Let E = S(R) be the Schwartz space of rapidly decreasing R-valued functions on R and let E* be a dual space of E . Then there exists a n orthonormal basis {e,},20 of L 2 ( R ) contained in E such that Ae, = 2(v+ l)e,, v = 0 , 1 , 2 , . . . , with A = --$ u2 1. For p E R define a norm 1 . IP by l f l P = I A P f l L z ( R ) for f E E and let E p be the completion of E with respect to the norm 1 . Ip. The canonical bilinear form on E* x E is denoted by (., .). We denote the complexifications of L2(R),E and E p by L&(R), E c and E C , ~respectively. ,
+ +
Set
Then by the Bochner-Minlos Theorem, there exists a probability measure pU,,,x on E* x E* such that r
where
hE)
= (z1,El)
+ (22,[2),
3:
= (z1,22)
E E* x E * ,
I
= (El,&)
E
E x E. Let ( L 2 ) u , a , x s L2(E* x E*,P,,,~,~) be the Hilbert space of C-valued square-integrable functions on E* x E' with L2-norm 11. ( ( u , a ,with ~ respect to p , , , , ~ .The Wiener-It6 decomposition theorem says that: w
(L2)u,a,x= @ H n ,
(2.2)
n=O
where Hn is the space of multiple Wiener integrals of order n E N and HO = C by definition. According to (2.2), each 4 E ( L 2 ) u , ais, ~represented as w
'p
=
X I n ( f n )fn, E L & ( R ) 9 n=O
where L&(R)6n denotes the n-fold symmetric tensor power of L&(R) (in the sense of a Hilbert space) and I n ( f n ) is given by
152
K . Saito and K . Sakabe
= C,"=oIn(fn) E ( L 2 ) u , aif, ~(2.3) exists in ( L 2 ) u , a ,If~ .p 2 0, let ( E ) p ; u , abe, ~the domain of I'(A)p. If p < 0, let ( E ) p ; u , abe, ~the completion of ( L 2 ) u , a with , ~ respect to the norm 11 . \ I p . Then ( E ) p ; u , a ,p~ E , R, is
for cp
a Hilbert space with the norm 11 . l i p . The projective limit space of spaces ( E ) p ; u , a ,p~E, R, is a nuclear space. The inductive limit space (E);,,,A of spaces ( E ) p ; u , a ,p~E, R, is nothing but the strong dual space of ( E ) u , a ,We ~ . call an element of (E)z,a,xthe generalized Livy functional or the Livy distribution. The U-transform on (E)Z,a,A is defined by using the canonical bilinear x ( - q u , a , A as form ((., .))u,a,X on U[@l(E)= ((@.I
where cp&)
(PE))u,a,A,
E
E
E x E,
= C(E)-l exp((z,J)).
3. The LQvy Laplacian a c t i n g on the LQvy d i s t r i b u t i o n s
Fixing a finite interval T of R, we take an orthonormal basis {& = ( ~ ) } ~ =c, E x E for L 2 ( T )x L 2 ( T )which is equally dense and uniformly bounded:
(A,
~
Let
IDL
N-1
denote the set of all cp E (E);,a,Asuch that the limit
-
N-1
exists for any E E E x E and i ~ ( U c pis) in U[(E)Z,a,A]. The L6vy Laplacian AL on V L is defined by ALP = U - l i ~ U c p ,cp E IDA. For any (T 2 0, X > 0 , a 2 0, and n E N let E u , a , ~ ,denote n the linear span of cp E (E)Z,a,X given by the form:
An Infinite Dimensional Fourier-Mehler Transform
153
of which the U-transform is given by n
~ ; ~ , ~ ( =t )z.-u2 (5 -~t 1 (ju)j ) + iU-2~2 2 ( u j ) + a where f belongs to L & ( R ) Q nwith supp f
X(e"E1(U3)-,-"Ez(2l3)
c T n and p1
+ . . . + pk
) > PJ
= n,
P I , . . . , P k E N.
Set Eu,a,x,O = C for any u 2 0,X > 0 , a 2 0. Then E,,,,A,~ is a closed linear subspace of ( J ! Z ) : , ~ , ~By . direct computations through the Utransform, we have the following.
Theorem 3.1. (cf. Ref.28) For each u 2 0 , n E N, X > 0 and a 2 0 the Le'vy Laplacian A, becomes a scalar operator o n EU,o,x,,U E o , ~ ,such ~ , ~ that ALP = 0 for all 'p E E,,o,x,, and A L ' = ~ -I& 'p for all 'p E E ~ , ~ , x , n . ITI
For N E N and X > 0 let D",;"' be the space of 'p E (E)E,a,xwhich admits an expression cp = C,"==, V n , Yn E Eo,a,x,n,such that I I I'pII I$,O,a,x =
By the Schwartz inequality we see that D",;"" is a subspace of (E)E,,,x and becomes a Hilbert space equipped with the new norm 111 . I / ~ N , o , ~ , J . Moreover, in view of the inclusion relations:
(E)G,a,x2
1
2
. . . 2 Do@)' 3 Do,a,X 3 ... N N+1
we define
n M
D
~
=Jp r o j l i m D y ~ =~ N+M
D ~ J .
N=l
DP"
Then AL is a self-adjoint operator densely defined in for each N E N and X > 0. In view of the action in Theorem 3.1, for each z E C with Re z 2 0 and X > 0 we consider an operator G$ on DZix defined by
n= 1
n=l
We also define G: on (L2),,0,x as an identity operator I by Iq
(L2)u,o,x.
= 'p, 'p E
154 K. Saito and K. Sakabe
Theorem 3.2. Let X > 0. Then the family {G:; t >_ 0 ) on DZi’ i s an equi-continuous Co-semigroup generated by AL. Proof. With help Theorem 3.1, the proof is a modification of the proof of Proposition 5 in Ref.29. 0
4. Extensions of the LQvy Laplacian Let du(X) be a finite Bore1 measure on R satisfying
Fix N E N and a 2 0. Let 9%be the space of (equivalent classes of) measurable vector functions cp = ((p’) with (p’ = C,”==, cp; E Dor;”” for all X > 0, and cpo E ( L 2 ) c , ~such , ~ , that
Then 9& becomes a Hilbert space with the norm given in (4.1). In view of the natural inclusion c 9%for N E N, which is obvious from construction, we define D& = proj limN,, 9%= 9%.
nF=,
The LQvyLaplacian A, is defined on the space 9& by
ALY = (ALP’), cp = (P’)E 9;. Then AL is a continuous linear operator from D& into itself. Similarly for z E C with Re z 2 0 we define
Gzcp = ( G W ) , cp
= ((PA) E
92.
Then Theorem 3.2 implies the following result through the similar method in the proof of Theorem 3.3 in Ref.24.
Theorem 4.1. The family {Gt; t 2 0 ) is an equi-continuousCo-semigroup generated b y A, as an operator acting on 92.
A n Infinite Dimensional Fourier-Mehler Transform
155
5. Associated infinite dimensional stochastic processes
For p E R let El'"' be the linear space of all functions X H Ex E Ep x Ep, X 2 0, which are strongly measurable. An element of EF'") is denoted by E = (&)x20. Equipped with the metric given by
the space El'"' becomes a complete metric space. In view of dp p 2 q1 we introduce the projective limit space
5 d, for
~ [ o ) " ) = proj limEF>"). P+c=
Similarly, let C[O*")denote the linear space of all measurable function X zx E C equipped with the metric defined by
H
z = ( z x ) , u = (ux).
Then C[O>") is also a complete metric space. 9 ;
The U-transform can be extended to a continuous linear operator on by
U p ( [ )= ( U ( P X ( E x ) ) x > 0 , E = ( E X ) X > O E @O;"), for any p = ('px)x>0 E 9 ; .The space U[D&] is endowed with the topology induced from DL by the U-transform. Then the U-transform becomes a homeomorphism from DL onto U[Di)",].The transform U p of p E 9 ; is a continuous operator from E[oi") into C[O>O0). We denote the operator by the same notation U p . Let { X j } , j = 1 , 2 , be independent Cauchy processes with t in [ O , o o ) , of which the characteristic functions are given by
E[eizx:]= e-'lZI, z E R,j = 1,2. Take a smooth function
VT E
E with
VT =
1/15" on T . Set
qx= (XktVT, - x & V T ) E
2 0.
Define an infinite dimensional stochastic process { Y t ;t 2 0 ) starting a t = (Ex)x>o E E[O)") by Yt = (Ex
+ qA)x>o,
t 2 0.
Then this is an E['@)-valued stochastic process and we have the following
156
K. Saito and K. Sakabe
Theorem 5.1. If F is the U-transform of an element in a&, we have
-
GtF(E) = E [ F ( Y t ) I Y o= E ] , t 2 0,
where
= UGtU-l.
Proof. We first consider the case when F E U [ D & ]is given by
F(E) = (FA(rx))x>o,F0 E ~[(L2)c7,0,x1,
with f E L;(R)@’”.Then we have
-
(e-tnA2/lT(FA
-
(E
=
A20
(G,XFX(Ex))x20 =Z F ( 0
+
Next let F = (FoGx,o C,”==, F,”)x>o E U[D&]. Then for v-almost all X > 0 and for any n E N , F,” is expressed in the following form:
and ] F,” E U [ D ~ ~there x ] , exist ‘po E (L2),,0,x and Since F o E U [ ( L 2 ) , , o , ~ p; E DC>A such that F o = U[po] and F,” = U[p;] for v-almost all X and each n. By the Schwarz inequality, we see that
n=l
n=O
A n Infinite Dimensional Fourier-Mehler Transform
157
where ptx = G(Jx)-le("Ex) for v-almost all X 2 0 and each N E N. There-
-
fore by the continuity of G f ,X 2 0, we get that
Thus we obtain the assertion. For any
q~ E
0
E[O>")we define a translation operator T,, on !D& by (UT,,cp)(<)= (Ucp)(t+ q I ) .
The translation operator and Theorem 5.1 imply a stochastic expression of Gt as follows.
Theorem 5.2. For all cp in 9 ;
we have
Gtcp = E[T(Y~)x20cp1, t 2 0. 6. A relationship between KUO'SFourier-Mehler transform
and the LQvyLaplacian
The operator K is a homeomorphism from DLovl onto K[DL0>l] c D!kl"ll. This means that the operator K is also a Gauss-Poisson correspondence. Let N be the number operator on (E)Y,O,l given by 00
n=O
n=O
158 K. Saito and K. Sakabe
The operator - N is the Laplace-Beltrami Laplacian. Since n ALKP, = -T?;i'pn, p n E E0,1/2,1, n = O , 1 , 2 , . . . , from Theorem 3.1, we have the following relationship between the Laplace Beltrami Laplacian and the LQvy Laplacian through the operator K. Proposition 6.1. For all
'p
Dk0>'we have
K[-Np] = lTlALK['pl. Moreover, by Proposition 6.1, we have a relationship between the semigroup generated by the Laplace-Beltrami Laplacian and that generated by the LQvy Laplacian as follows.
Theorem 6.1. For all
'p
E
Dk0v1 we have
K [ e - 6 N ' p ]= GiK['p]. The operator K also implies an interesting relationship between KUO'S Fourier-Mehler transform and the one-parameter group generated by the Levy Laplacian. The KUO'SFourier-Mehler transform 3 0 , 0 E R, depending on T is defined by
[
S [ . F ~ ' ~ ]= ( ES'p(ei0C) ) exp ieio sine
S,
[ ( t )d ~ t] , 6 E
EC.
Let g be a Gaussian white noise distribution given by
Through Theorem 6.1, we see that G1-&IT1 . K'p = K[eieNp],
'p
E
DLop"
for all 0 E R. Since
and
we have the following relationship between 3 0 and GLielT,= e-ielTIAL :
An Injinite Dimensional Fourier-Mehler Transform
Theorem 6.2. For all
'p
159
E DLo>',we have
K[.%4 = G-iO(T(KPo 1 G?i6'lT/Kg
Gi'3(T/K9?
where o means the Wick product defined by @ o Q = U-' [U@. UQ],a, Q E KID&o*l].
Theorems 5.2 and 6.2 imply a stochastic expression of K[.F@9]. Moreover, since AL is a derivation under the Wick product, we have the following.
Acknowledgments This work was written based on a talk at the special session on infinite dimensional analysis honoring Professor H.-H. Kuo in the 2007 AMS Spring Annual Meeting. The authors would like to express their deep thanks to Professors A. Sengupta and P. Sundar for inviting the first author to this stimulating session, and their support and warm hospitality. This work was also partially supported by JSPS grant 19540201. The authors are grateful for the support.
References 1. P. Lbvy: LeCons d'Analyse Fonctionnelle, Gauthier-VilIars, Paris, 1922. 2. L. Accardi and V. Bogachev: The Ornstein-Uhlenbeck process associated with the L6vy Laplacian and its Dirichlet form, Prob. Math. Stat. 17 (1997), 95-114. 3. L. Accardi, P. Gibilisco and I. V. Volovich: Yang-Mills gauge fields as harmonic functions for the Lkvy Laplacian, Russ. J . Math. Phys. 2 (1994), 235250. 4. L. Accardi and 0. G. Smolyanov: Trace formulae for Levy-Gaussian measures and their application, Mathematical Approach to Fluctuations Vol. I I ( T . Hida, Ed.), 31-47, World Scientific, 1995. 5. L. Accardi, A.Barhoumi and H. Ouerdiane: A quantum approach t o Laplace operators, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 9 (2006), 215-248. 6. D. M. Chung, U. C. Ji and K. SaitB: Cauchy problems associated with the Lbvy Laplacian in white noise analysis, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 2 (1999), 131-153. 7. M. N. Feller: Infinite-dimensional elliptic equations and operators of Levy type, Russ. Math. Surveys 41 (1986), 119-170.
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K. Saito and K. Sakabe
8. A. Ishikawa and K. Sait6: The LBvy Laplacian and the LBvy Process, R I M S Kokyuroku 1507 (2006), 14-25. 9. U. C. Ji and K. Sait6: A similarity between the Gross Laplacian and the LBvy Laplacian, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 10 (2007), 261-276. 10. H.-H. Kuo, N . Obata and K. SaitB: Levy Laplacian of generalized functions on a nuclear space, J . Funct. Anal. 94 (1990), 74-92. 11. H.-H. Kuo, N. Obata and K. Sait6: Diagonalization of the LBvy Laplacian and related stable processes, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 5 (2002), 317-331. 12. R. LBandre and I. A. Volovich: The stochastic LBvy Laplacian and Yang-Mills equation on manifolds, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 4 (2001) 161-172. 13. N. Obata: A characterization of the LBvy Laplacian in terms of infinite dimensional rotation groups, Nagoya Math. J. 118 (1990), 111-132. 14. N. Obata: Quadratic quantum white noises and LBvy Laplacian, Nonlinear Analysis 47 (2001), 2437-2448. 15. K. SaitB: A stochastic process generated by the LBvy Laplacian, Acta Appl. Math. 63 (ZOOO), 363-373. 16. H.-H. Kuo: Fourier-Mehler transforms of generalized Brownian functionals, Proc. Japan Acad. 59A (1983), 312-314. 17. D. M. Chung and U. C. Ji: Transformation groups on white noise functionals and their applications, Appl. Math. Optim. 37 (1998), 205-223. 18. T.Hida, H.-H. Kuo and N. Obata: Transformations for white noise functionals, J. Funct. Anal. 111 (1993), 259-277. 19. H.-H. Kuo: White Noise Distribution Theory, CRC Press, 1996. 20. K. Sait6: A (Co)-group generated by the LBvy Laplacian 11, Infin. Dimen. Anal. Quantum Probab. Rel. Top. 1 (1998) 425-437. 21. K. Sait6: An infinite dimensional Laplacian acting on multiple Wiener integrals by some LBvy processes, Infinite Dimensional Harmonic Analysis 111 (2005) 265-276. 22. K. SaitB and A. H. Tsoi: The LBvy Laplacian as a self-adjoint operator, Quantum Information 159-171, World Scientific, 1999. 23. N. Obata and K. Sait6: Cauchy processes and the LBvy Laplacian, Quantum Probability and White Noise Analysis 16 (2002), 360-373.
161
THE HEAT OPERATOR IN INFINITE DIMENSIONS Brian C. Hall* University of Notre Dame, Dept. of Mathematics, Notre Dame IN 46556-4618 USA bhallQnd. edu Let ( H , B ) be an abstract Wiener space and let pLsbe the Gaussian measure on B with variance s. Let A be the Laplacian ( n o t the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H . I will show that the heat operator exp(tA/2) is a contraction operator from L z ( B , p s )to L2(B,p,-t), for all t < s. More generally, the heat operator is a contraction from P ( B ,pa) to Lq(B,ps-t) for t < s, provided that p and q satisfy
p - 1 I S . q-1
s-t
I give two proofs of this result, both very elementary.
1. Introduction The heat operator, both on Euclidean space and on Riemannian manifolds, is a basic tool in finite-dimensional analysis. In infinite-dimensional analysis, the Laplacian (i.e., the most naive infinite-dimensional generalization of the finite-dimensional Laplacian) cannot be defined as a self-adjoint operator, because there is no such thing as Riemannian volume measure in infinite dimensions. If one replaces the nonexistent volume measure with a Gaussian measure or something similar on a nonlinear manifold, the Laplacian not only fails to be self-adjoint but fails to be closable in L 2 . This makes it difficult to define the heat operator as a reasonable operator in L2. In this paper, I will consider only the case of an infinite-dimensional Euclidean space. In that case, I present one possible way of making the heat operator into a well-defined, bounded operator. The proof hinges on a relationship between the heat semigroup and the Hermite (or OrnsteinUhlenbeck) semigroup. This relationship in turn hinges on the close relationship between the Laplacian and the number operator. Specifically, the *Supported in part by NSF grant DMS-0555862
162 B. C. Hall
two operators differ by a first-order term which is simply the generator of dilations. Of course, these relationships are not new. For example, the relationship between the Laplacian and the number operator, in the setting of white noise distribution theory, is clarified in Kuo [l,Theorem 10.181. Nevertheless, the particular application I make of these relationships in Theorem 5.1 is new, so far as I can tell. I thank Professor Leonard Gross for valuable discussions, especially in pushing me to understand the relationship between the heat semigroup and the Hermite semigroup in terms of commutation relations but also in making me aware of the work of Nualart and Ustunel. I also thank the referee for valuable comments and corrections. 2. Gaussian measures
Let H be an infinite-dimensional, real, separable Hilbert space. Since there does not exist anything like Lebesgue measure in infinite dimensions, we may consider instead a Gaussian measure. Let us attempt to construct, then, a “standard” (i.e., mean zero, variance one) Gaussian measure on H . This should be given by the nonrigorous expression
where Vx is the nonexistent Lebesgue measure on H and Z is a normalization constant. Unfortunately, this formal expression does not correspond to any well-defined measure on H . Specifically, (1) can be used to assign a “measure” to cylinder sets, but this set function does not have a countably additive extension to the generated g-algebra. (A cylinder set is a set that can be described in terms of finitely many continuous linear functionals on
H.1 To rectify this situation, we follow the approach of L. Gross.2 We introduce a Banach space B together with a continuous embedding of H into B with dense image. If B is “enough bigger” than H , in a sense spelled out precisely in Ref. 2, then there is a measure p on B that captures the essence of the formal expression in (1). One rigorous way to characterize the measure p is to use (1) to define a set function on cylinder sets in the larger space B. This set function, unlike the one on cylinder sets in H , has a countably additive extension to the generated o-algebra. (This result, of course, assumes Gross’s condition on the embedding of H into B.) The original space H turns out to be a set of p-measure 0 inside B. The book3 of Kuo is an excellent standard reference on this material.
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A pair ( H ,B ) satisfying Gross’s condition is called an abstract Wiener space. The prototypical example is the one in which H is the space of H1 functions on [O, 11, equaling 0 at 0, with inner product given by
JO
In this case, one may take B to be the space of continuous functions on [0,1] equaling 0 at 0. In this case, p is the (concrete) Wiener measure, describing the behavior of Brownian motion. As an alternative to using (I), one can characterize p in terms of its Fourier transform. The measure p is the unique one such that for all continuous linear functionals $ on B , we have
where l l $ l l H is the norm of $ as a linear functional on H ( n o t B ) . (That is, llq511H is the norm of the restriction of q5 to H.) In light of the standard formula for the Fourier transform of a Gaussian, (2) is formally equivalent to (1). Note from either (1) or (2) that it is the geometry of H , rather than the geometry of B , that is controlling the Gaussian measure p. One should think of p a s the standard Gaussian measure “on” H , where the larger space B is a technical necessity, needed to capture the measure. We can also introduce Gaussian measures with variance s. For any s > 0, there is a unique measure p s on B such that
for all continuous linear functionals 4 on B. The measure p , is simply a dilation of p . (Since B is closed under dilation, we can use the same Banach space B , independent of s, for all of the measures p s . )
3. The Laplacian and the heat operator
If we keep our focus on the geometry of H (rather than B ) , then we can introduce a Laplacian on B as follows. Let {en}:=l be an orthonormal basis for H , with the property that the linear functionals z -+ (e,,z) extend continuously from H to B. Then let { x , } ~ =be ~ the coordinate functions (on B ) associated to this basis; that is, 2, = ( e , , z ) for z E B . Then we
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define the Laplacian to be the operator given by
A = C - ax; . a2
n= 1
This operator is defined, for example, on polynomials, that is functions that can be expressed as polynomials in some finite collection of the xn’s. Note that this operator is really the Laplacian and not the frequently considered number operator (also known as the Ornstein-Uhlenbeck or Dirichlet form operator). That is, (1) is the naive infinite-dimensional generalization of what is usually called the Laplacian on Rd. (Those who are emotionally attached to the number operator should not lose heart; that operator will have its role to play later on.) We would like to try to define the Laplacian as an unbounded operator in the Hilbert space L 2 ( B ,p) or more generally L 2 ( B ,,us).In fact, A can be defined densely in L 2 ( B ps), , by, for example, defining it on polynomials. Unfortunately, though, the Laplacian defined in this way is a nonclosable operator. (This is equivalent to saying that the adjoint operator is not densely defined. Note that A in not self-adjoint in L2 with respect to a Gaussian measure, even in finite dimensions.) Nonclosable operators are generally considered to be pathological; not only does such an operator fail to have a densely defined adjoint, but it is difficult to make any canonical choice of what its domain should be. The nonclosability of the Laplacian can easily be seen by example. Define functions fn E L 2 ( B , p s )by 1
fn(x) = n - - y ( . 2 k - s). k=l
Since, as is easily verified, (xi - s , x; that
-
s ) ~ ~ ( ~ ,= , , 0, )for Ic
# 1, we can see
Thus, f n + 0 in L2(B,p,) as n tends to infinity. On the other hand, a simple calculation shows that
Afn = 2 (constant function). Thus, the pair (0,2) is in the closure of the graph of A , which shows that A is not closable. Since the Laplacian A is not closable, we cannot expect the heat operator eta/’ to be any sort of reasonable semigroup in L 2 ( B ,us). , One can
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define etAI2 on polynomials as a (terminating) power series in A, but the resulting operator is again not closable. (This shows, in particular, that the heat operator is not bounded.) The same example functions f n demonstrate the nonclosability of the heat operator. After all, since A f n is constant, A2fn is zero and so etA12fn = f n tAfn/2. Thus, etA12fn tends to t , whereas f n tends to zero.
+
4. The Segal-Bargmann transform
The discussion in the previous section shows that we cannot regard the heat operator as a reasonable operator from L 2 ( B ,p s ) t o itself. If, then, we are going to make etA12 into a reasonable (preferably bounded) operator, then we must regard it as mapping from L 2 ( B ,p s ) to some other space. One way to do this is to look at the Segal-Bargmann transform. This transform (from one point of view) consists of applying the heat operator to a function and then analytically continuing the resulting function etA12f in the space variable. Even in the infinite-dimensional case, this makes sense at least on polynomials. One can then prove an isometry formula that allows one to extend etala to a bounded operator from L 2 ( B , p s )into an appropriate Hilbert space of holomorphic functions on the complexification of B .
Theorem 4.1. Let P denote the space of polynomials inside L 2 ( B , p s ) . For all t < 2s, there is a Gaussian measure ps,t on Bc := B+iB such that the map f
+
analytic continuation of e t A 1 2 f ,
as defined o n polynomials, is isometric f r o m P
c L 2 ( B , p s ) into
L2(B@, ps,t). This is Theorem 4.3 of Ref. 4. The measure ps,t is the product of a variance-s/2 Gaussian measure in the real directions and a variance-t/2 Gaussian measure in the imaginary directions. Here, again, the analytic continuation is in the space variable (from B to B c ) with t fixed. This theorem shows that extends continuously to an isometric map of L 2 ( B , p , ) into L2(Bc,p,,t).In Ref. 4 , it is shown that the image of this extended map is precisely the L2 closure of the holomorphic polynomials in L2(Bc,p s , t ) .
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5. “Two wrongs make a right”
In the preceding section, we saw that the heat operator eta/’, followed by analytic continuation, can be regarded as a bounded (even isometric) map of L 2 ( B ,p s ) into a Hilbert space of “holomorphic” functions on Bc, provided that t < 2s. (Here “holomorphic” means “belonging to the L2 closure of holomorphic polynomials.”) We may ask, however, whether it is possible to regard the heat operator itself, without the analytic continuation, as a bounded operator from L 2 ( B p,) , into some space of functions on B. The answer, as we shall see in this section, is yes. The key is to regard the heat operator as mapping from L 2 ( B ,p,) into a space defined using a Gaussian measure with a different variance. Specifically, we will see that for t < s, etA12 is a bounded operator from L 2 ( B ,p s ) into L 2 ( B ,p s -t). (Something somewhat similar to this was done in a paper of Nualart and Ustune15 . They consider a Hilbert space obtained by integrating over a range of values for the variance. In this Hilbert space, the Laplacian becomes a closable operator.) Now, ordinarily, such a “changing of the variance” is not a good idea. That is, the identity map, which simply regards a function f E L 2 ( B , p s ) as an element of L 2 ( B , p s - t ) ,is highly ill defined. After all, it is known that the measures ps and p,-t (0 < t < s ) are mutually singular; each measure is supported on a set that has measure zero with respect to the other measure. Thus, two functions that are equal ps-almost every where may not be equal p,-t-almost everywhere. Thus, the map f + f is not well defined from L 2 ( B , p s )to L 2 ( B , p s - t ) ,because elements of L2 are not functions but rather equivalence classes of almost-everywhere equal functions. Alternatively, one can define the identity map on polynomials (mapping a polynomial in L2(B,p s ) to the same polynomial in L 2 ( B ,p s - t ) ) and then check that this map is not closable (consider again the functions fn).
We see, then, that the heat operator is not well defined from L 2 ( B ,p,) to itself and that the identity map is not well defined from L 2 ( B , p s ) to L2(B,ps-t).Nevertheless, when we put these two maps together, two wrongs turn out to make a right: the heat operator is a well-defined and bounded map from L 2 ( B , p s )to L 2 ( B , p s - t ) .Somehow, for f E L 2 ( B , p s ) , if we regard eta12f as belonging to L2(B,p,-t) rather than L2 (B ,p ,), things work out better. With this point of view, eta/2 actually becomes a bounded operator. Actually, more than this can be said. The heat operator is actually bounded (even contractive) from Lp(B,p s ) to Lq(B,p S w t )for , certain pairs
The Heat Operator in Znjinite Dimensions
(PI
with 4
167
> P.
Theorem 5.1. Fix t < s and numbers p , q > 1 such that
q-l,-, S p-1-
s-t
Then the heat operator, initially defined on polynomials, extends to a contractive operator from LP(B,p s ) to Lq(B,ps-t). Note that since s is greater than s - t , the condition on p and q allows q to be greater than p . In particular, p = q is always permitted. Theorem 5.1 is the main result of this paper. I will present two different proofs, in the two following sections.
6. Proof using Hermite polynomials Let (I! = (QI, ( ~ 2a3, , . . .) be an infinite multi-index, in which all but finitely many of the aj’s are zero. A polynomial is then a finite linear combination of functions of the form xQ := xP‘x;~ . . . . We let P denote the space of all polynomials. Then inside L 2 ( B ,p s ) we define a Hermite polynomial to be a polynomial of the form
hQ,s(x):= e - s A / 2 ( x a ) .
(1)
Here, e-sA/2 is the backward heat operator, which is defined on any polynomial by a terminating power series in powers of A. The formula (1)is one of many equivalent ways of defining the Hermite polynomials. It is known , that the Hermite polynomials form an orthogonal basis for L 2 ( B , p s )as Q varies over all multi-indices of the above sort. The normalization is as follows:
where a! = Q I ! Q ~ ! .. . Furthermore, for 1 < p < 00, the span of the Hermite polynomials is dense in LP(B,p s ) . We note that for any t and s we have, on the space of polynomials, e t A / 2 e - ~ A / 2 - ,-(~-t)A/2 7
by the usual power series argument. Thus for s
< t , we have
etA12(h,,,) = ha,s-t. This means that the time-t heat operator maps the Hermite polynomials that go with the Hilbert space L 2 ( B ,p s ) to the Hermite polynomials that
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go with the Hilbert space L 2 ( B ,p,-t). This simple observation provides the first indication that the “right” way to think of eta/‘ is as an operator from a function space defined using ps to a function space defined using ps-t. Actually, eta/’ maps an orthogonal basis for L 2 ( B ,p,) to an orthogonal basis for L 2 ( B , p s - t ) .Furthermore, since (s - t)lal 5 sI”I, it follows easily that eta’’ extends to a contractive mapping of L 2 ( B p,) , t o L 2 ( B ,p s - t ) . To establish the LP to Lq properties in Theorem 5.1, we use scaling. We have said that the identity map (i.e., the map f --$ f ) is not well defined from L 2 ( B ,p,) to L 2 ( B ,p s - t ) , or vice versa, because the measures p s and ps-t are mutually singular. There is, however, a nice map from L 2 ( B ,,u,-t) to L 2 ( B ,p,), or more generally of LP(B,p s - t ) to L P ( B p,), , consisting of dilation. That is, if we define D,,t : L 2(B,ps-t)+ L2(Blps)by
then this map is well defined and isometric from L p ( B ,p s - t ) to L p ( B ,p,), for all 1 5 p 5 a. This amounts to saying that p, can be obtained from ps-t by a dilation of B. Meanwhile, how do Hermite polynomials transform under this dilation? Well, using the formula (1)for the functions h,,,, it is not hard to see that
Now, it makes sense to start with a Hermite polynomial ha,+E LP(B,p,), apply the heat operator to get h,,,-t E LP(B,ps-t), and then apply the dilation Ds,t to get back to LP(B,p,). We have, by (2),
Let us introduce the ‘‘number operator” N , defined on polynomials by the condition
Then (3) can be rewritten as
T h e Heat Operator in Infinite Dimensions
169
where 7-
= -log 1
2
(L). s-t
Here e-rNs is defined on polynomials by setting e-rNsh,,s = e-Tlalha,slin accordance with (4). Now, every polynomial is a finite linear combination of Hermite polynomials. What ( 5 ) says, then, is that on the dense subspace P of L 2 ( B ,p s ) we have 0
eta/’ = e-T*s.
(7)
Note that e-TNs is a bounded operator on P c L 2 ( B ,,us),since its action on the orthogonal basis { h a } consists of multiplying by e-Tlal. Thus (7) tells us that D,,t o has an extension which is a bounded operator from L 2 ( B ,p s ) to itself. We can, however, say more than this. Nelson‘ has shown that e-TNs is contractive from Lp(B,ps) to Lq(B,ps),provided that
1 q-1 r 2 -log2 p-1’ More precisely, this can be interpreted as saying that for r satisfying the above condition, ebrN has extension from the space P of polynomials to LP(B,p s ) that maps contractively into Lq(B,p s ) . (This contractivity property of eCrNs, where q > p is permitted, is referred to as hypercontractivity.) The condition (8) is, in light of (6), equivalent to s
q-1
2 p-, - 1 s-t
(9)
Thus, whenever (9) holds, Ds,t o eta/’ has an extension that is a contractive map of LP(B,p s ) to LQ(B,ps-t). Since Ds,t is an isometric isomorphism of Lq( B, ps-t) onto Ln( B , ps), this amounts to saying that eta/’ has an extension that is a contractive map of Lp(B,p s ) to Lq(B,p s - t ) . This establishes Theorem 5.1. 7. The Hermite semigroup and the heat semigroup Looking back on the argument in the preceding section, we see that the role of the Hermite polynomials is not essential. Rather, we used the Hermite polynomials to obtain the identity (7), at which point Theorem 5.1 is seen to be a consequence of Nelson’s hypercontractivity theorem. That is, our result really hinges on a relationship between the heat semigroup (the operators
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B. C. Hall
eta/') and the Hermite semigroup (the operators e-rNs). What (7) is saying is that (at least on polynomials) the Hermite semigroup at time T is the same as the heat semigroup a t t , modulo a dilation, where t and r are related as in ( 6 ) . What we want to do in this section is explore two other ways (besides the Hermite polynomial argument of the previous section) of understanding this relationship between the two semigroups. We begin by looking a t the integral kernels for the two semigroups. In the finite-dimensional case, the heat semigroup eta/' can be computed as integration against the heat kernel, which is a Gaussian. Meanwhile, in the finite-dimensional case, the Hermite semigroup e--7Na can be computed using the Mehler kernel, which is also a Gaussian, though of a slightly more complicated variety. See, for example, Sjogren [7, Thm. 11, which derives the formula for the Mehler kernel in a way that emphasizes the relationship with the heat kernel. From these formulas for the kernels, one can easily read off the identity (7) in any finite number of variables. This is sufficient to establish (7) on polynomials, since each polynomial is a function of only finitely many variables. As an alternative to using the Hermite polynomial argument of the previous section or the argument in this section based on formulas for the integral kernels, we can explore the relationship between the heat semigroup and the Hermite semigroup using commutation relations. Although we have defined the number operator by its action on Hermite polynomials (N,h,,, = I C Y[ ha,,), N, can also be expressed as a differential operator, as follows. Let
Then Drc" = IaJx". From this it follows, in light of (l),that
e--SA/2DeSA/2ha,s = la1 ha+. We now use the standard identity eABe-A = eadA( B ) ,where adA(B) = [A,B ] .A simple calculation shows that
[A,D] = 2A. Thus if we set
N , --e
-Sada/2
=D
(D)
- sA,
where (adA)"D = 0 for n 2 2, N, defined in this way will have the correct behavior on Hermite polynomials.
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Our task, then, is to compute e - ~ N s - exp { r ( s A- D)}, with the aid of the commutation relation (1).We will apply a special case of the Baker-Campbell-Hausdorff formula. Suppose that A and B are linear operators on a finite-dimensional vector space and that [ A ,B] = a A . Then we have
To prove (3), let 2 ( r )denote the quantity on the right-hand side. Using the identity ,TBA,-TB
= eradB ( A ) = e-rQA,
it is not hard to show that Z satisfies the differentialequation dZ1d.r = ( A + B ) Z ( T ) Since . the left-hand side of (3) clearly satisfies the same differential equation and since the two sides are equal when r = 0, we conclude that the two sides are equal for all T . (See, for example, [8, Section 41, where a slight variant of (3) is analyzed in the context of unbounded operators.) We wish to apply (3) with A = sA and B = -D,in which case we would have a = -2. Of course, we should not blindly apply results for operators on finite-dimensional spaces to operators on infinite-dimensional spaces. Fortunately, however, there is no problem in this instance, since any polynomial p E P is contained in a finite-dimensional space that is invariant under both D and A and hence under N,. (A polynomial p , by definition, involves only finitely many monomials 5". Thus there is some n such that p is contained in the space of polynomials of degree at most n in some finite collection of variables x1 , .. . , xm.) On each finite-dimensional invariant subspace, then, we have
where we may compute using ( 6 ) that e-2T - 1 s--t 1-t - -2 2 2s' Thus we get e - ~ N ,=e-~DetA/2
(4)
It remains only to understand the factor of e P r Don the right-hand side of (4).Recall that Dx" = la1 xa. Thus, e-rD x - e - ~ l " l x ~ = (e - T x ) " , (5) Q
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d m .
where e-r = From this we can see that e--7D = D,)t on polynomials and thus (4) is equivalent (on polynomials) to the identity (7). 8 . Concluding remarks
The boundedness properties of the heat operator given in Theorem 5.1 can be deduced from the relationship (7) between the heat semigroup arid the Hermite semigroup. That relationship, in turn, can be understood in various ways, using Hermite polynomials, using the integral kernels, or using commutation relations. In the last approach, the identity (7) follows from the commutation relation (1) together with the special form (3) of the Baker-Campbell-Hausdorff formula. We have seen in (5) that D is the generator of dilations. Thus the commutation relation between D and A in (1) reflects that the Laplacian transforms in a simple way under dilations. This in turn reflects that the metric on Euclidean space transforms in a simple way under dilations. What prospect, then, is there for proving some analog of Theorem 5.1 in some other setting, that is, on some (possibly infinite-dimensional) manifold other than Euclidean space? One possibility is to consider manifolds where there is some natural sort of dilation operators. For example, on the Heisenberg group there is a sub-Laplacian that behaves in a nice way with respect to certain nonisotropic “dilations.” Thus the Heisenberg group, whether in its finite- or infinite-dimensional form, is a natural candidate for proving a theorem similar to Theorem 5.1. On the other hand, even if an infinite-dimensional manifold has no natural dilations, it is still conceivable that something like Theorem 5.1 might hold. Specifically, suppose there exists on an infinite-dimensional manifold M some natural sort of heat kernel measure p,, based a t a fixed point in M . (One may think, for example, of path or loop groups and the heat kernel measures considered by Malliaving and Driver-Lohrenz, l o or various infinite-dimensional limits of finite-dimensional groups and the heat kernel measures of Gordina.l19l2) One might hope that in some cases, the heat operator eta/’ could be a bounded operator from LP(M,p,) to Lq(M,p,-t), for t < s and appropriate pairs ( p , 4 ) . One might begin by studying only the Hilbert space case and try to see whether is bounded (or contractive) from L 2 ( M , p , ) to L 2 ( M , p s - t ) .Of course, the methods of proof used in the present paper would not carry over to such a setting. Nevertheless, the proposed result is of a simple enough form that some other method of proof may be found. The conclusion I would like t o draw from all of this is that one should
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not give up on studying the heat operator associated to the true Laplacian, even in the infinite-dimensional setting. Here by “true” Laplacian I mean something like the Laplace-Beltrami operator associated to some Riemannian metric on an infinite-dimensional manifold, that is the V*V operator associated to the (fictitious) Riemannian volume measure. This is to be contrasted with something like a number operator or Ornstein-Uhlenbeck operator, which is the V*V operator associated to a Gaussian or heat kernel measure. Even though the Laplacian is bound to be a pathological operator, this should not cause us to give up on defining the heat operator. One merely has to look for the right interpretation of the heat operator, an interpretation that will allow it to make sense. One candidate for such an interpretation is to view the time-t heat operator as an operator from L 2 ( M ,p s ) to L 2 ( M ,p3-t). References 1. H.-H. Kuo, “White noise distribution theory.” Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. 2. L. Gross, Abstract Wiener spaces. In: “1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66),” Vol. 11: Contributions to Probability Theory, Part 1, pp. 31-42. Univ. California Press, Berkeley, Calif. 3. H.-H. Kuo, “Gaussian measures in Banach spaces.” Lecture Notes in Mathematics, Vol. 463. Springer-Verlag, Berlin-New York, 1975. 4. B. K. Driver and B. C. Hall, Yang-Mills theory and the Segal-Bargmann transform, Comm. Math. Phys. 201 (1999), 249-290. 5. D. Nualart and A. Ustiine1,Une extension du laplacien sur l’espace de Wiener et la formule d’It6 associke, C. R. Acad. Sci. Paris SLr. I Math. 309 (1989), 383-386. 6. E. Nelson, The free Markov field, J . Funct. Anal., 12 (1973), 211-227. 7. P. Sjogren, Operators associated with the Hermite semigroup-a survey. Proceedings of the conference dedicated to Professor Miguel de Guzmbn (El Escorial, 1996). J . Fourier Anal. Appl. 3 (1997), Special Issue, 813-823. 8. D. Di Giorgio, Sums and commutators of generators of noncommuting strongly continuous groups, J. Math. Anal. Appl. 275 (2002) 165-187. 9. P. Malliavin, Hypoellipticity in infinite dimension, In: “Diffusion processes and related problems in analysis,’’ Vol. 1 (M. Pinsky, Ed.), Birkhauser, 1989, pp. 17-31. 10. B. K. Driver and T. Lohrenz, Logarithmic Sobolev inequalities for pinned loop groups, J. Funct. Anal. 140 (1996), 381-448. 11. M. Gordina, Holomorphic functions and the heat kernel measure on an infinite-dimensional complex orthogonal group, Potential Anal. 12 (2000), 325-357. 12. Gordina, Maria Heat kernel analysis and Cameron-Martin subgroup for infi-
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nite dimensional groups, J . Funct. Anal. 171 (2000), 192-232.
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QUANTUM STOCHASTIC DILATION OF SYMMETRIC COVARIANT COMPLETELY POSITIVE SEMIGROUPS WITHUNBOUNDEDGENERATOR Debashish Goswami Stat-Math Unit, Indian Statistical Institute, 209, B. T. Road, Kolkata-35, India, email:
[email protected] Kalyan B. Sinha J.N. Centre for Advanced Scientific Research, Bangalore, India, email: kbs-jayaQyahoo.co.in This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or C* algebra, here with unbounded generators. The additional assumption of symmetry with respect to a semifinite trace allows the use of the Hilbert space techniques, while the covariance gives rise to better control on domains. An Evans-Hudson flow is obtained, dilating the given semigroup. Keywords: completely positive semigroups, quantum stochastic dilation
1. Introduction
The problem of constructing quantum stochastic dilation for quantum dynamical semigroups (q.d.s. for short) is an interesting and important problem from physical as well as mathematical viewpoints. Physically, quantum dynamical semigroups are often taken as mathematical models for the timeevolution of open quantum dynamical systems, and a (quantum stochastic) dilation to a flow of homomorphic maps may be interpreted as the construction of a bigger system, by adding some random noise to the original one, so that the bigger system is reversible, i.e. conservative. In a series of by us and other collaborators, and the book Ref. 4, we achieved a complete theory of such dilation for q.d.s. with bounded generators. There the computations involved C* or von Neumann Hilbert modules, using the results of Christensen and Evans,5 map-valued quan-
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tum stochastic processes on modules and stochastic integration with respect to them.6>7It is then natural to consider the case of a C P semigroup with unbounded generator and ask the same questions about the associated stochastic dilations. As one would expect, the problem is too intractable in this generality and we impose some further structures on it, viz. we assume that the semigroup is symmetric with respect t o a semifinite trace and covariant under the action of a Lie group on the algebra. This additional hypothesis enables us t o control the domains of the various operator coefficients appearing in the quantum stochastic differential equations so that the Mohari-Sinha conditions8 can be applied. The covariance is exploited again as in the work by Chakraborty and US,^ along with the assumption that the crossed-product von Neumann algebra A X G is isomorphic with the von Neumann algebra generated by A and the representation ug of G in the GNS Hilbert space associated with the trace, to obtain the structure maps, and finally the Evans-Hudson (EH) flow is constructed essentially along the lines of the proofs in our previous work.' As precursors of this work, we may mention those in Fagnola and Sinhag and Accardi and Kozyrev." While the first one deals with a general EH flow with unbounded structure maps under some additional hypotheses, the second one treats the problem in a different spirit. We also refer the reader to our work" for the theory of Hudson-Parthasarathy (H-P) dilation under the assumptions of smoothness and symmetry. A closely related but slightly different theory of E-H dilation has been presented in the book Ref. 4, where the assumption of symmetry is relaxed, but compactness of the group and a stronger version of smoothness (called 'complete smoothness') has been imposed. 2. Preliminaries
Let A be a separable C*-algebra and r be a densely defined, semifinite, lower semicontinuous and faithful trace on A. Let A, E {z : ~ ( x * x<) m}. Let h = L'(T),and A is naturally imbedded in B(h). We denote by d the von Neumann closure of A with respect to the weak topology inherited from B ( h ) . Clearly A, is ultraweakly dense in d. Assume furthermore that G is a second countable Lie group with (xi,i = 1, ...N ) a basis of its Lie algebra, g H as E Aut(d) a strongly continuous representation. Suppose = T ( Z * Y ) for x E A,,y E A, g E G (by that ag(d,)C A, and r(ag(z*y)) polarization this is equivalent to the assumption that T ( C X , ( Z * X ) ) = T ( ~ * Z ) for x E A, ). This allows one t o extend as as a unitary linear operator (to be denoted by u g ) on h and clearly as(%)= u,xu; for x E A. It is indeed easy to verify this relation on vectors in A, and then it extends to
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the whole of h by the fact that h is the completion of A,. For f E C p ( G ) (i.e. f is smooth complex-valued function with compact support on G) and an element 3: E A, we denote by a ( f ) ( x )the norm-convergent integral J , f ( g ) a , ( x ) d g , where d g denotes the left Haar measure on G.
Lemma 2.1. g +-+ ug is strongly continuous with respect t o t h e Hilbertspace topology of h.
Proof :Let A1 z {x E Al~(lx1)< m}. It is known that A1 is dense in h in the topology of h. Furthermore, for z E A, and y E A1, I T ( ( u g ( I c ) - x ) * y ) ] I II(ug(z)- x ) * 1 1 ~ ( l y l ) ,which proves that g H T ( ( c x ~ ( I ~ ) - ~is) continuous, * ~ ) by the strong continuity of a with respect to the norm topology of A. But by the density of A1 and A, in h and the fact that ug is unitary, we conclude that for fixed E E h, g H ugc is continuous with respect to the 0 weak topology of h, and hence is strongly continuous. The above lemma allows us to define a (f ) ( l= ) J f ( g ) u , ( t ) d g E h for f E C r ( G ) , JE h. Furthermore, from the expression ag(x) = ugxu;, it is possible to extend as to the whole of B ( h ) as a normal automorphism group implemented by the unitary group ug on h and we shall denote this extended automorphism group too by the same notation. Let A, = {x E A : g H ag(x) is infinitely differentiable with respect to the norm topology}, i.e. A, is the intersection of the domains of tli1ai2...tlik; k 2 1, for all possible i l l i2, ... E {1,2, ...iV}, where tli denotes the closed *-derivation on A given by the generator of the one-parameter automorphism group a,xp(tXi) , where e x p denotes the usual exponential map for the Lie group G. The following result is essentially a consequence of the results obtained by Garding,12 Ne1s0n.l~ Proposition 2.2. (i) A, is a dense *-subalgebra of A. (ii) If w e denote by d k t h e self-adjoint generator of t h e u n i t a r y group u,xp(tXk) o n h s u c h t h a t u,,.(~~~) = eitdk, a n d let h , = ,,,,Dorn(di,di, ...di,;k = 1 , 2 , ...), t h e n h , is dense in h. (iii) If w e equip A, w i t h a f a m i l y of n o r m s JJ. l l, , n ; n = 0 , 1 , 2 , ... g i v e n by:
nil,i2
fo r n 2 1, a n d 1 1 ~ 1 1 ~ , = 0 11x11, and similarly define a f a m i l y of Hilbertian n o r m Il.llz,n; n = 0 ,1,2, ... on h, by:
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il
,iz,...ik;k
o n h,, then A, and h, are complete with respect t o the locally convex topologies induced by the respective (countable) family of norms as defined above. I n other words, A, and h, are Frechet spaces in the topologies (to be called '(Frechet topologies" f r o m now o n ) described above. (iv) ag(A,) C A,, ug(h,) C h, f o r all g E G. Furthernlore, g +-+ a g ( x )g, c--) us(()are smooth (C") in the respective Frechet topologies f o r x E A,,( E h,. (v) Let A,,, = A, h,. I t is a *-closed two-sided ideal in A, and is dense in A, A,, h and h, with respect t o the relevant topologies.
n
Proof :The proof of (i) and (ii) will follow immediately from the references cited before the statement of this proposition. The proof of (iii) is quite standard, which uses the fact that di,di's are closed maps in A and h respectively. Next we indicate briefly the proof of (iv) for A, only, since it is similar for h,. First of all, by the definition of A, and the fact that G x G 3 ( g I , g 2 ) H g1g2 E G is C" map, we observe that for x E A, the map (g1,g) a g , ( a g ( x )= ) a g l g ( x )is Cm on G x G, hence in particular for fixed g , G 3 g1 H agl(as(x)) is Coo,i.e. a g ( x )E A,. Similarly, for fixed x E A, and any positive integer k, the map F : Rk x G 4 A given by F(t1, . . . t k , g ) = a , , p ( t l X i l ) . . . e z px( )~isi kC". ~k~ Byg (differentiating F in its first k components a t 0, we get that dil...&,(ag(z))is C , in g . To prove (v), we need to note first that the elements of the form a(f ) ( ( ) , with f E CF(G) and 5 E A, are clearly in A,,,. Let us first consider the density in h and h,. Since the topology of h, is stronger than that of h and since h, is dense in h in the topology of h, it suffices to prove that the set of elements of the above form is dense in h, in the Frechet topology. For this, we take ( E h,, and choose a net x, of elements from A, which converges in the topology of the Hilbert space h to (, and then it is clear that a ( f ) ( x , ) -+ a ( f ) ( ( )Vf E CF(G) with respect to the Frechet topology of h,, since di ,...d i , a ( f ) ( x , - E ) = ( - l ) k a ( ~ i , . . . ~ i k f )( xE v) . Thus, it is enough to show that { a ( f ) ( ( )f , E C F ( G ) , ( E h,} is dense in h, in the Frechet topology. For this, we choose a net f p E C r ( G ) such that f p d g = l V p and the support of f p converges to the singleton set containing the identity element of the group G, and then it is simple to see that a(f p ) ( ( ) + ( in the Frechet topology. Finally, the norm-density of
s,
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A,,, in A and the Frechet density in A, will follow by similar arguments. 0
Remark 2.3. It may be noted that for x E A,,,, ai1...aik(x)= di, ...di, (x)E A n h. This follows from the fact that if yp is a net in A h which converges both in the norm topology of A as well as in the Hilbert space topology of h, then the norm-limit belongs t o h and the two limits must coincide as vectors of h. Now we shall introduce some more useful notation and terminology and prove some preparatory results. If 7-1 is any Hilbert space with a strongly continuous unitary representation of G given by U,, we denote by 7-1, the intersection of the domains of the self-adjoint generators of different one-parameter subgroups, just as we did in case of h. We denote the corresponding family of “Sobolev-like” norms again by the same notation as in case of h and consider 7-1, as a Frechet space as earlier. We call such a pair (7-1, U,) a Sobolev-Hilbert space and for two such pairs ( K ,U,) and (K,V,), we denote by B(7f,, K,) the space of all linear maps S from 7-1 to K such that 7-1, is in the domain of S, S(7-1,) K,, and S is continuous with respect t o the Frechet topologies of the respective spaces. We call a linear map L from 7-1 to K to be covariant if 7-1, C Dom(L) and LU,(O = V,L(EWg E Gl t E 7-1,.
Lemma 2.4. If L fromIFI to K is bounded (in the usual Hilbert space sense) and covariant in the above sense, then L E B(K,, Km). Proof :Let d y and d f be respectively the self-adjoint generator of the one parameter subgroup corresponding to xi in 7-1 and K. From the relation LU, = V,L it follows that (since L is bounded) L maps the domain of d y into the domain of d f and Ldy = dFL. By repeated application of this argument it follows that Ldz ...d c ( < ) = d: ...d k L ( t ) @ E ’ N,, and thus
llLt112,n I IILIIIIE112,n. We shall call an element of B(’,Jf,l K,) a “smooth”map, and if such a smooth map L satisfies an estimate IILEl12,n 5 Clltl12,n+pfor all n and for some integer p and a constant C , then we say that L is a smooth map of order p with the bound 5 C . From the proof of the above lemma we observe that any bounded covariant map is smooth of order 0 with the bound 5 11L11. By a similar reasoning we can prove the following :
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Lemma 2.5. Suppose that L is a closed (an the Hilbert space sense)), covariant m a p f r o m 'H to K and x , is in the domain of L . Under these assumptions, L is smooth of the order p for some p .
P r o o f : For simplicity of notation, we shall use the same symbol di for both d y and d f , and also we use the same symbols for the corresponding one parameter groups of unitaries acting on 'H and K.Let L be a map as above. Since L is closed in the Hilbert space sense, and the Frechet topology in If, is stronger than its Hilbert space topology, it follows that L is closed as a map from the Frechet space H ', to the Hilbert space K, and being defined on the entire If,, it is continuous with respect t o the above topologies. By the definition of Frechet space continuity, there exists some C and p such that llL(J)llz,~ 5 C11J11~,~. Now, for any fixed k , let ut = u,,p(tXk).Since ut maps h, into itself and L is covariant, we have that L( = ut(L$)-Lc. Now, since + dk(<) as t O+ in the Frechet topology, we have that L( u t ( < ) - E ) = ..t(LE)--LE it converges to LdkJ in the Hilbert space topology of zt Ic, and so by the closedness of dk LJ must belong to the domain of d k , with LdkJ = d k L t . Repeated use of this argument proves that L('H,) K, and L ( d i , ...di,J) = di ,...dik(LJ)VJ E H '., Now, a direct computation enables one to show that L is of order p with the bound 5 C . 0
q)
---f
Theorem 2.6. Let ('H, U g ) ,( K ,V,) be two Sobolev-Halbert spaces as in earlier discussion), and L be a closed (not as Frechet space m a p but as Hilbert space m a p ) linear m a p from If to K . Furthermore, assume that H ' , is in the domain of [LIZand as a core for ILlZ, and Lug = V,L o n 3-1,. Then we have the following conclusions :
(i) L is a smooth covariant m a p with some order p and bound 5 C for some C ; (ii) L' (the densely defined adjoint in the Halbert space sense ) will have K , in its domain; (iii) L" is also a smooth covariant m a p f r o m Ic, to H ;', with order p and bound 5 C as in (a).
Proof :Let the polar decomposition of L be given by L = WILI. We claim that both W and ILI are covariant maps. First we note that If, is also a core for L (being a core for lL12) and since Ug is a unitary operator that maps H ', into itself, clearlyH ' , is a core for Lug and also for V,L. Thus the relation Lug = VgLon If, implies that the operators Lug and VgL have the same domain and they are equal. Now, note that L being closed and
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V, being bounded, we have that (V,L)* = L*Vl = L*V,-I. Furthermore, since U,-' maps the core 3-1, for L into itself, one can easily verify that ( L u g ) *= u;L* Thus, we get that U,L* = L*V,Vg. It then follows that Ug(Ll2= IL12Ug and hence by spectral theorem U, and ILI will commute. By Lemma 2.5, we get that ILI(3-1,) 2 X,, and ( L (is a smooth covariant map of some order. Now, if P denotes the projection onto the closure of the range of lLl, then P clearly commutes with U, for all 9,hence in particular U,Ran(P)I C Ran(P)'. Thus WU,PL = WP-'U, = 0 = V,WPL. On the other hand, V,WP = WU,P, because V,WlL = V,L = LU, = WlLlU, = WU,ILI. Hence we have that W is a bounded covariant map, and thus by 2.4, it follows that W* is covariant too, and in particular W*(K,) F'1, C Dom(lLI),so that K, Dom(L*) = Dom(lLIW*).Furthermore, from the fact that W and W * are smooth maps of order 0 with bound 5 1 (as 11 WII = IIW*ll = 1)and ILI is a smooth covariant map of some order p with bound 5 C for some C , clearly both L = WlLl and L* = ILIW* are smooth covariant maps of order p and bound 5 C, which completes the proof. 0
Lemma 2.7. Let ('Hi, U i ) ,i = 1,2 and (Xi, v,Z),i= 1 , 2 be Sobolev Hilbert spaces and k be any Hilbert space. T h e n we can construct Sobolev Halbert spaces ('Hi@ Ki,U i @ Vg) and (3-1% @ k , U i @ I ) (with the symbols carrying their usual meanings) and if L E f3(3-11w,'H2,), M E 1 3 ( K 1 w l K ~ - ) ,then we have the following : (2) L @ M E f3((3-11@ Kl),, ( f i z @ K z ) , ) , and (ii) (3-11 @ k ) , is the completion of 3-11,&lgk under the respective Frechet topology and the map L galgI o n 3-11, galgk extends as a smooth map o n the respective Frechet space (we shall denote this smooth m a p by L @ I or sometimes 2). Furthermore, i f L is of order p with some constant C , so will be 2.
Proof : (i) is straightforward. To prove (ii), we fix any orthonormal basis { e l } of k and let E = C &@ el be a vector in the domain of the self adjoint generator of the one parameter unitary group ut @ I , where ut is as in the proof of Lemma 2.5 and the summation is over a countable set since = 0 for all but countably many values of 1 . So, without loss of generality we may assume that the set of 1's with ('1 nonzero is indexed by 1 , 2 , .... Since 8 el is Cauchy (in the Hilbert space topology) suppose that C u t(( E l ) -7 El )@ el -+ Cql @ el. Clearly, for each 1 , ql =
x(w)
l i m t - + o ( ~ which ) , implies that
E D o m ( d k ) and d,&
= ql.
Thus, if
182 D. Goswami and K. B. Sanha
d;, denotes the self
adjoint generator of the one parameter unitary group
ut @ I , then we have proved that the domain of it consists of precisely the vectors C3 el such that each E Dom(&) and Ildk(&)I12 < ca. Repeated use of this argument enables us to prove that (El 8 k), consists
ctj
of the vectors E = C 51 8 el with the property that ( 1 E E l a V l and for any n, \\<\\g,n = El < 00. From this, it is clear that CEl &@elconverges (as m --t 00) to ( in each of the II.I12,n norms, i.e. in the Frechet topology. The rest of the proof follows by observing that for any $, = CfiniteJlbel E 7-ll,m&$,
c
llmll;,n = llGlll;,n. 0
3. Assumptions on the semigroup and its generator Let Tt be a q.d.s. on A which is .r-symmetric (i.e. .r(Tt(z)y)= .r(zTt(y))for all positive z,y E A, and for all t 2 0). We refer the reader to the article Ref. 14 for a detailed account of such semigroups from the point of view of Dirichlet forms. We shall need some of the results obtained in that reference. As it is mentioned in that reference, Tt can be canonically extended to a normal .r-symmetric q.d.s. on d as well as to Co-semigroup of positive contractions on the Hilbert space h. We shall denote all these semigroups by the same symbol Tt as long as no confusion can arise. Furthermore, we assume that Tt on d is conservative, i.e. Tt(1) = 1Vt 2 0. Let us denote by C the C' generator of Tt on A, and by C2 the generator of Tt on h. Clearly, C2 is a negative self-adjoint map on h. We also recall14 that there is a canonical Dirichlet form on h given by, ~ o m ( v=) ~ o m ( ( - ~ z ) +q(u) ) , = / / ( - ~ 2 ) ~ ( u ) l l & , Eu ~ o m ( 7 )w. e recall from Ref. 14 that B := A n Dom(v) is a *-algebra, called the Dirichlet algebra, which is norm-dense in A. We now make the following important assumptions : Assumptions: ( A l ) Tt is covariant, i.e. Tt commutes with ag for all t 2 0 , g E G. (A2)C has A, in its domain, (A3) C2 has h, in its domain.
v
The assumption that Tt is covariant in particular implies that Tt leaves A, invariant, hence by Lemma 8.1.19 of the this domain is a core for C,and clearly agC = Lag on A,. It follows that (by arguments similar to those in the proof of 2.2) C(A,) A,. Similarly, &(h,) C_ h,. Since the actions of C and C2 coincide on A,,,, one has that C(A,,,) C Am,T.
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Furthermore, we have the following :
Lemma 3.1. Am,, is stable under the action of Tt and hence is a core f o r both C2 and L . The proof of the lemma is straightforward and hence omitted. By the Lemma 3.1, h, is also a core for Cz, as d,,, C h,. It is important to remark here that the assumption A3 is the only hypothesis on the generator of the semigroup which involves the generator at the L2level, not the norm generator. However, we shall later on see that in an important special case, where the group G is compact and acts ergodically on the algebra d, the assumption A3 will follow automatically from the other hypotheses. Modifying slightly the arguments of the articled4 and15 (see also4), we describe the structure of C.
Theorem 3.2. (i) There is a canonical Hilbert space K equipped with an A-A bimodule structure, in which the right action is denoted by (a,<) H
<
<
B. (iii) For a , b E A,,,, Ildo(a)bllx: 5 Callb112,0, where Il.lln: denotes the Hilbert space norm of K , and C, is a constant depending only o n a. Thus, for any fixed a E A,,,, the map d,,r 3 b H &&(a)b E K extends to a unique bounded linear map between the Hilbert spaces h and K , and this bounded map will be denoted by 6(a). (iv) dC(a,b, c ) 3 6(a)*7r(b)6(c)= C(a*bc) - C(a*b)c- a*C(bc)
+ a*C(b)c,
for a , b, c E A,,r. (v) K is the closed linear span of {S(a)b : a, b E d,,,}. (vi) n extends as a normal *-homomorphism o n A.
Proof : We refer for the proof of (i) and (ii) t o the papers Sauvageot15 and Cipriani and S a ~ v a g e 0 t . Now, l ~ we note that d,,, is contained in the “Dirichlet algebra” (c.f. Cipriani and Sauvageot14) and in fact is a formcore for the Dirichlet form r] mentioned earlier. Using the calculations made
184 D. Goswami and K. B. Sinha
in the proof of Lemma 3.3 of [14, page 81, we see that for a , b E A,,,, 1 -7(-b*C(a)*ab - b * U * C ( U ) b b*L(a*a)b). 2 Here, we have also used the fact that a,a*,a*a E Dorn(L). From the above expression (iii) immediately follows. We verify (iv) by direct and straightforward calculations, which we omit. To prove (v), we first recall from14 that K can be taken t o be the closed linear span of the vectors of the form bo(a)b,a , b E B. Now, by Lemma 3.3 of the article,14 Il6o(a)bll; 5 ~ ~ b ~ ~ ~ , oSince ~ ( a A,,T , a ) . is on one hand norm-dense in A and also form core for v on the other hand, (v) follows. Let us now prove (vi). It is enough to prove that whenever we have a Cauchy net a, E A,,, in the weak topology, then (<, 7r(a,)<) is also Cauchy for any fixed belonging to the dense subspace of K spanned by the vectors of the form 6(b)c,b, c E A,,,. But it is clear that for this, it suffices t o show that a H (S(b)b’, 7r(a)b(b)b’)is weakly continuous. Now, by the symmetry of L and the trace property of T , we have that for a E A,,,,
Ilso(a)bll;
+
=
<
(S(b)b’, 7r(a)6(b)b’) =
( b , abL(b‘b’*))- ( b , UL(bb’b‘*)) - (L(bb’b’*),ab)
+ (C(bb’(bb‘)*),a )
The first three terms in the right hand side are clearly weakly continuous in a , so we have to concentrate only on the last term, which is of the form ~ ( L ( z z * ) for a ) z E Am,,. Now, we have,
+
+
.r(L(zz’)u) = 7(C(z)z*a) 7 ( z L ( z * ) a ) -r(S(z*)*b(z*)a), and since L(A,,,) A,,,, the first two terms in the right hand side of the above expression are weakly continuous in a , so we are left with the term T ( ~ ( z * ) * ~ ( x * )Let u ) .us choose an approximate identity e, of the C* algebra A such that each e, belongs to A, (this is clearly possible, since A, is a norm-dense *-ideal, and for z E A,, one has that IzI E A,). By normality of T , T ( s ( z * ) * s ( z * ) )= sup,T(e,S(z*)*S(z*)e,) = 2sup,1160(z+)enIIg I 2~up,Ile,Il~,~v(z*,z*) < 00, since Ile,Il,,o 5 1 and x* E Am,, C Dorn(7). Thus, S(z*)*6(z*)= y2 for some y E A,, hence ~(6(z*)*6(x*)a) = ~(yay), which proves the required weak continuity.
0 Now we obtain the Christensen-Evans type form of the generator L. Theorem 3.3.
Let R : h
4
K be defined as follows :
D ( R ) = A,,,,
RX 3 &SO(Z).
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Then R has a densely defined adjoint R*, whose domain contains the linear span of the vectors G(x)y, x , y E d,,, and R * ( G ( X ) Y ) = 4 Y > - L(.>Y - L(xCy).
We denote the closure of R by the same notation R. For x , y
E
A,,,,
1 1 (R*T(x)R- -R*Rx - - x R * R ) ( y ) = C ( X ) Y . 2 2
Furthermore, ~ ( z ) Y= (RE- ~ ( ~ ) R ) ( YY )E, Am,,, X,
1 C2 = --R*R. 2
Proof :For x , y, z E A,,,, we observe by using the symmetry of C that (S(X)Y,Rz) = 2(6O(X)Y,GO(Z) = T ( Y * c ( Z * Z ) - y*c(X*)Z - y * Z * c ( Z ) ) = .(C(y*)x*z -
( C ( z ) y ) * z- C ( x y ) * z )
= ({.C(Y) - C(Z)Y - C ( w ) } , z ) .
This suffices for the proof of the statements regarding R*.It can be verified by a straightforward computation that ( R * r ( x ) R - ~ R * R-x ; x R * R ) ( y ) = C ( x ) y holds for x , y E d,,,. The remaining statements are also verified in a straightforward manner. 0
4. Evans Hudson type dilation We now study sufficient conditions for the existence of Evans-Hudson (EH) dilation for the q.d.s Tt considered in the previous section. Let denote the generator of the one-parameter group d 3 x H u ~ ~ x uwith ; ~ ,respect to the operator norm topology of A, where gt = exp(tXi), i.e. x E d beU g t X U i t -x longs to the domain of 8, if and only if converges as t 4 O+ in the operator norm topology of d inherited from B ( h ) . Let A, denote the intersection of the domains of 8Jl...8ik for all possible choices of i l , ...ak. w e shall make the following additional assumptions (either A4,A5, A6,A7; or A4,A5,A6,A7’) on the algebra and the group action: A4: do = { x E A , : 3Cz,CL > Os.t.llxll,,n 5 CLCz V n } is norm-dense in d A5: ho = { u E h, : 3Mu,ML > O s . t . l l ~ l l 25 , ~ MLMz V n } is L2-dense in
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h; A6: The canonical homomorphism of the crossed-product von Neumann algebra d > a G onto the weak closure of the *-algebra generated by {A, us;g E G} in B ( h ) is an isomorphism; and either A7: Assume that the trace r is finite (hence A, coincides with A,,T) and that d , = A,, i.e. any element of d which belongs t o the intersection of all the domains (“smooth”), then it must belong to A. or A7‘: The trace r is not finite, and in this case we assume that {d, ug;g E G}” is a type I factor isomorphic with B ( h ) . We note that A4 and A5 hold for G compact or G = R“. For compact groups, the verification of A4,A5 can be done by the Peter-Weyl decomposition, and for R“, the space {f : f^ E C r ( R n ) } can be taken as a candidate for both A0 as well as ho. Similar conclusions will be valid for more interesting noncompact Lie groups. Lemma 4.1. Under the assumption (A6), it follows that A‘ XG is isomorphic with {A’,us;g E G}”.
Proof
: The proof is an easy consequence of the Tomita-Takesaki modu-
lar theory. Let J denote the closed extension of the canonical anti-linear isometric involution of that theory which sends x E A,,T to x*. Since ug(x) = a,(.) for 5 E A,,,., and ag is *-preserving, it is clear that u g J = J u g and thus (since we also have J 2 = i d , ) J u J = u g . But we know from the Tomita-Takesaki theory that JdJ = A’, hence the von Neumann algebra B = { d , u , ; g E G}” is anti-isomorphic under the map 3.3 with C = {A‘,u,;g E G}”. But by assumption A6, B is isomorphic with d X G , which is nothing but the von Neumann algebra generated by d@l and ug@Lgin B ( h @ L 2 ( G ) ) ,where L, is the left regular representation. Clearly d > a G is anti-isomorphic with A’ x G , under the obvious anti-isomorphism ( J . J ) @ i d . Thus A’ XG is isomorphic with C, since the composition of two anti-isomorphism is an isomorphism. 0
Theorem 4.2. Under the assumption A 6 above, there exist a Halbert space ko, a partial isometry : h@ko -+ h@ko and a closed linear map ?!, from h into h@ko with h, in its domain, such that the followings hold : (i) C is covariant, i.e. C ( u g @ i d )= (ug@id)2; (ii) E(x) f %(x@lko)g* is a covariant normal *-homomorphism of d into B(h@ko), which is structural in the sense of Goswami-Sinha,’ i.e.
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E ( d ) C d@B(ko) and we also have E ( a g ( x ) ) ( u g @ i d=) (u,@id)E(x); (iii) from h to h@ko is smooth covariant map (with respect to the same Gaction as in (i)),i.e. s u , = (ug@id)s; and d(x) = $x--E(x)s for x E extends as a bounded map from h to h@ko which is also structural, i.e. d ( x ) E d@ko, x E A,,,, and covariant in the sense that d ( a g ( x ) )= y,(d(x)), where yg : d @ k o -+ d @ k o by yg(.) = (u,@id).(u;@id).; (iv) L ( x ) = $*E(x)$ - :$*sx - ax$*$ f o r x E A,,,, in the sense that the LHS is a bounded extension of the RHS, which is defined o n its natural domain. (v) Furthermore, if we also make the assumption A7, then we have a stronger structure relation in the following sense : For x E A, and 5, r] E ko, < €.,d(x) > E A, and
s
Proof: First of all we proceed in the line of the proof of [4, Theorem 8.1.21, page 202-2041, and consider the covariant normal *-homomorphism 7r as in that theorem and lift it t o a normal *-homomorphism ?i of the crossed product, which is isomorphic by the assumption with { d , u , ; g E G}”, and thus we take 7i to be a normal *-homomorphism on { d l u g ; gE G}” satisfying 7i(x) = ~ ( z for ) x E d and 7i(ug)= V,. Then the construction as in [4, Theorem 8.1.211 ensures that there exist some Hilbert space kl and isometry C from K t o h@kl (where K is as in the section 3) such that C*(x@lk,)C = 7i(z),C*(u,@lkl)C = V,. Let K1 = C(K) C h @ k l , and let PI = CC* and 61(x) = C d ( x ) , x E A,,,. Clearly {61(x)w,x E A,,TIv E h } is dense in K1. As in Ref. 3, we now construct a normal *homomorphism 7r’ of A’ in B(K1) by setting 7r1(a)8l(x)w= 61(x)av, and extending by linearity and continuity (details can be found in Ref. 3 or Ref. 1).We extend this d ( a ) on the whole of h@kl by putting it equal to 0 on P t ( h @ k l ) , and we denote this trivial extension also by d ( a ) . Clearly, 7r’ is covariant with respect to g H (u,@l), and thus we can extend 7r’ to a normal *-homomorphism of {A’,us;g E G}” (which is isomorphic with the crossed product von Neumann algebra A’ xG), say T ” , satisfying 7r”(ug)= u g @ l k l . Hence there exist a Hilbert space k2 and a partial isometry C1 : h@kl -+ h@k2 such that C;(a@lk,)C1 = P1(a@lkl) and C;(ug@lk2)C1 = P ~ ( U , @ ~We I~C take ~ ) ko . = k1 @ k2, 2 = C1 @Olhg,kz : h@ko = ( h @ k l )@ (h@k2) --+ h@ko, and Sv := ~ ( C R @ V 0). The remaining is verified as in the paperI3 and hence the details are omitted. Finally, t o prove (v), we first note that since % , is covariant and bounded
188 D. Goswami and K. B. Sinha
(in Hilbert space sense),
.
Ugt
< E 7 4 4 T I > U l t =< El?(%7t(z))TI>,
(where gt = e z p ( t x i ) , for some fixed i ) and thus we have that for z E A,,
rt
which goes t o 0 in norm as ii is a norm-contractive map. This shows that < 6 , ii(z)q > belongs to the domain of & for any i, and repetition of similar arguments proves that it belongs t o d, = A,. We can prove similar fact about 8 by using the covariance of 8. We first note that C,being covariant, norm-closed and having A, in its domain, is continuous in the Frechet topology. Using the Frechet continuity of C and the cocycle identity
8(2)*8(z) = L(x*x)- C ( x ) * x- z * L ( x ) for x E A,, (by Assumption A 7 = A,) we conclude that 8 : A, -+ B(h,h@ko) is continuous with respect to the Frechet topology on the domain and the operator norm topology on the range. Thus, if we denote by Q the map A, 3 z 4 (E,8(z)) (for fixed E E k o ) , then Qj : A, 4 B ( h ) is continuous with respect to the Frechet topology of A, and the norm topology of B ( h ) . Since a g t c ~ ) - z -+ &(x) (as t -+ 0+) in the F’rechet topology for z E A,, it follows that
as t + O+ in norm, i.e. Q ( x ) E Dorn(&) with & ( Q j ( x )= ) Q ( & ( x ) ) .Repeated use of arguments proves that Q ( x ) E d, = d,, which completes the proof. 0
We make the assumptions Al-A7(or A l - A7’) from now on for the rest of the section and proceed t o prove the existence of an Evans-Hudson dilation. Our first step is to introduce a map 0 , as in the work of Goswami and Sinha,’ which combines all the “structure maps” L , 8, ii into a single map. Let 3 C @ ko and 0 : + B(h@&) given by
where it(,)
= 8(x*)*and
a(z) = ii(z) - ( z @ l k , )
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Lemma 4.3. There exist a Hilbert space kh and two covariant smooth maps B : h@& -+ h@kh, A : h@kh 4 h&d& such that O(z) = A(x@lk;)B,where covariance is with respect to the representation g H U g @ ' l k & in all cases.
Proof : We take kh = g0@C3=
f3
@
& @ &. Let
A = ( A l , A z , I ) , B=
, where Ail Bj s are covariant smooth maps from h@&to itself given
by (with respect to the direct sum decomposition h@& = h @ (h@ko)): A i = ( 0 S*2 - ) , A Z = ( - lS*' ~ , B1 = AT, B2 = A;. We note 0 -c s -ilh@ko that all the maps above are defined with their usual domains and from the results of the previous sections (since 2 is a bounded covariant map and S is smooth covariant, and satisfies the condition of 2.6, so that its adjoint is smooth covariant too, and furthermore composition of smooth covariant maps is again smooth covariant) it follows that A and B are indeed smooth covariant. That O(z) = A(z@lk&)B can be verified by direct and easy 0 computation.
)
We now extend the definition of the map 0 , taking advantage of the fact that ( A @ I )and ( B @ I )are also smooth covariant maps with the same order and bounds as A and B respectively where I denotes identity on any separable Hilbert space with trivial G-action (see Lemma 2.7 ).
Definition 4.4. For any two Hilbert spaces X 1 , X 2 and a smooth (not necessarily covariant) map X from the Sobolev-Hilbert space (h@X1,ug@l)to (h@'Hz,ug@l) we define O ( X ) to be a smooth map from ( h @ & @ X l , u g @ l @ l to ) (h@&1@'H2, ug@l@l),given by,
@(x)= ( A ~ 1 7 - r , ) P ~ 3 ( X @ 1 k 6 ) P 2 3 ( B @ 1 ~ ~ ) i where P23 : h@kh@?ll -+ h@X1@kl, denotes the operator which interchanges 2nd and 3rd tensor components, and a similar definition is given for Pi3.
It is clear that for ' H I = X 2 = C, we indeed recover the definition of O ( x ) for 5 E A,,,, and furthermore we make sense of the same symbol even for x which are not necessarily bounded operator on h, but are smooth map on h with respect to the G-action given by ug. The above extended definition enables us t o compose 0 with itself, i.e. we can make sense of O ( O ( X ) )and so on. We shall denote O(...O ( X ) ) (n-times composition) by
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W ( X ) for X as in the definition 4.4. The following estimate will be a fundamental tool for proving the homomorphism property of the EH dilation which we are going to construct. Theorem 4.5. For x E A,, 6 E (h@60n),, (where 60" denotes n-fold tensor product of 60 with itself, and G-action is taken to be ug@lion) we have that
of G ) , C2 = c 1 c 2 ( 2 f i ) P 1 , c1,c2,p1,p2 are such that A is a smooth map of order p l and bound 5 c1, and B is a smooth map of order p2 and bound 5 cq. I n particular, let x E &, [n = V@Wl@...@Wn with v E ho, wi E LO, llwi112,o 5 K for some K , and let V be a smooth covariant map of order 0 and bound 5 1 o n (h@60n@K'), (where K' is some separable Hilbert space, with trivial G-action) and r] E K'. Then we have that II(On(X)@1)V(En@r])ll2,o 5 11r]112,oK1K~ f o r some constants K2, K2 depending on x,v,K only. Furthermore, f o r x , y E A0 and v,V,r] as above, we get constants K i , K; depending on x , y , v only such that II((W(~)*@~(y))@l)V(v@r])Il2,o 5 Ilr]l12,0KiK;n Proof: First we note that for x E A, and v E h,,
we have that
JN)m+ 1 II x II,m II II 2
I I x u II 2 ,m 5 &2
(2) . To prove this estimate we note that for any fixed k-tuple i l , ...i k with k 5 m, where each i j E { 1, ...N } , we have the following: 21
,m
C
~ ~ d i ~ . . . d i ~ ( x=~ II) ~ ~ E , o ~ J ( x P J ~ ( ~ ) I I ; , ~ ~ J C { I , ...k }
where for any subset J of (1, ...k } , we denote by d J the map aipl...aipt, with pl < p2 < ...pt being the arrangement of the elements of J (I JI = t ) in the increasing order; and a similar definition is given for d p , J" being the complement of J . Clearly, for any 1 nonnegative numbers al,...al, one has that (a1 ...ul)' 5 21(a: ...a;) and using this we see that
+
+
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191
The proof of the estimate then follows by noting that the number of possible k-tuples as above is N k , and Cr=o 22k+1Nk = "-((4N)"+' 4N-1 - 1). Now we come to the proof of the main estimate in the present theorem. It is easy to see that the estimate (2) holds even when z E A, is replaced by ( ~ 8 1 % for ) any separable Hilbert space 'FI and w E h, is replaced by v E (h@'FI), where 'FI carries the trivial G-representation. Hence we have
II @ n ( 4 c I 1 2 , m IC~(I(~@1)~23(B@'1)...~23(B~l)Jl(z,m+np1 5
, / Z ( 24fN l )-"1' + " P ( + '
I I ~ l l m , m + n PIl I ( ~ ~ ~ ) . . . ~ 2 3 ( ~ ~ ~ ) 1 1 1 2I , m + n P l
from which the desired estimate follows. (Note that P 2 3 and 1 are used here as generic notation, without distinguishing among different spaces on which they act, and also we have made use of the fact that P 2 3 , Pi3 are trivially smooth covariant maps of order 0 and bound 5 1.) The last two assertions of the result follows by noting that for z E do,'u E ho, we have that ))zl),,npl+m 5 M:plfm and a similar estimate is valid for w , and furthermore we have that l l V ( c , @ ~ ) I l ~ , ~ ( ~ ~5+ l 1 4 1 2 , n ( p l + P z ) + r n ~ n llr1112. 0 We now prove an important algebraic property of 0.
Lemma 4.6. For z, y of the f o r m < [, O n ( a ) > ~ f o r some a E d,,<,q E A
&In
(h@ko ,,)
we have that
+ (z@l)O(y)+ O(z)QO(y),
0 ( z y ) = O(z)(y@l)
where Q =
(0,
Ih@ko
) as in the paper.'
Proof :We extend the definition of iiin the obvious manner to make sense of ii(z) for any z E B(h,, h,) so that).(fi is also a smooth map. However, this extended iineed not be a homomorphism on whole of the B(h,, h,). For proving the lemma, it suffices to verify that ii(z9) = ii(s)if(y) for z, y
..&In
of the form < (,On( a ) q > for some a E d,,c,r] E (h@iko .,) We now consider the cases corresponding to the assumptions A7 and A7' separately. In the first case, we have already shown that < c , W ( a ) >E ~ A, for a , 1 , as ~ in the statement of the lemma, and then it is trivial to see that the above algebraic relation holds, since iiis indeed a homomorphism
~ ~ ) +
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D. Goswami and K . B. Sinha
on A,. Now, we consider the case when A?' is assumed. In this situation, the proof of the theorem 4.2 enables us t o see that k l , k 2 , ko (in the notation of the theorem mentioned above) can be chosen in such a way - that PI = I h B k l , and hence C*C = I h B k l @ O h B k z . But then it follows that B(h,, h,) 3 x H F(z) = C ( x @ I h B k l @ I h B k z ) % * is a homomorphism. This completes the proof of the required algebraic property of 0 in the present case. 0
C,s,
Theorem 4.7. Let ko be as in the statement of theorem 4.2. Then the ko)) admits a contraction-valued following operator q.s.d.e. in h@l?(L2(R+, solution.
a$(dt)
+
- ag:,S(dt)-
Furthermore, V, is covariant with respect to the G-action ug@l on [email protected](L2(R+,k o ) ) . Proof : The existence and uniqueness of the solution V, can be obtained essentially as in [4,Theorem 8.1.23, pages 204-2061 by verifying the MohariSinha conditions for solution of q.s.d.e. with unbounded coefficients as in Mohari-Sinha' (see also Ref. 16). Covariance of V, is straightforward t o show. It is important to note that V, is not unitary since 2 is a strict partial isometry. Now, let us recall the map-valued quantum stochastic calculus developed by us in Ref. 1 and also in the book Ref. 4. We want t o define integrals of the form Y, o M ( d s ) where M stands for (a,(ds) aA(ds) R,(ds) I c ( d s ) ) (see notations in the work of Goswami-Sinha'), where 5 ( x ) = ?(z) - ( z @ l k o ) . The ideas and construction will be almost the same as those in Goswami-Sinha,' and hence we only briefly sketch the steps, omitting details. We shall adopt the following convention throughout the rest of the paper : unless otherwise stated, G-action on a Hilbert space of the form h@K' will be taken t o be ug@l. In our article,' the integrator 0 was a norm-bounded map, and thus (I? = the above integral could have been defined on the whole of d@l? I'(L2(R+,ko))).But here we are dealing with unbounded 0, and thus we shall make sense of the above integral only on a restricted domain. Let D denote the algebraic linear span of the elements of the form x @ e ( f ) , with x E B(h,, h,), i.e. x is a smooth map from h , to itself, f is a bounded continuous ko-valued function on [O,w),and let S denote the
+
Jot
+
+
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193
space B(h,, (h@I')w).Let (Y,)t>o - be a family of maps from 2, t o S , with the adaptedness in the obvious sense as in the work of Goswami and Sinhall and also satisfying the following condition (an analogue of 3.12 of the paper'):
for some Hilbert space N', where r 1 , q are smooth maps between appro-priate spaces, in contrast t o their being bounded in the work of GoswamiSinha.' We call such a process Y, regular, and we define 2, = Y, o M ( d s ) exactly in the same way as in the paper Ref. 1. Indeed, O f ( s ) ( x (with ) the already given definition of 0, and notations as in the article Ref. 1) is a densely defined closable operator on h, with h, in its domain, and also < f ( s ) ,0 ( x ) g ( s ) > clearly is an element of B(h,, h,) (i.e. smooth map), hence the conditions required by Goswami-Sinhal for defining the maps S_(S)l T ( s ) by S ( S > ( 4 f s ) ) = J?s(6ijc)@e(fs))v and T ( s ) ( v e ( g s ) @(s)) f = K ( 5 ( z ) f ( , ) @ e ( g s ) ) was in the paper1 (with similar notation) are satisfied. Then an analogue of 11, Proposition 3.3.51 can be applied to 2, (see Cor. 2.2.4 (ii) of the paper Ref. l), and we conclude that 2, is well-defined and regular. However, in the present situation Z t ( x @ e ( f ) ) need not be a bounded operator from h to h@r,and need not belong t o d@I'.Nevertheless, a t least in case A7 is assumed, we shall show that the EH flow Jt to be constructed by us will map d@I' to itself.
Ji
Theorem 4.8. W e set j t : d -+ B(h@I') by jt(x) = K(x@lr)V,* (where vt as in theorem 4 . 7 ) and also we define Jt : dhalgI' B(h,h@I') by Jt(x@e(f ) ) w = j t ( x ) ( v e (f ) ) and extending by linearity. Furthermore, the above definition of j t ( x ) can be extended to x E B(h,, h,) (i.e. for x which are possibly unbounded as Halbert space map) and Jt : 2, -+ S also similarly. T h e n we have the following : (i) Jt is a regular process and satisfies the q.s.d.e. dJt = J t o M ( d t ) ;Jo = id; (ii) j t : A -+ B(h@I') is a normal covariant (with respect to cr,@id) *-homomorphism which dilates Tt in the sense of Goswami, Sinha and Chakrabarty; and (iii) I f A 7 (and not A?) i s assumed, then jt(A) C d@B(I'),and < e ( f ) , j t ( + ( f ' ) >E -4, f o r x E A,, f , f' E L2(R+,ko). ---f
' 1
Proof: Since and V,* are smooth covariant of order 0, the regularity (even in the sense of Goswami-Sinha') of Jt easily follows, and thus J s o M ( d s )
6
194
D. Goswami
and K.
B. Sanha
makes sense. Furthermore, by It6 formula and the q.s.d.e. satisfied by Vt, it is simple to verify that Jt satisfies the required map-valued q.s.d.e. This proves (i). For proving (ii), our strategy is as in the paper Ref. 1. We define a family of maps @t as follows. First we extend the definition of Jt to make sense of J t ( X @ e ( f ) )for X E B(h,,(h@I'f,),) where rf, is the free Fock space over & (see Ref. l), by setting J t ( X @ e ( f ) ) w = (Vt@lr,,)P23(X~lr)V,*(we(f)) and then we define for fixed u,w E ho and f,f' being bounded continuous ko-valued functions on R+, Q t ( X ,Y ) =< J t ( X @ e ( f ) ) uJ, t ( Y @ e ( f ' ) ) v > - < u e ( f ) , J t ( X * Y @ e ( f ' ) ) v>, for all X , Y E B(h,, (h@rf,),) satisfying that X * E B((h@I'f,),, h,) (and thus X * Y E B(h,,co)), and in case it is A7 ,not A7', the assumption taken by us, we also require that for any vector p in r f , < p, X >, < p,Y > E A,. With this definition, we now fix x , y E A0 too. We identify I&" as canonically embedded subspace of rf, and it is clear that for w E LO",O"(z), is a smooth map with its adjoint being smooth too (which follows from the explicit structure of Q(.) = A(.@l)B, where A , B are smooth covariant, hence have smooth adjoints). Thus, for any n, and w, w' E i o n , @t(On(x),, O"(y),t) is well-defined and furthermore one can easily verify relation [l,(3.18)] with X = W ( x ) , , Y = O"(y),, by using Lemma 4.6. We can now iterate this relation arbitrarily many times, and by noting the estimate ( 1 ) of Theorem 4.5, (and also by using the fact that ll@t(X,Y)II I Ilve(f)Ilz,oll(X*Y@l)(V,*(ue(f')))llz,o}, and V,. is a smooth covariant map of order 0, bound 5 1) conclude that @t(x,y) = 0 for z, y E Ao. This proves the weak homomorphism property of j t . Since jt(x) is by the very definition in terms of V, is a bounded map for all z E d and lljt(z)II 5 11x(1,,0, the strong homomorphism property follows. The covariance and other properties of j t as in (ii) of the statement of the present theorem are straightforward to see. Finally to prove (iii), first of all we show that jt(d)C_ d@B(r).For this, we construct iteratively J P ' , by setting J,"'(z@e(f))v = ( z @ e ( f ) ) v and , JP+" = J s ( n ' o M ~ ( d sfor ) n 2 0, and furthermore define J I ( x @ e ( f ) ) v=
+
6
En
J,'"'(x@e(f))v, for all x,w such that the above sum is convergent in the Hilbert space sense. One can verify that for z E A o , E~ha, J l ( x @ e ( f ) ) w exists and satisfies the same q.s.d.e. as Jt with the same initial condition, hence by the standard iteration argument used to prove the uniqueness of solution of q.s.d.e., it follows that J l ( x @ e ( f ) ) vequals J t ( z @ e ( f ) ) w . Now, ( u @ l ) J P ' ( x @ e ( f ) ) v= J i " ' ( ( x @ e ( f ) ) u v for z E d o , v E ho and
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195
a E A' of the form J'yJ' for some y E do, where J' is the anti-unitary operator of Tomita-Takesaki theory mentioned before. Clearly such an a maps h, into h, and also J'doJ' is strongly dense in A'. From this we obtain that ( a @ l ) J t ( x @ e ( f ) ) v= J t ( x @ e ( f ) ) a vfor x , a , v as before. Note that we needed x E &,v E ho for showing the summability of J,'"'(&e(f))v.Thus jt(x) commutes with all ( a @ l ) , aE J'&J'; and since j t ( x ) is bounded operator, the same holds for all a E A'. This proves that j t ( d 0 ) G d@B(I'), and then due to the normality and boundedness of the map x H j t ( x ) , the same thing will follow for all x. Now, take x E d". Since Vt is covariant contractive map, we can easily verify that < e ( f ) , j t ( z ) e ( f ' )> E d, for z E A,, f , f ' E L2(k0),, and hence by the assumption A7, < e( f ) ,j t ( z ) e (f') > E A,, which completes the proof. 0 5. Applications and examples
In this section we shall show that it is indeed possible t o accommodate many interesting classical and noncommutative semigroups in our framework. First of all we prove that the assumption A6 regarding crossed product is valid in a typical classical situation.
Theorem 5.1. Let X be a locally compact separable Hausdorff space with a regular Bore1 measure p o n it, and let a locally compact group G act o n X freely and transitively, and the measure p is G-invariant. Then the von Neumann algebra L,(X,p) satisfies the assumption A6, i.e. the crossed product von Neumann algebra L " ( X , p ) X G is isomorphic with the weak closure of the *-algebra generated by L " ( X , p ) and ug in f 3 ( L 2 ( X , p ) ) , which is in fact the whole of B ( L 2 ( X , p ) ) ,where ug denotes the unitary representation of G in L 2 ( X , p )induced by the G-action o n X .
Proof: Let xo be any point of X . It is clear that the bijective continuous map G 3 g H gxo E X is a homeomorphism (because G is locally compact and X is Hausdorff, so that any continuous bijection from G to X is automatically a homoemorphism). Let us denote L" ( X ,p ) and L2( X ,p ) by d and h respectively. We recall that d > a G can be defined to be the von Neumann algebra generated by f @ lfl E d and ug@Lg,gE G in B(h@L2(G))(where L, is the left regular representation). So, the commutant of this von Neumann algebra is the intersection of d @ B ( L 2 ( G ) )and {ug@Lg,gE G}' (since d is maximal abelian). But d@B(L2(G)) can be identified with the direct integral of copies of B ( L 2 ( G ) )over ( X Ip ) . In this direct integral picture, we can view any element B of d@B(L2(G)) as a
196
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D. Goswami and K. B. Sinha
measurable map B : z B ( z ) E B(L2(G)); and then it is easy to see that B also commutes with all ug@Lgif and only if B(gz) = L;B(z)Lg ’dx,g.Thus, the map B(.) is determined by the value of B(.) at any one point of X (since the action is free and transitive). It is now easily seen that (d > a G ) ’ is isomorphic with l@B(L2(G)).To verify this, we denote the inverse of the map g ++ gzo (which is a homeomorphism as noted earlier) by \I, and consider the unitary U on h@L2(G)given by ( U 4 ) ( z , g ) = +(z,Q(z)g), for 4 E h@L2(G)E L 2 ( X x G). Clearly, for any B E (d xG)’, i.e. B ( z ) = LzB(zO)Lg,with g = @(z),we have that UBU* = 1@B(zo),and since B(z0) can be allowed to be an arbitrary element of B(L2(G)),we have shown that U ( d >aG)’U* = (1@B(L2(G))), hence U ( A >aG)U* = B(h)@l. Now it is enough to prove that the von Neumann algebra generated by d and u g , g E G in B ( h ) is the whole of B ( h ) ,which is clear. 0
Remark 5.2. 1. The assumption of transitivity is not so crucial and can be weakened to the assumption that the bundle ( X , X / G , r ) (where T : X + X/G is the canonical quotient map) is trivial (which will be true in particular whenever X / G is contractible). Now, we proceed to give classical examples where our theory works. Let G be a separble Lie group, equipped with an invariant (with respect to the action of G on itself) Riemannian metric, A be the C*-algebra of continuous functions on G vanishing at LLoo”,and let G act on itself by the left regular action, which is trivially free and transitive, so that A6 will be valid. In case G is compact, we check A7 by using the fact that any element of L2(G) which is almost everywhere differentiable and all the partial derivatives are again in L2 obviously belongs to the Cm-class.For noncompact G, A7’ follows by Theorem 5.1. The Laplace-Beltrami operator on G commutes with the isometry group associated with the G-invariant Riemannian metric, and hence the heat semigroup generated by it satisfies the covariance and symmetry (with respect to the left Haar measure) conditions. As far as the assumptions A4, A5, A6, A7 (or A7’) needed for the EH dilation theory are concerned, we have already verified A6,A7 (/ A7’). However, one has to verify A4 and A5 case by case, and in many cases (e.g. compact groups, R” etc.) these can indeed be verified. In the context of Rn, we now discuss two interesting classes of q.d.s. First, we consider the expectation semigroup of a diffusion process, such that the generator C is of the form C(f) = - Cijaijaiajf for smooth f with
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compact support, where ai denotes the i-th partial derivative, and aij are smooth functions, with the matrix ( ( a i j ) )being pointwise nonsingular and positive definite, and 2 H ( ( u i j ( z ) ) ) - *is a smooth bounded function. If aij are non-constant functions, then the above L is not covariant with respect t o the action of translation group; however, we now show how t o choose a different group acting on R" such that L can be written as LO+SO for some covariant 2nd order operator LO and a derivation SO. To achieve this, we change the canonical Riemannian metric of R" and equip it with a different metric given by < ail > = b i j ( z ) ,where ( ( b i j ( z ) ):= ) ((aij(x)))--I. Let us denote by X the Riemannian manifold R" with this new metric and let G be the group of Riemannian isometries of X. It is easy to verify that if we choose LO t o be the generator of the heat semigroup on X, then Lo is G-covariant and symmetric with respect to the Riemannian volume measure, and moreover L is indeed the same as LO up to some first order operators, which can be written as &(.) for some suitable closed derivation SO which (under some further assumptions on a i j ) may generate a 1-parameter automorphism group of the underlying function algebra. In such a case, one can apply the present results t o Lo to construct dilation, and then use standard techniques to obtain the dilation of exp(tL) from the dilation of etLo using some standard perturbation techniques. However, we must point out here that one needs to verify A1-A? case by case. A sufficient condition for verifying these assumptions is that there is a nice and large enough Lie subgroup of G which acts freely and transitively on X , (which in particular will imply that X is a Riemannian homogeneous space) and such that A, and h, will coincide as sets with those as in the case of Rn with the action of itself. For the simple case when n = 1, we can show the existence of such a subgroup by direct computation which gives us an explicit description of the group of isometries of X. We can also give examples from noncommutative geometry where our theory gives E-H dilation of q.d.s. on a noncommutative C*-algebra. Example 1: Fix irrational number 8 and consider A = As, the irrational rotation algebra which is generated by two unitaries U and V satisfying the commutation relation U V = eZKieVU.We take the T2-action on A given by CY(,~,~~)(U :=~zVr z"g)U m V n for m,n E Z,21,z2 E T.The so-called heat semigroup coming from the canonical noncommutative geometric structure on A is given by T,(UmVn) = e - t ( m 2 f n 2 ) U m V ,n and it is easy t o see that this q.d.s. satisfies the assumptions of our theory. The trace r with respect t o which Tt is symmetric is the canonical trace of de given by r ( U m V n ) = 0
a,
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if ( m 1 n )# (O,O), and r(1)= 1.
Example 2 (counterexample) : In this example, we do not quite have all the assumptions of our theory satisfied, but we do have the covariance and symmetry. However, some of the technical conditions regarding the crossed product von Neumann algebras are violated. Nevertheless] one can obtain a dilation by other techniques] namely those discussed in Refs. 4 and 11. We consider noncommutative 2ddimensional plane described in [4, Chapter 91, which we briefly explain here. Let (Ua)aERd and (Vfi)pEp denote the d-parameter groups of unitaries on L 2 ( R d )given by
(Uaf)(t) = f(t + a ) , (Vfif)(t) = exp(it. P)f(t), for f E C r ( R d )and where For f E C r ( R 2 d ) ,define
b(f) :=
. denotes the Euclidean scalar product of Rd.
1
f(z)W,dz E B(L2(Rd)),
W2d
where W, = U,lV,zexp(-~zl . zz), for z = (z1,52) E Let A be the C*-subalgebra of B ( L 2 ( R d ) )generated by {b(f), f E Cr(R2d)}.There is a canonical action of R2d on A given by
4z(b(f)) := b(f3))I where f, is defined by f,(t) = exp(it. z ) ( t ) .We denote by S j , j = 1,...)2d the derivations obtained by differentiating at 0 the canonical one-parameter subgroups of 4,. Thus, Sj is given by S j ( b ( f ) ) = b ( a j f ) , where 8, is the partial derivative with respect to the j - t h coordinate of R2d. The functional r given by r ( b ( f ) ) = f(0) for f E CF(R2d)extends to a faithful semifinite trace on A. We claim that the q.d.s. (Tt)generated by the ‘Laplacian’ 6; of R2dl and it is also symmetric is covariant with respect to the action with respect to the canonical trace r on the noncommutative 2d-plane. To verify the covariance] we observe the following:
x;tl
Moreover] we have, T (Tt (b(
f)* )b(u))=
which proves symmetry.
Lzd
e- f
T(z)G( z)dx = r (b( f )* T‘ (u))
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However, the assumptions A6 and A?’ are not valid for this example. To see why A7‘ fails, let us observe that the C*-algebra A is nothing and the trace coincides with the usual trace. Thus, in the but IC(L2(Rd)), of our theory, the GNS Hilbert space h is L2(Rd)@L2(Rd), with 2 notation being B ( L 2 ) @ 1 ,and the RZd-action on h is given by u, = L x l @ L x 2where , x = ( x l , ~ xi ) , E Rd and L, denotes the unitary operator of translation by y on L2(Rd).So, the von Neumann algebra generated by 2 = B(L2(Rd))@l and {u,,xE R2d} is isomorphic with
B(L2(Rd))@{L,, y E W d } t t zB(L”Rd))@P(Wd), which has nontrivial centre and so is not a factor. Thus, this example does not fit into the framework of the present paper, and we refer to Refs. 4,11 for the construction of a dilation (through a Hudson-Parthasarathy q.s.d.e.) for Tt.
Acknowledgment Debashish Goswami’s research is partially supported by the INSA project on ‘Noncommutative Geometry and Quantum Groups’. Kalyan Sinha’s research is partially supported by the Bhatnagar Project of C.S.I.R., India.
References 1. D. Goswami and K. B. Sinha, Hilbert modules and stochastic dilation of a
quantum dynamical semigroup on a von Neumann algebra, Commun. Math. Phys. 205, no. 2(1999), 377-403. 2. D. Goswami, A. Pal and K. B. Sinha, Stochastic dilation of a quantum dy-
namical semigroup on a separable unital C* algebra, In& Dim. Anal. Quan. Prob. Rel. Topics 3,no. 1 (2000), 177-184. 3. P. S. Chakraborty, D. Goswami and K. B. Sinha, A covariant quantum stochastic dilation theory, Stochastics in finite and infinite dimensions, Trends Math., Birkhauser Boston, Boston, MA (2001), 89-99. 4. D. Goswami and K. B. Sinha, “Quantum Stochastic Processes and Noncommutative Geometry”, Cambridge Tracts in Mathematics 169, Cambridge University Press, Cambridge, UK (2007).
5. E. Christensen and D. E. Evans, Cohomology of operator algebras and quantum dynamical semigroups, J . London Math. SOC.( 2 0 ) (1979), 358-368.
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6 . R. L. Hudson and K. R. Parthasarathy, Quantum Ito’s Formula and Stochastic Evolutions, Commun. Math. Phys. (93)(1984),301-323.
7. K. R. Parthasarathy, “An Introduction t o Quantum Stochastic Calculus”, Monographs in Mathematics, Birkhauser Verlag, Bessel (1992). 8. A. Mohari and K. B. Sinha, Stochastic dilations of minimal quantum dynamical semigroup, Proc. Indian Acad. Sc. ( Math. Sc. ), 1 0 2 (3)(1992), 159- 173. 9. F. Fagnola and K. B. Sinha, Quantum flows with unbounded structure maps and finite degrees of freedom, J . London Math. SOC.( 2 ) 4 8 (no. 3 ) (1993), 537-551. 10. L. Accardi and S. Kozyrev, On the structure of Markov flows. Irreversibility, probability and complexity(Les Hoches/Clausthal, 1999); Chaos, Solitons and Fractalsl2:14-15(2001),2639-2655. 11. D. Goswami and K. B. Sinha, Dilation of a class of quantum dynamical semigroups, Communications on Stochastic Analysis 1 (no. 1 ) (2007), 87101. 12. L. Garding, Note on continuous representations of Lie groups, Proc. Nut. Acad. Sci. U.S.A. 33(1947), 331-332. 13. E. Nelson, Analytic vectors, Ann. of Math. ( 2 ) 7 0 , 1959, 572-615. 14. F. Cipriani and J-L. Sauvageot, Derivations as Square Roots of Dirichlet Forms, J . Funct. Anal. 2 0 1 ( 1 ) (2003), 78-120.
15. J-L. Sauvageot, Tangent bimodule and locality for dissipative operators on (?*-algebras, Quantum Prob. and Applications IV, Lecture Notes in Math. 1396(1989), 322-338. 16. F. Fagnola, On quantum stochastic differential equations with unbounded coefficients, Probab. Th. Rel. Fields, (86)(1990),501-516.
201
WHITE NOISE ANALYSIS IN THE THEORY OF THREE-MANIFOLD QUANTUM INVARIANTS Atle Hahn
Institut fur Angewandte Mathematik der Universitat Bonn Wegelerstrape 6, 531 15 Bonn, Germany E-mail: [email protected] The heuristic Chern-Simons path integral has played an important role in Mathematical Physics in the last two decades. In particular, the so-called Wilson loop observables have attracted a lot of attention since they provide a fascinating bridge between Quantum Field Theory and Knot Theory. It is an open problem if and how one can make rigorous sense of the heuristic path integral expressions in terms of which the Wilson loop observables are given. In the present paper we consider a closely related problem, namely the problem of finding a rigorous realization of the modified path integral expressions for the Wilson loop observables which arise after a suitable gauge fixing has been applied. In two recent papers1-2 we proposed a strategy for the solution of the latter problem which is based on “torus gauge fixing” and white noise analysis. In the present paper we will fill in several technical details which were omitted in Ref. 2. In particular, we will make some of the informal claims in Ref. 2 more explicit by formulating them as mathematically rigorous conjectures.
Keywords: Chern-Simons models; Quantum invariants; White noise analysis
1. Introduction In 1988 E. Witten succeeded in defining, on a physical level of rigor, a large class of new 3-manifold and link invariants with the help of the heuristic Chern-Simons path integraL3 Later Reshetikhin and T u r a e ~ ~found -~ a rigorous definition of these “Jones-Witten (quantum) invariants” which is based on the theory of quantum groups. Unfortunately, it is very difficult to relate this quantum group versiona of the Jones-Witten invariants t o classical topology and geometry, which might explain why these invariants have not proved to be as useful for making progress towards the solution of some of the classical open problem in topology as many people had hoped acalled “Jones-Reshetikhin-Turaev-Witteninvariants” in the literature
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(cf. the introductions in Refs. 7 and 8). It would therefore be desirable to find a rigorous “path integral definition” of the Jones-Witten invariants, i.e. a definition which is obtained by making rigorous sense of Witten’s path integral expressions, either before or after a suitable gauge fixing has been applied. It is reasonable to expect that for such a rigorous path integral definition of the Jones-Witten invariants the relation to classical topology & geometry would be considerably more explicit. The results in Refs. 1,2,9 suggest that at least for special manifolds M of the form M = C x S1, for which “torus gauge fixing”b is available, it should indeed be possible to find a rigorous path integral definition of the JonesWitten invariants in the framework of white noise analysis. Since the main emphasis in Refs. 2,9 was on explicit computations rather than conceptual completeness some arguments and claims in Refs. 2,9 were kept rather informal. The aim of the present paper is to give fully rigorous versions of some of the main claims in Refs. 2,9 by formulating them as rigorous conjectures. The paper is organized as follows: In Sec. 2 we recall the relevant heuristic formulas for the Wilson loop observables in Chern-Simons theory, first the original formula before a gauge fixing has been applied (cf. Eq. (1) in Subsec. 2.1) and later the modified formula which was obtained in Refs. 2,13 by applying torus gauge fixing in the special case where M = C x S1 (cf. Eq. (15) in Subsec. 2.3). In Sec. 3 we then reformulate & extend the approach in Refs. 1,2,9 for making rigorous sense of Eq. (15) in such a way that several informal claims in Refs. 2,9 can be formulated as rigorous conjectures. 2. The heuristic Chern-Simons path integral in the torus gauge 2.1. Chern-Simons models
Let G be a simply-connected compact Lie group and g its Lie algebra. Moreover, let T be a maximal torus of G with Lie algebra t. Without loss of generality we can assume that G is a Lie subgroup of U ( N ) ,N E N,so G c u(n)c Mat(N, C). Let M be an oriented compact 3-manifold and A the space of smooth g-valued 1-forms on M . The Chern-Simons action function SCS associated bthis gauge fixing was first used in the context of Chern-Simons theory in Refs. 10-12 where the partition function of Chern-Simons models on these manifolds was studied
White Noise Analysis an the Theory of %manifold Quantum Invariants
to M , G, and the “level” k E
203
N is given by
with Tr := c . T r ~ ~ t ( N where , @ ) the normalization constant c is chosen‘ such thatd T r ( H . H ) = -2 if H E it c Mat(N, C ) is a long (complex) coroot. For example, if G = S U ( N ) then c = 1 so in this case Tr coincides with nMat(N,@).
From the definition of Scs it is obvious that SCS is invariant under orientation-preserving diffeomorphisms. Thus, at a heuristic level, we can expect that the heuristic integral (the “partition function”) Z ( M ) := Jexp(iScs(A))DA is a topological invariant of the 3-manifold M . Here DA denotes the informal “Lebesgue measure” on the space A. Similarly, we can expect that the mapping which maps every sufficiently “regular” colored link L = ( ( 1 1 , 1 2 , . . . l n ) , ( p l , p2, . . . p n ) ) in M t o the heuristic integral (the “Wilson loop observable” associated to L )
is a link invariant (or, rather, an invariant of colored links). Here Trfi is the trace in the representation pi, and P e x p ( h i A ) denotes the holonomy of A around the loop li. 2.2. Some notation related to G and T
For G , g, T , and t as above we introduce the following notation. - Let
(., .) denote the Ad-invariant scalar product on g c U ( N )given by (A, B ) = -&Z
nMat(N,@)(A ’ B,
and I . I the corresponding norm. With the help of (., .) we can identify t with t* in the obvious way. - t ise the (.,.)-orthogonal complement of t in g and 7rt : g 4 t is the ( . l -)-orthogonal projection. - cg will denote the dual Coxeter number of g. - We set I := ker(explt) c t and denote by I* c t’ the lattice which is dual to I . =such a normalization is always possible because by assumption g is simple so all Adinvariant scalar products on g are proportional to the Killing metric dHere “.” is, of course, the standard multiplication in Mat(N, C ) enote that in1!2,9 we use the notation go instead of e
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A . Hahn
- We set beg := exp-l(T,,,)
where T,,, denotes the set of “regular” elements of T, i.e. the set of all t E T which are not contained in another maximal torus T‘ # T . Moreover, let us fix a connected component P of beg(in other words: P is a “Weyl alcove”).
2.3. Torus gauge fixing applied to Chern-Simons models
During the rest of this paper we will set M := C x S1 where C is a closed oriented surface. Let 7rx (resp. TSI) denote the canonical projection C x S1 4 C (resp. C x S1 --+ Sl).For every curve c in C x S1we set cc := 7rx o c and CSI := 7rs1o c. Moreover, we will fix an arbitrary point 00 E C and set t o := 1 E s1. By Ax (resp. Ax,t)we will denote the space of smooth 0-valued (resp. tvalued) 1-forms on C. will denote the vector field on S1 which is induced by the curve is1 : [0,1] 3 t H earit E S1 c C and dt the 1-form on S1 which is dual t o We can lift and dt in the obvious way to a vector field resp. a 1-form on M , which will also be denoted by resp. dt. Every A E A can be written uniquely in the form A = A’ Aodt with A* E A* and Ao E C”(M, 0) where A’ is defined by
&
&
&.
+
A’
:= {A E
&
A I A(&) = 0)
We say that A E A is in the “T-torus gauge” if A E A’ @ {Bdt
(2)
1
B E
C”(& t>>. By computing the relevant Faddeev-Popov determinant one obtains for every gauge-invariant function x : A 4 C (cf. [l,Eqs. (4.10a) and (4.10b)l)
s,
X(APA
/ [LL
X(A*
1
+ Bdt)DAL
C-Gt)
x A ( B )det(idc - exp(ad(B))le)DB (3)
+
where A ( B ) := Idet(a/at ad(B))I. Here DA’ denotes the (informal) “Lebesgue measure” on dl and D B the (informal) “Lebesgue measure” on C”(C, t). In the special case where x(A) = TrPi ( P exp(h, A)) exp(iScs(A)) we then get
ni
WLO(L)
x A ( B )det(ide - exp(ad(B))le)DB (4)
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
205
Above and in the sequel denotes equality up to a multiplicative constant independent of x resp. L. Observe that N
Scs(A
I +Bdt) =
&
lM
[Tr(AL~dA1)+2Tr(A'r\Bdt~A')+2
+
n(Al~dB~dt)]
Bdt) is quadratic in A' for fixed B , which means In particular, Scs(A' that the informal (complex) measure exp(iScs(A* Bdt))DA' appearing above is of "Gaussian type". This increases the chances of making rigorous sense of the right-hand side of Eq. (4)considerably.
+
So far we have ignored the foIlowing three "subtleties": 1)When one tries to find a rigorous meaning for the informal complex measure e x p ( i S c s ( A l B d t ) ) D A l (or the corresponding integral functional) appearing in Eq. (4) above one encounters certain problems which can be solved by introducing a suitable decomposition d l = d' @ d:, which we will describe now (for a detailed motivation of this decomposition, see Sec. 8 in' and Sec. 3.4 in2): Let us first make the identification d l Z C"(S1,dc) where C"(S1,dc) denotes the space of all "smooth" functions a : S1 -+ dc, i.e. all functions Q: : S' -+ with the property that every smooth vector field X on C the function C x S1 3 (c, t ) +-+ Q:(t)(X,)is smooth. Then we set
+
d' := {A' d,I := {A'
C"(S1,dc) 1 nAc,t(A'(to)) = 0}, E C"(S1,dc) I A' is constant and dc,t-valued} E
(5)
(6)
: d c 3 dc,t is the projection onto the first component in the where decomposition d c = dc,t @ Ac,e. Using
Scs(A'
+ A: + Bdt) = S c s ( A l + Bdt) + 2",
and setting dfi$(A')
:= -J--- exp(iScs(A' Z(B)
exp(iScs(A'
ffor step (*) and the definition of here, see,12
L
+ Bdt))DA'
Tr(dA,I . B )
(7)
wheref
+ Bdt))DA-1(5)I d e t ( g + ad(B))I-1/2 -
& +ad(B): C w ( S 1 , A ~ )
+
Cw(S1,A~ appearing )
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A . Hahn
we obtain
WLO(L)
(8)
A more careful analysis shows that in the formula above one can replace t by begor, alternatively, by the Weyl alcove P . This amounts to including the extra factor l c - ( ~ , ~ ,or , , )lc-(z,p) in the integral expression above. 2) Naively, one might expect that we have
&+ &+
since the operator ad(B) in the numerator is defined on C,”(C x S1) and the operator ad(B) in the denominator is defined on C”(S1, dz). However, the detailed analysis in Sec. 6 of” suggests that, already in the simplest case, i.e. the case of constantg B 3 b, b E P , the expression det (ide - exp(ad(B)IF)) n [ B ] Z ( B )
(10)
should be replaced by the more complicated expression (det(idc - exp(ad(b)le)))x(z)/2x e x p ( z g
Tr(dAb . b))
(11)
where x ( C ) is the Euler characteristic of C. In2 we suggested that in the special situation where the set DP(L) of “double points”h of L is empty and where the set C\(U, arc(l&))has only finitely many connected components R1, Ra,. . . , R,, m E N the expression (10) for arbitrary B E C”(C, t) should be replaced by det,,,(ide
- exp(ad(B)le)) x exp(zg]z Tr(dA$ . B ) )
(12)
gwhich is the only case of relevance in” hp E C is a “double point” of the link L if there are (at least) two different x , y E arc(Zi) with p = T C ( Z ) = ~ c ( y )
Ui
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
207
where det,,,(ide
- exp(ad(B)lt)) :=
Here we have fixed an auxiliary Riemannian metric g := gc on C. pg denotes the Riemannian volume measure on C associated t o g and BE(a0) is the closed €-ball around 00 with respect t o g . 3) If one studies the torus gauge fixing procedure more closely one finds that - due t o certain topological obstruction^^^^^^^^ - in general a 1-form A can be gauge-transformed into a 1-form of the type A' Bdt only if one uses a gauge transformation R which has a singularity in a t least on point. Concretely, in Ref. 2 we worked with gauge transformations R of the type = Qsmooth . Rsing(h) E Cm((Z\{a~})x S1, G ) with a s m o o t h E C"(C x S1,G) and Rsing(h) E C"(C\{CJO},G)c Cm((C\{a0}) x S 1 , G ) where a0 E C is the point fixed above and where the parameter h is an element of [ C , G / T ]i.e. , a homotopy class of mappings from C to G / T . O,ing(h) is obtained from h by fixing a representative g(h) E C"(C, G / T ) of h and then lifting the restriction g(h)lc\(go}: C\{ao} 4 G / T t o a mapping C\{ao} 4 G. The use of the singular gauge transformations OSing(h)gives rise t o an extra summation ChEIC,G/Tl and to extra terms
+
Ah,g(h) := nt(Rsi,g(h)-ldRsi,g(h)),
(14)
(observe that Ai,,(h) is t-valued 1-form on C\{ao}). Accordingly, we will in Eq. (8) above and we will replace A: include a summation ChEIC,G,T1 by A: AAng(h) (for a detailed description and justification see2?l3).
+
Taking into account the three points above we obtain the following heuristic formula, which will be the starting point for our considerations in the next section
lc-
( C , P )( B )
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A . Hahn
where
Remark 2.1. The mapping n : [C,G / T ]--+ t given by
is independent of the special choice of the Riemannian metric g, fixed above, and of the special choice of g(h) and OSing(h),cf. Ref. 13. Moreover, n is a bijection from [C,G / T ] onto I = ker(exp,,). In particular, we have
I h E [C,G/TI) = I
(17)
3. Towards a rigorous realization of the r.h.s. of Eq. (15)
In order to make rigorous sense of the path integral expression appearing on the right-hand side of Eq. (15) we will proceed in four steps:
Step 1: We make sense of the integral functional Step 2: a) We make sense of the inner integrals
. . .d j i i
+
for all fixed A: E , ' d B E C"(C,t), h E [C,G/T].Here P e x p ( h . Ak+AA,,(h)+Bdt) denotes the function d'- 3 'A H P e x p ( h J A ' - + A: A;, (h) Bdt)) E Mat(N, C). b) We compute IL(A:, B;h) explicitly Step 3: We make sense of the integral functional
+
+
for a suitable subspace V c A: x B. Step 4: We make sense of the total expression on the right-hand side of Eq. (15) and compute its value.
Remark 3.1. One could try to postpone part b) of Step 2 to Step 4, making sense of the full r.h.s. of Eq. (15) first before one begins to evaluate any terms explicitly. At least in the special case where DP(L) = 0 this should indeed be possible, which would mean that in this special case we
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
209
really have a full rigorous path integral representation for the WLOs. On the other hand, postponing part b) of Step 2 t o Step 4 would lead to expressions that are even more complicated than those appearing below. In order to maintain the readability of the present paper at an acceptable level we will use the structure given above.
Step 1 In Sec. 8 in' we gave a rigorous implementation of the integral functional . . d f i i as a generalized distribution on a suitable extension d' of d' using a similar strategy as in the context of Chern-Simons models on &t3.
s.
i) First we chose a convenient Gelfand triple
( N ,' H N , n/*) and set dL:=
Jv. More precisely, we chose'
N
:= A'(equipped
with a suitable family of semi-norms)
(s',d t )
'FIN := L k C
(2)
(3)
wherd := L2-l?(Hom(TC,g),pg),i.e. Xx is the Hilbert space of L2sections of the bundle Hom(TC,g) w.r.t. the measure pg and the fiber metric on Hom(TC, g) E T*C @ g which is induced by g and (., .). Using "second quantization" and the Wiener-Ito-Segal isomorphism F O C k y m(
XN)
Lc(nr* ,W * )
where "/N* is the standard Gaussian measure on N* we then obtained a new Gelfand triple ( ( N )L, g ( f l , Y N * ) ,( N ) * ) . ii) Next we evaluated the Fourier transform 3fii of the "measure" fii a t a heuristic level obtaining
'at first look it would be more natural t o choose
N
:= d l ( e q u i p p e d with a suitable family of semi-norms)
However, it turned out in' t h a t it is considerably more convenient t o work with the slightly larger (test function) space N = d l jIn other words: 7 - t is ~ the space of 'HE-valued (measurable) functions on S' which are square-integrable w.r.t. d t . Observe that 'HE is a rather natural extension of d c and L k c (Sl,d t ) = HN is a rather natural extension of Cm(S1, d E ) d l 3 dl
210
A . Hahn
where m ( B )is the heuristic LLmean” and C ( B )the heuristic ‘‘covariance operator” of the Gauss-type heuristic measure iii.In Ref. 1 it was shown how one can make sense of m ( B ) as a well-defined element of 7 - l and ~ of C ( B )as a (well-defined) continuous symmetric linear operator N 4 XN. After doing so we obtained a well-defined continuous function U : N -+ C. given by ~ ( = j )exp(i << j , m ( ~~ )7
<< j , C ( B ) j>xN)
- l exp(-$ ~ )
(4)
for every j E N. iii) It is straightforward to show that the function U : N 4 C. is a “Ufunctional” in the sense of Refs. 17,18. In view of the Kondratiev-PotthoffStreit Characterization Theorem (cf. again Refs. 17,18)the integral functional @$:= J . . . d j l i can therefore be defined rigorously as the unique of (N)* such that element
@i
=U(j)
@$(exp(i(.,j)” holds for all j E N.Here (., .)N: n/* x
N
-+
(5)
R is the canonical pairing.
Step 2 Let us now make rigorous sense of the heuristic integrals IL(A$, B ; h) appearing above for fixed A: E A : , B E Cw(C, t), h E [C,G / T ] .We already have a rigorous version @; of the heuristic functional J . . . d@;. However, clearly we can not just set
since the function
P exp (J i. + A$ + A$,g
+
(h) Bdt) was defined as a func-
tion on d’ and not as a function on all of d’ = N* = (A’)*. Moreover, it is well-known3 that for making sensek of the WLOs in Chern-Simons theory it is necessary t o fix a framing of the manifold M and a framing of the link L. The first problem can be resolved/circumvented by using “loop smearing” as we will now explain briefly. (The second problem will be addressed below). Observe that for every smooth 1-form A, i.e. A E d and every smooth loop 1 in M we have Pexp(1A) =
c] m=O
nA(E’(ui)) du E Mat(N,@)
Arni=1
kthis is even true if one is only working at a heuristic level as in Ref. 3
White Noise Analysis in the Theory of &manifold Quantum Invariants
211
where
A,
:= {u E [O, 11,
I u1 2 u2 2 . . . 1 u m )
a'
In particular, we can apply this formula t o the case where A = or AAng(h) Bdt with A' E d' . However, for A = A' where
A = A' +A:
+
+
-
is a general element of d' = N* = (,aL)* the expressions A(l'(ui)) appearing above do not make sense any more. In order to get round this complication we replace for each u E [0,1]the point l(u) by a suitable test function l'(u), E > 0, whose support lies in the €-ball around 1(u) (for more details, see2). Then we set'
'A
a
where is any fixed (., .)-orthonormal basis of g and (., .)N: JVx N -+ E% is as above. The expression
(AL+ A:
P,xp(/
+ Ai,,(h) + Bdt)) :=
1'
is then well-defined for general A' functions Pexp(
4: + (.
A:
E
dL= N* and we obtain well-defined
+ A i n g ( h )+ Bdt)) : JV-+ Mat(N, C )
Using similar methods as in the proof of [16, Proposition 61 it should not be difficult t o prove the following Conjecture 1. For every
E
>0
As mentioned above framing is necessary if one wants to succeed in making sense of the WLOs. In1i2 this framing procedure was implemented in the following way: Firstly, a suitable family ( d s ) s , of ~ diffeomorphisms of C x S1was fixed which was assumed to fulfill the following conditionsm lif one makes suitable identifications then TaZ'(u)Z'(u)will indeed be an element of N, cf. [2, Sec. 51 where the notation a€(. - l ( u ) )is used for F(u) mIn2 the definition of the term "horizontal" was somewhat broader but also more complicated.
212
A . Hahn
- ($,)*dl= dl for all s > 0. This condition implies that each $,,
s
> 0,
is of the form $s(g,
t ) = ($,(a),%(C, t ) ) V(a, t ) E
c x s1
for a uniquely determined diffeomorphism $, : C + C and a uniquely determined w, E C” (C x S1,Sl). - 4, -+ idM as s -+ 0 uniformly w.r.t. to the Riemannian metric g M on which is induced by g = gc. - Each $, s > 0, is volume preserving w.r.t. g m . - ($,),>o is “horizontal” in the sense that it can be obtained by integrating a smooth vector field X on C x S1, which for all i 5 n, u E [0,1] is orthogonal to the tangent vector Z:(u) and “horizontal” in li(u) (i.e. d t ( X ( l i ( U ) ) )= 0 ) . Secondly, for each diffeomorphism 4, a “deformation” of @$was introduced and used to replace the Hida distribution @$introduced above. More precisely, was defined to be the unique element of (N)*such that @&4, (exP(i(*, j)N))
= exp(i K j , m ( B )B
~XP(-;
<< ( $ s ) * j , C ( B ) jB N N )
(6) holds for all j E N where ($,)* : N + N is a certain linear automorphism which is induced by 4, and the Riemannian metric g.” Conjecture 2. There is a
SO
w N )
> 0 such that
exists for all s E (0, S O ) and is independent of the special choice of s. In the special case DP(L) = 0 it should not be difficult to prove Conjecture 2 along the lines of Sec. 5.2 in2 and to derive the following explicit formula for I P ( A : , I?;h) (cf. [2, Eq. (5.33)])
“note that
(4s)* really depends on g. In particular, we have (dS)*# (q$;)-l
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
where 1; := (
lil :=
l j ) ~ ,
( l j ) ~ Ij~ ,:= (li1)-'({to}),
aj,. :=
I&(.)
213
E C, and
crosses t o a t u in the positive direction crosses t o at u in the negative direction does not cross but only touches t o a t u for j I n, u E I j . Observe that Eq. (8) implies
I p ( A f , B ;h) = I p ( A i , B ;h) for all A,:
A; E A: with dAf
(9)
= dAi.
Speculation 1. Eq. (9) might also hold (for A f , A + E A: with dAf dA+) in the situation where DP(L) # 0.
=
Step 3 For simplicity we will assume in the sequel that C E S2 and therefore H 1 ( C ) = (0). In this case the Hodge decomposition of Ax (w.r.t. the metric g fixed above) is given by
AX = Aez CB A:,
(10)
where
A,, A;,
I f E C"(C,g)) := {*df I f E C"(C,g)) := {df
(11)
(12)
*
being the relevant Hodge star operator. According t o Eq. (10) we can replace the s . . . D A , ' integration in Eq. (15) by the integration . . . DA,,DAZ, where DA,,, DAZ, denote the heuristic "Lebesgue measures" on A,, and AZz. Combining this with Eq. (15) (and replacing Ih(A:, B;h) by I F ( A ; , B ;h)) we arrive a t
ss
xexp(i9
Tr(dA:,
. B))(DA;,
@ DB)
(13)
214
A . Hahn
+ A:,
According t o Eq. (9) we have I?(A,, informally] I?(A,,
+ A;,]
B; h) DA,,
I;g(AZ,l
B; h) so,
IP(A:,l B; h)
(14)
B ;h)
-
Moreover] if we introduce the decomposition
B’ := { B E B I
B = B‘ @ B, where
S,
B, := {B E B I B
=
B(g)dpLg(c)= 0)
is constant}
t
(15) (16)
s s.
s.
we can also replace . . DB by ’ . DB‘db where DB‘ is the heuristic “Lebesgue measure” on B’ and db is the rigorous Lebesgue measure on B, S t. According t o Remark 2.1 above and the definition of ( . I .) we have
Combining Eqs. (13), (14), and (17) we obtain
where
and where we have inserted the factor
,s
in order to ensure that the ’ . . db-integral exists. Observe that the operator *d : B’ -+ dZ, is a linear isomorphism and we can therefore consider the change of variable Bi := (*d)-lA$, and
Bh := B’.
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
If we introduce the function J h ( b , h) : B’ x
for all Bi, B$ E
B‘
215
B‘ -+ C by
and take into account
then we can rewrite Eq. (19) as
where
+
d v ( B i , Ba) := e x p ( - 2 ~ i ( S ce) << *d
* d B i , BL >Lt(C,dp,))(DBi18DB;) (24)
Clearly, u is of “Gauss type” with covariance operator
In order to make rigorous sense of the heuristic integral functional . . . d u as a generalized distribution on a suitable extension B’ x B’ of the space B’ x 8’we now proceed in a totally analogous way as in Step 1 above. i) First let us choose a convenient Gelfand triple ( E , ‘FIE, E * ) and set
B’ x
a‘ := €*.
More precisely, we choose
E := B’ x B’(equipped with a suitable family of semi-norms) (25) ‘FIE := LB(C,dpg)‘ x L f ( C ,dpg)‘ (26)
s
where L?(C,dp,)’ := {f E L?(C,dpg) I f d p g = 0). In the sequel we will make the identification
I*= (B’x
,I)*
2
(f3’)* x (B’)*
(27)
Using second quantization and the Wiener-Ito-Segal isomorphism
Fo&,m(%)
“=
Lt(E*,YE’)
where is the standard Gaussian measure on I*we obtain a new Gelfand triple ( ( E ) , L ; ( E * , T E * ) ,( E ) * ) .
216
A . Hahn
ii) Nest we evaluate the Fourier transform Y at an informal level. We obtain b’j E & :
324)=
S
Fv
of the heuristic “measure”
exp(i << . , j >>xHE)dv = exp(-$ << j , C j B E & )
where C is the heuristic “covariance operator” above. In fact, C is already a well-defined symmetric linear operator & -+ 7 - l ~and it is well-known that C is bounded so the function U : & 4 C given by
U ( j ) = exp(-$ << j , C j
(28)
>>NS)
for every j E & is continuous. iii) Clearly, U : & + C is a “U-functional” in the sense so using the Kondratiev-Potthoff-Streit Characterization Theorem the integral functional 9 := x B , . . . dv can be defined rigorously as the unique element Q of (&)“ such that 0f17118
sB,
Q(exP(q.,j)&))= U ( j ) holds for all j E
Step
(29)
E . Here (., . ) E : €* x & -+ R is the canonical pairing.
4
Finally, let us make rigorous sense of the heuristic expressions on the righthand sides of Eq. (23) and Eq. (18) above. In the previous step we have found a rigorous version Q of the heuristic functional JB,x B , . . . dv but, clearly, we can not expect 9 ( J L ( ~h)) , to be a rigorous realization of the r.h.s. of Eq. (23) since the function J L ( b , h) was defined as a function on & = B’ x B’ and not on all of &*. Moreover, we must again take into account the need for framing. The latter point is quite straightforward. Recall that above we have fixed a family ($s)s>o of diffeomorphisms of C x S1 with certain properties and that each $s induces a diffeomorphism $s of C which is volume-preserving w.r.t. the Riemannian metric g on C. Clearly, each such $s induces a linear automorphism (Bs), : & -+ & in a natural way. Using the Kondratiev-Potthoff-Streit Characterization Theorem we can define a “deformation” Q J ~ s, > 0, of 9 as the unique element of ( I ) * such that Q$3(exp(i(.,j)c))= exp(-;
<< ( B s ) * j ,cj B E , )
(30)
holds for all j E &. The other problem mentioned above is solved by finding a “suitable” regularization of JL(b, h), i.e. a family (Jf’”)(b,h))E>o,nGNof functions
White Noise Analysis an the Theory of 3-manifold Quantum Invariants
Jf’”’(b, h) : E*
+
CC with Jf’”’(b, h)
n-cc lim E-0 lim
E
( E ) such that
Jf’”’(b,h)(B‘,,Ba) = J ~ ( b , h ) ( B i , B a )
for all (Bi,Ba)E B’ x B‘ = E . Recall that J L ( ~h), is the product of four functions JL,i(b, h) : & i = 1 , 2 , 3 , 4 ,given by
J ~ , l ( bh)(Bi, , Bh)= I p ( * d B i , Ba
J ~ , 3 ( bh)(B:, ,
Ba) = det,,,(ide
+ b, h)
- exp(ad(Bb
JL,4(brh)(BLBh) = lc-(C,p)(Bh
217
+ b)
(31)
4
@,
(324
+b ) , ~ ) )
(32c) (324
for all ( B i ,Bh) E B’ x B’ = &. As a first step towards the construction of ( J f ’ ” ) ( b h))E>O,nEN , let us introduce “point smearing”: For each a E C choose a Dirac family” (Se,u)t>~around a w.r.t. pg and set S;,+,:= St,,, -
J,
hTdPL,.
Moreover, let us assume that the family ( b t , u ) e > ~ , u Ewas ~ chosen “smoothly in a” in the sense that for each B’ E (B’)* the function C 3 a ++ B’(S&) E t is smooth. Then, clearly, the function B’(E): C 3 t given by B’WJ) = W:,J -
J
B’(S:,,)dPg(4
is an element of B’. We can then introduce smeared versions J t { ( b ,h) : E* 3 @, i = 1 , 2 , 3 , 4 ,given by
J:m)(B;,
Ba) = J L , i ( b , h)(B;(E),
W))
for every ( B i , B ; )E E* E (B’)* x (.a’)*. The function J g l ( b , h) needs additional regularization. According to the definitions we have
Jg@, h)(B:, for every ( B i ,B;) E E*
E
= lc-(C,P)(Ba(4
+ b)
(a’)*x (B’)*.But J t i ( b , h) is not
an element of
( E ) so we will regularize J t i ( b , h) further. 1.e. 6,,,, is non-negative, smooth, and i6,,,dpg = 1 for all fixed u we have 6L,,+ 6, weakly a s E + 0
0‘
E
> 0, u E
C , and for each
218 A . Hahn
i) For each fixed n E N we fix a finite subset C(") of C such that V a E C : d(a,C("') < l/n. ii) We approximate the indicator function l p : t 4 [0,1] by a suitable sequence of analytical functions ( l g ) ) , E ~ . For example, in the special case
G = SU(2), T = (exp(8. T ) I 8 E [0,24}, P = { e S rI o E (o,+
(33a) (33b) (33c)
where
,I2")
where 1 : t we can choose lg'(b) = exp(-ll(b) form given by I ( . T ) = 1. iii) For each n E N we introduce J t r ' ( b , h) : €* -+ R by
5
J t c ' ( b , h)(B:, B;) =
n
-+
IR is the linear
lg'(B;(~)(a) + b)
(34)
U€C(n)
Remark 3.2. The results in9 suggest that it might me necessary to modify the strategy above somewhat. Since P is a connected component of begwe have lc-(E,P)(B)= lp(B>(ao)x 1C..(C,t,,,)(B)
(35)
This motivates the following alternative ansatz
n
JEf'(b,h)(B;,Bi) := ~ ~ ' ( B ; ( E ) ( ( Tx o ) + ~l)$ r J , ( B s ( ~ ) ( ~ ) (36) +b) U€C(n)
where
lg' is as above and (l{r)JnE~ is a suitable pointwise approximation
of Itreg. E.g., for G, T , P as in (33) above we can take l$;j,(b) C O S ( $ ~ ( ~ ) ) ~ where " 1 : t -+ R is as above.
:= 1 -
Conjecture 3. i) J t ) , ( b ,h), J t $ ( b ,h) E ( E ) for each E ii) J t r ' ( b , h) E (€) for each iii) In the special case where E > 0.
> 0.
> 0, n E N. DP(L) = 0 we have JE\(b, E
h) E ( E ) for each
White Noise Analysis in the Theory of 3-manafold Quantum Invariants
219
Speculation 2. If Conjectures 1 and 2 are true J t \ ( b , h ) , E > 0, can be defined as above also for general L.P In this case one should expect Jt\(b, h) E ( E ) to hold also for general L. Since ( E ) is closed under pointwise multiplication Conjecture 3 implies that J f ' " ' ( b , h) E ( E ) where we have set
n 3
$'"'(b,
h) :=
J t { ( b ,h) . J f , r ' ( b , h)
(37)
i=l
for all
E
> 0 , n E N.
Conjecture 4. i) There is a
SO
> 0 such that F p ' ( b , h) := im 9qs( J f ' " ) ( b , E-+O
exists for all s E (0, SO) and all n E N. ii) For each n E N,h E [C, G / T ] and fixed s < so the function F F ) ( . ,h) : t -+ C is smooth with compact support. Moreover,
F y g ( b ,h)
:= n-+m lim F p ' ( b , h)
(39)
exists for all b E t. iii) In the special case where DP(L) = 8 we should haveq (with equality up to a multiplicative constant independent of L )
c (n n c,
-
denoting
n
Fyg(b,h)
mp3(a,))exP(-2Kz(n(h),C
,lY,EI'
lY1,a2,
(
3
a31yQN
3=1
n
l ( t r e g )(B)lP(B(.O)) c
exp(274a.7,
EL;
[B(~s(.~))+B(~,l(a~))))
3=1
detreg(ide - exp(ad(B),d))) IB=b+&
C3n,la3 1'
(40)
R:
where mp3( a 3 )and 1'
(for j 5 n ) are given as in the Appendix below.
R:
Clearly, the r.h.s. of Eq. (40) does not depend on the special choice of s < SO if SO was chosen small enough. pthis is even the case when Speculation 1 turns out t o be wrong. However, if Speculation 1 is wrong then the approach in Step 3 breaks down for general L and the question whether Jt,)l(b,h) is in (&) or not will be irrelevant qusing Eq. (36) or Eq. (34)
220
A . Hahn
Part i) and ii) of Conjecture 4 are easy to believe. Part iii) of Conjecture 4 is strongly suggested by the heuristic considerations in the Appendix (which are based on2>’). Thus if Conjecture 4 is true FLig(b, h) as defined above can be considered to be a rigorous realization of the r.h.s. of Eq. (23) and setting
we also obtain a rigorous realization of W L O ( L ) .It is tempting t o use the Poisson summation formula (cf. Eq. (44) below) t o simplify the r.h.s. of Eq. (41). However, since @(., h) is not a smooth function Eq. (44) below can not be applied. We will circumvent this technical problem by using the following alternative definition of WLO,i, ( L ) instead
(42) for fixed s < so. From Eq. (45) below it will follow that WLO,ig(L) does not depend on the special choice of s provided that SO was chosen small enough. Let us now evaluate the r.h.s. of Eq. (42) explicitly in the special case DP(L) = 0. In order to do so we will assume for simplicity‘ that the Riemannian metric g on C was chosen such that the following condition is fulfilled.
Condition 1. vol(Ri)
= p L g ( R iE)
N for all i 5 m.
Observe that if Condition 1 is fulfilled it follows that 11.; and that exp(-2si(n(h),Cjolj1’R:(ao)))
aj
E
=
Z, j 5 n, [C,G/T]and
(00) E
1 for every h E
I* so Fiig(b,h) will then be independent of h. Setting E.0
:= [ I T ] E
‘if one does not demand that this condition is fulfilled one should still be able to derive Eq. (46) below. However, the computations will then be more complicated making the paper more difficult to read
W h i t e Noise Analysis in the Theory of 3-manifold Q u a n t u m Invariants
221
[C, G / T ]we then obtain
Step (*) follows from
which should not be too difficult t o prove and which is easily believed in view of the relation Fpg(b,h) - F p g ( b , ho) = 0 for every h E [C, G / T ] . In step (**) we have used that
C
,-2ri(k+ce)(n(h),)
-
C e2ri(k+cg)(~,’) XEI
h E [GG/Tl
c
-
hb/
(44)
b/€&I*
in the sense of tempered distributions on t. Note that step (+) follows from Eq. (17) above. Step (++) is nothing but the Poisson summation formula for tempered distributions. By combining Eqs. (43) and (40) and taking into account Condition 1 above one now obtains n
n
C,,,
exp(2Ti(at.,,
x
[B(&(&)) + ~(6;’(4,~)]))
j=1
x det,eg(idt
- exP(ad(B),t)))
IB=b+--Lk+c,
(45)
cy=l
aJ’l’R+
J
In9 the r.h.s. of Eq. (45) was evaluatedS explicitly (cf. [9, Theorem 5.11) and found to agree with JLI up to a multiplicative constant. Here ILI is Scf. Eqs. (53)-(55) and Sec. 5 in.9 Observe that the r.h.s. of [9, Eq. (54)] almost coincides
222
A . Hahn
the “shadow invariant” of (the “shlink” corresponding to) L in the sense of Refs. 7,9. Thus
WLOri&)
-
ILI
(46)
Let us summarize this: Assuming the validity of Conjectures 1-4 we have proven that in the special case where DP(L) = 0 relation (46) holds.
Speculation 3. Also for general L the limits (38) and (42) exist and relation (46) is again fulfilled. It remains t o be seen if Speculations 1-3 withstand closer scrutiny. As for the Conjectures 1-4 we plan t o give full proofs in the not too distant future. Acknowledgments It is a pleasure for me to thank Prof. Dr. T . Hida for several interesting and useful discussions.
Appendix A. Motivation for Conjecture 4, part iii) In order to motivate Conjecture 4 iii) we will evaluate F L ( ~h), at a heuristic level, following c l o ~ e l y . ~ ~ ~ First we combine Eq. (19) with Eq. (8) and replace the function Ic..(=,p)(B) appearing in in Eq. (19) by the function lp(B)(ao) x l(treg)C(B), which is a non-smooth analogue of the r.h.s. of Eq. ( 3 5 ) above. Then we obtain, informally,
with r.h.s. of Eq. (45) above. The only difference is that the function lShift, j 5 n, appears R :
instead of 1;.
From Condition 1 it follows easily that the value of the r.h.s. of Eq. (45)
does not change if we replace 1’
R :
by lshift R:
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
223
Let Rjf denote the unique element of { R I ,Ra, . . . ,Rm} which “lies to the left”t of 1; and let us set
where 1 + is the indicator function of RT. From Stokes’ Theorem and Rj
the definition of the Hodge star operator * we then obtain S j A:, = 1, JR; dA:, = JRf *dA:,dpg = J *dAz, . lkTdpg and therefore, for every aEtEt*.
Setting 6 := 27ra,
for each
cy E
I*uwe then obtain from Eq. (22) and Eq. (A.2)
Here mpj( a j )is the multiplicity of the weight aj in the character of pj and (&, Akng(h)) denotes the obvious real-valued 1-form. Plugging this into Eq.
tHere “to the left” is defined using the orientation of C; cf. Ref. 9 for a rigorous definition of Rj’ %ate that since G was assumed to be simply-connected the lattice I* coincides with the lattice A in9
224
A . Hahn
(A.l) above we get F L ( b , h)
x
[/
n
exp(i << *dA$,, X G j .1' + j=1
4,
R3
-
27r(k
I
+ cg)B' >>xs)DA2, DB'
(A.4) Using a suitable the infinite dimensional analogue of the well-known informal equation JRd exp(i(z,y ) ) d z 6(y), namely N
n
and taking into account
for each j 5 n (cf. the second equation before [9, Eq. (53)] and observe that 12; - l;;(ao) = lR+- 1 +(go) = ? :l) we obtain j
R3
n
White Noise Analysis in the Theory of 3-manifold Quantum Invariants
225
So far we have ignored t h e “framing” procedure mentioned above. Including the framing procedure at t h e heuristic level we work on in this appendix amounts to replacing (by hand) t h e expressions B(ok)appearing above by i [ B ( J s ( o k ) )B($;’(ok))].If we d o this we obtain Eq. (40).
+
References 1. A. Hahn, J . Geom. Phys. 53,275 (2005). 2. A. Hahn, An analytic Approach to Turaev’s Shadow Invariant, arXiv:mathph/0507040, to appear in J. Knot Th. Ram. 3. E. Witten, Commun. Math. Phys. 121,351 (1989). 4. N. Reshetikhin and V. Turaev, Commun. Math. Phys. 127,1 (1990). 5. N. Reshetikhin and V. Turaev, Invent. Math. 103,547 (1991). 6. V. Turaev, J. Diff. Geom. 36,35 (1992). 7. V. Turaev, Quantum invariants of knots and 3-manifolds (de Gruyter, 1994). 8. S. Sawin, Jones-Witten invariants for non-simply connected Lie groups and the geometry of the Weyl alcove, arXiv: math.QA/9905010, (1999). 9. S. de Haro and A. Hahn, The Chern-Simons path integral and the quantum Racah formula, Preprint, arXiv:math-ph/O611084. 10. M. Blau and G. Thompson, Nucl. Phys. B408,345 (1993). 11. M. Blau and G. Thompson, Lectures on 2d Gauge Theories: Topological Aspects and Path Integral Techniques, in Proceedings of the 1993 Pieste Summer School on High Energy Physics and Cosmology, ed. E. G. et al. (World Scientific, Singapore, 1994). 12. M. Blau and G . Thompson, Commun. Math. Phys. 171,639 (1995). 13. A. Hahn, Chern-Simons models on S2 x S1, torus gauge fixing, and link invariants 11, in preparation. 14. S. Albeverio and A. N. Sengupta, Commun. Math. Phys. 186,563 (1997). 15. S. Albeverio and A. N. Sengupta, Nonlinear AnaLTheor. 30,329 (1997). 16. A. Hahn, Commun. Math. Phys. 248,467 (2004). 17. T . Hida, H.-H. Kuo, 3. Potthoff and L. Streit, White Noise. A n infinite dimensional Calculus (Dordrecht: Kluwer, 1993). 18. Y . Kondratiev, P. Leukert, J. Potthoff, L. Streit and W. Westerkamp, J. Funct. Anal. 141,301 (1996). 19. S. Albeverio and J. Schafer, J. Math. Phys. 36,2135 (1994). 20. S. Axelrod and I. Singer, J. Differ. Geom. 39,173 (1994). 21. D. Bar-Natan, J. Knot Theory and its Ramifications 4,503 (1995). 22. R. Bott and C. Taubes, J. Math. Phys. 35,5247 (1994). 23. E. Guadagnini, M. Martellini and M. Mintchev, Nucl. Phys. B 330, 575 (1990). 24. M. de Faria, J. Potthoff and L. Streit, J. Math. Phys. 32,2123 (1991). 25. H.-H. Kuo, J. Potthoff and L. Streit, Nagoya Math. J. 121,185 (1991). 26. P. Leukert and J. Schafer, Rev. Math. Phys. 8,445 (1996). 27. T . Hida and L. Streit, Stochastic Process. Appl. 16,55 (1984).
226
A NEW EXPLICIT FORMULA FOR THE SOLUTION OF THE BLACK-MERTON-SCHOLES EQUATION J. A . Goldstein Department of Mathematical Sciences, The University of Memphis, 373 Dunn Hall, Memphis TN 381 52-3240, USA, E-mail: jgoldsteQmemphis. edu.
R. M. Mininni’ and S. Romanelli Dapartimento di Maternatica, Universitb di Bari, Via Orabona 4, 70125 Bari, Italy, * E-mail: mininniQdm.uniba.it, E-mail: romansQdm.uniba.it The Black-Merton-Scholes equation plays a fundamental role in the option pricing theory. Our main purpose is to derive an explicit formula for its solution, using simple tools from operator semigroups. The paper includes also an expository treatment of how the equation arises. Keywords: Black-Merton-Scholes equation, stochastic differential equation, translation group and operator semigroup, explicit solution.
1. Introduction The application of stochastic differential equations to financial mathematics has undergone tremendous growth in recent years. The fundamental work of F. Black, R.C. Merton and M. S ~ h o l e s ~led> to ~ ~a >Nobel ~ ~ Prize in Economics. The Black-Merton-Scholes (BMS) equation is now a fundamental part of option pricing theory. Some “explicit” formulas have been derived for its solution. Our main purpose in this paper is to derive another such explicit formula, using simple tools from operator semigroups. This is done in Section 3. Section 2 is a completely expository treatment of how the equation arises. We include Section 2 to make the paper self-contained. Going from the basic stochastic differential equation to the BMS equation is a non obvious and nontrivial calculation worth emphasizing. In the following Co(R) (resp. Co(O,+m)) will denote the space of all real-valued continuous functions on R (resp. on (0, +m)) having compact support, while C[-cq +m] (resp. C(0,+m]) will be the space of all realvalued continuous functions on R (resp. on (0, +m)) having finite limits
A New Explicit Formula for the Solution of the Black-Merton-Scholes Equations
227
at f m (resp. at +m).
2. Background The density function of the normal distribution N ( p l a 2 )with mean p E and variance a2 > 0 is given by fp,&)
=
2 -112
e
-(x-p)2/22
,
IR
2 E R .
Brownian motion ,B = { P ( t ) : t 2 0 ) is a stochastic process with independent increments (i.e. p(u) - P ( t ) , P ( s ) - P(T) are independent if 0 5 T < s 5 t < u ) satisfying p(0) = 0, p(t) - p ( s ) has the normal N(0, t - s) distribution for 0 5 s < t , and p ( . , w ) is continuous on (0, +a) for a.e. w E R, where ( R , 5 , P) is the underlying probability space on which p lives. By considering (in the kinetic theory of gases) many small particles having lots of independent collisions, A. Einstein3 appealed to the Central Limit Theorem and “derived” Brownian motion. Since u(t,z) = fo,,. (x) is the fundamental solution of the heat equation
probability theory could thus be applied to problems of diffusion of heat. In fact, Einstein’s derivation of (1) led to an expression for a involving Avogadro’s number a. Later J. Perrin performed diffusion experiments and thus “measured” a 2 ,thereby determining a. For this he received a Nobel Prize in Physics. A key property of the sample paths p(.,w ) is that they are nowhere differentiable, and in fact, not of bounded variation on any interval of positive length (for a.e. w ) . Scientists wished to model phenomena using white noise, the time derivative of p; this did not exist in the usual sense (although we now know how to interpret it). K. It6’ showed in 1944 how to get around this problem in a useful way. Let Ft (C 3) be the a-algebra generated by { p ( s ) : 0 5 s 5 t } . Let g = { g ( t ) : t 2 0 ) be a stochastic process such that g ( t ) is Ft-measurable for each t 2 0. It6 showed how to define s,bg(s)d,B(s). The point is that g(s, w ) @(s, w ) cannot in general be computed classically for (almost) b every fixed w E R (think of the case g = p). The integral u d u , for continuous real functions u,u,exists if either of u,u is of bounded variation, but not in general.
s,”
s,
228
J. A . Goldstein, R. M. Mininni, and S. Romanella
Recall Its's construction. Suppose g be a step process, so that g ( s ) = g(ti-1) on [ti-l,ti),where a = t o < tl < . . . < tn = b. Here g ( t i - 1 ) must be in L2(R,Fti-l,IF'). Then n
[gd@
:=
dti-1)
( P ( t i ) - @(ti-1)).
(2)
i=l
Since @ has independent increments, the two factors in each summand of (2) are independent, whence E (Jab g d@) = 0. Moreover, since
E [ ( p ( t i )- /3(ti-1))~] = ti - t i - 1 ,
(3)
it follows that
-
b
Thus g J, g d @ extends by continuity to a linear isometry from the adapted (to {Ft : t 2 0 ) ) stochastic processes of mean zero in L 2 ( [ 0 , T ]L2(R, ; 5,IF')) to L2(R,3, IF'). This defines the It6 stochastic integral. The formula (3) leads to the conclusion (d@)2 = d t and Its's multiplication table
The stochastic differential equation
dX (or X
=
m(X)
+
=
m(X)dt
+ a(X)d@,
X ( 0 ) = Xo
(5)
a ( X )8) is interpreted as
the first integral being a sample integral (i.e. it is evaluated with w E R fixed) and the second an It8 stochastic integral. If u is a (nice) real function on R,then Taylor's formula implies
du(X)
=
d(X)dX
u" ( X ) (dX)2 + . ... +2
A New Explicit Formula
for
the Solution of the Black-Merton-Scholes Equations
229
Using (5) and replacing ( d X ) 2 by O ( X )dt~ (by Itb's multiplication table), we obtain ItB's formula
u ' ( X ) m ( X ) +u " ( X ) o ( X ) 2 } dt + .(X) d ( X )dp, with the unexpected term
C2) ~
that makes Itb's calculus so interesting.
Standard existence-uniqueness arguments from ordinary differential equations apply to It8 stochastic differential equations with easy modifications. Here is an easy but useful example of a linear stochastic differential equation. Let S ( t ) be the price of a stock at time t >_ 0. Assume the relative rate of change of S is given by
dS S where p and a are positive constants ( p = drift, u = volatility). The fact that p > 0 reflects confidence in the stock, and the a d p term reflects the uncertainty of the market. The resulting linear equation - = pdt+ad,B,
dS = p S d t + a S d p is solved by
S(t)
=
so exp
{a P ( t )+
(P -
;> t}
where So = S(0) > 0 is the initial price. S ( t ) itself is not an increasing function, but E [ S ( t ) = ] So e p t is. A (European) call option is the right to buy one share of the stock at the (given) strike price p at the expiration time t = T ( > 0 ) . What is the fair price to pay for this option at time t = O? It is assumed that bonds can be purchased at a no risk fixed interest rate r > 0. Thus if D (dollars or euros) arc paid for a bond at time t = a, the bond will be worth D e '( t - a ) at time t > a. The option should be priced to avoid arbitrage (i.c., opportunities for risk-free profits). Hedging means that one can duplicate the option by a portfolio of changing holdings of the risk-free bond and of the stock. The price function u(t,x) is the proper price of the option at t = 0, given S ( t ) = 2 . We seek to find u(0,x). The partial differential equation satisfied by u is usually called the Black-Scholes equation (see Ref. 2), although it seems fair to call it the BMS equation because of Merton's contribution (see Refs. 12,13).
230
J. A . Goldstein, R. M. Mininni, a n d S. R o m a n e l l i
The function u should be nonnegative and satisfy the terminal condition
u(T,x)
= (X-P)
+
since the stock option is worthless if x 5 p. Let
P ( t ) := u(t,S(t)) be the current price of the option a t time t. Using dS = p S d t and omitting arguments, Itb’s formula implies
dU
+ uSdP
dU 1 d2U -dS+--(dS)2 dX 2 ax2
dP=-dt+ at
The bond value is determined by the differential equation dB = r B dt, B(0) = 1, with solution B ( t ) = e r t . The BMS idea, based on hedging, is to find (adapted to {.?t : t 2 0)) stochastic processes (p, such that
+
P
=
(pS++B,
O
(7)
This eliminates the risk of a loss to the brokerage firm. If the firm sells a call option, there is the chance that S ( t ) a t the later time T will exceed p and the buyer will cash in on the option. But if the firm has the portfolio given by ( 7 ) ,the profits from it will equal the funds needed to pay the customer. If S ( t ) 5 p, the option is worthless and the portfolio will have no profit. It is desired to have (7) hold without using new money. This will be valid if
+ +dB
=
(pdS
=
(p(pSdt
by (8)
+ uSd/3) + + r B d t .
Multiply (9) by d/3 and use Itb’s multiplication table; the result is
dU
ffsdX
= asp,
(9)
A New Explicit Formula for the Solution of the Black-Merton-Scholes Equations
231
whence we choose $0
=
2 (=-d(Xt , S ( t ) ) ) d X dU
.
Thus after dropping the d,B terms, (9) becomes
(g+$S2$) dt = r $ B d t = by (8) and (10). Now replace S by any x parabolic problem, for u = u(t,x ) :
> 0; this leads to the
backward
which includes a terminal condition (TC) and a boundary condition (BC). Let v ( t ,x ) = u(T- t , x ) . Then v satisfies the (forward) parabolic problem
@,x) = (x -P)+, v(t,O) = 0 ,
on [O,T]x (0).
This is the “final form” of the BMS problem. We want to find w(T,x). This means solving (ll),for T > 0 arbitrary. We next derive an “explicit formula” for w. Note that, remarkably, the drift coefficient p in the stochastic differential equation for S plays no role in (11). In this section we have largely followed the beautiful presentation in the lecture notes of L.C. Evans.4 There are many useful references, too numerous to mention. But some particularly nice recent books
232
J . A . Goldstein, R. M . Mininni, and S. Romanelli
3. The Explicit Formula The real numbers under addition, (R,+), form the simplest Lie group arising in analysis. Let X be a space of functions on R. The translation group (with respect to +) is T = { T ( t ): t E R}given by (z E R,t E R). (T(t)f)(x) = f ( z + t ) It satisfies T ( t s ) = T ( t )T ( s )and has a generator G defined by Gf
+
=
d - T ( t ) f It=O, with maximal domain. Clearly G f ( z ) = f’(z) so that G = dt D = d / d x . (Here we proceed informally, without specifying X. We could take X to be L p ( R ) , 1 5 p < 03, Co(R), C[-m, m] or other choices). The heat equation problem
was solved by Gauss:
J -03
for many choices of f . This can be rewritten as
L
03
u(t,x) = ( 4 ~ t ) - l / ~
e-y2/4t
e P y Gf(z)dy.
Operator semigroup theory (cf. e.g.6) tells us that if G generates a strongly continuous (a (Co)) group T = { T ( t ) : t E IR} on a Banach space X, then the unique solution of du - = G 2 u (t 2 0 ) , ~ ( 0=) f , dt is given by
L
00
u(t)= (47~t)-l/~
e-y2/4t
T(-y) f dy
(3)
for t > 0. This solution is (strongly) continuous on [0,co)and continuously differentiable on [0,m) (resp. on (0, m)) if f E Dom(G2)(resp. if f E X). The next simplest Lie group is ((O,m),.),the positive real numbers under multiplication. The exponential map exp : (R, +)
-
((01 m),.)
is a Lie group isomorphism which carries properties of problems on into the corresponding properties for problems on ( ( 0 ,m), . ).
(R, +)
A New Explicit Formula f o r the Solution of the Black-Merton-Scholes Equations
Then the “translation group” acting on a Banach space on (0, m) is given by
Y of
233
functions
To(t) f ( x ) = f ( x e V t ) , where Y > 0 is a parameter. This corresponds to the translation group (for function on (R,+)) given by
(T(t)f)>.(
=
f(. + Y t >
with generator G = I/ dldx . Note that TOis a (CO)group on Y if Y is taken to be L p ( ( O , c a ) , m ) , 1 5 p < 00, C(0,+m], or Co(0,+m), where dm(x) = dx/x, so that m is Haar measure on ( ( 0 ,m), . ). d Let Go be the generator of To: Gof = --To(t)f It=O. Formally we dt have Gof(x)
=
v.f’(x)
so that Go = v x d / d x and
+
(G;f)(x) = v2x2f”(x) v2xf’(x). Now let G1 generates a (CO)group on X and let G2
=
G;+yGi+bI,
where y,S E IR.Then G2 generates a semigroup (analytic and ( C O )given ) by, using suggestive notation,
(4)
exp(tG2) = exp(tGt) e x p ( t y G l ) e J t . Thus, the unique solution of
is given by
L 00
w(t)
=
-y2/4t e t y GI
-Y GI
f dY
(5)
by (3) and (4), for t > 0. For the BMS problem (ll),we take X to be Co(0,+m) (or C(0,+m], or LP((0,m), m ) , 1 5 p < ca) and let G l f ).( with u = a / a , so that
=
Gof I.(
= v 3:
fY.)
234
J . A . Goldstein, R. M . Mininni, and S. Romanelli
and we take
Then
solves the (BMS) equation with a more general initial condition u(0, z) = f(z).For f(z)= ( z - p ) + , the integral in (6) converges and gives the unique nonnegative solution of (11).However, this (unbounded) f is not in any of the spaces X under consideration here. This is easily remedied as follows. Formula (6) gives the unique solution to (11) with (z - p ) + replaced by any f E X. Let f~ = min{f, M } for M = 1 , 2 , . . .. This fM belongs to Co(0,+00] and to G[O, +00] but not to X , = LP((0,+w), m ) ,1 5 p < 00. Let U M be the solution of (11) given by (6) with initial data f ~ As . M + 00, f ~ ( zconverges ) monotonically to f(z),and u~(z,t) converges monotonically to u(x, t ) ,the unique nonnegative solution to (11). Formula (6) works for many choices of nonnegative f because of the e-Y2/4t factor in the integrand. For instance, f can be in L;,JR) and can grow polynomially (or even faster) a t *co. Problem (11) can be generalized to
O
A systematic study of this problem on the above mentioned spaces of continuous functions (for all O 5 a I 1) is in our recent paper.7 References 1. W. Arendt and B. de Pagter, Pacific J . Math. 202 (2002). 2. F. Black and M. Scholes, J. Polit. Econom. 81 (1973). 3. A. Einstein, Ann. Physik 17 (1905) [Reprinted in R. Furth (ed.), Investigations o n the Theory of Brownian Movement (Dover, 1959)l. 4. L. C. Evans, An Introduction to Stochastic Differential Equations, Version 1.2, University of California (Berkeley) Lecture Notes (1995). 5. J.-P. Fouque, G. Papanicolaou and K. R. Sircar, Derivatives in Financial Markets with Stochastic Volatility (Cambridge, 2000). 6. 3. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs (Oxford, 1985).
A New Explicit Formula for the Solution of the Black-Merton-Scholes Equations
235
7. J.A. Goldstein, R.M. Mininni and S. Romanelli, Communications o n Stochastic Analysis l ( 2 ) (2007). 8. K. It6, Proc. Imperial Acad. Tokyo 20 (1944). 9. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. (Springer, 1991). 10. I. Karatzas and S. E. Shreve, Methods of Mathematical Finance (Springer, 1998). 11. H. H. Kuo, Introduction to Stochastic Integration (Springer, 2006). 12. R. C. Merton, J . Econom. Theory 3 (1971). Erratum: ibid. 6 (1973). 13. R. C. Merton, Bell J . Econom. Management Sci. 4 (1973).
236
VOLATILITY MODELS OF THE YIELD CURVE Victor Goodman
Department of Mathematics, Indiana University Bloomington, Indiana 47405, USA E-mail: [email protected] unuw.math.indiana. edu Models of forward interest rates often use high-dimensional Brownian motions to capture imperfect correlations between near term and long term rates. Several statistical analyses suggest the practicality of using a simpler model. Principal component analyses reveal a pattern of correlations which can be essentially explained with a properly chosen three-dimensional Brownian motion. We describe the well known mathematical difficulties in implementing this approach, and we provide a substantial resolution of these obstacles. By focussing on no-arbitrage models using a long bond as a numeraire, we avoid problems with infinities which appeared in conventional risk-neutral models.
Keywords: no-arbitrage, yield curve, Brownian motion
1. Principal Component Analysis of Forward Interest Rates
Interest rates are determined by trading results in several markets for U.S. treasury bonds. There are two common conventions used in discussing these results. Although the face value of each bond is $1000, we adopt this as the unit of currency. That is, when a bond matures, it pays the owner one unit. The other convention concerns coupon payments. These small, periodic payments are regarded as payoffs for several small bonds; the present value of the payments are stripped away from the market value of the treasury bond leaving its zero coupon price which is determined by the results of bond trading. The yield Y of a zero coupon bond is the solution of
P = exp( -Y . T ) when we are given the time t o maturity T and the current trading price P of the bond.
Volatility Models of the Yield Curve
237
Since a bond price changes daily, its yield varies with time. Longer maturity bonds typically have larger yields. Currently, these yearly yields are in the range .037 < Y < .046. A plot of the yields for a variety of zero coupon bonds, using current market prices, gives us the yield curve. The yield curve varies with time. During the last twenty-five years an enormous effort has produced a wide variety of probabilistic models for these curves. Most statistical analyses of yield curves begin with a further encoding of bond prices. If T and T AT are two closely spaced maturity dates, then we say the forward rate f for a maturity at T is given approximately as the solution of
+
P(T + AT) = exp(- f . AT)P(T) From this definition we see that a forward rate is the additional discounting necessary to convert the T-maturity bond price to the T+AT-maturity bond price. Additional information regarding properties of forward rates can be found in Carmona and Tehranchi.' Data sets typically consist off (T) values, measured simultaneously for a variety of T-values (see Rebonato2). The central question is the correlation of these quantities over a specific time period. We follow the discussion in chapter 3 of Rebonato.2 f (Ti) denotes data for a fixed set of maturity times, TI,. . . T,. We assume that time increments of f i have the form
dfi--
fi
pidt
+ aid&
where Zi is a set of correlated Brownian motions. The infinitesimal relations are E[dZi(t)dZi(t
+ At)] = 0
and
E[dZi(t)dZj(t)]= pijdt
One can produce estimates for variances {ai} and the correlation matrix { p i j } . Typically, results of a principal component analysis (PCA) are striking. Even though the correlation matrix is high-dimensional, a PCA shows that the first three principal components explain an overwhelming portion of the variance. The observed pattern for the principal components is More than 90% of the variance can be explained by three factors
238 0
0
0
V. Goodman
First factor loadings are all the same; this captures 70% - 80% of the variance . Second factor loadings are approximately linear in T with a positive slope; this captures approximately 10% additional variance. Third factor loadings are quadratic in T ; this factor explains approximately 5% of the variance
Results of several analyses are described in Rebonato2 and W i l ~ o n . ~ Also, Carmona and Tehranchi,’ page 39, find a similar pattern for the U.S. treasury yield curve using data for the period 2001-2005. Using this pattern for first three eigenvector entries, we restate the statistical model in Eq. (1) using only the first three principal components with independent Brownian motions Wi, i = 1,2,3. The three-factor model is
where C is a constant, L(u)is a fixed linear function] and Q(u)is quadratic. These coefficient functions are determined by PCA results. C is the common entry in the first eigenvector. It is the most important parameter in the model since it corresponds to the variance of the first principal component. L and Q correspond to the other eigenvectors. In Eq. (2) the parameter T now represents a fixed maturity date. Therefore, the coefficient functions depend on the time remaining to maturity]
T
- t.
2. The One-factor Model To develop these models, let’s first set L and Q equal to zero. The one-factor model has the form
df(t,T ) =
T)f(tIT)dt + C f ( t T , )dW(t)
(1)
In the midst of developing a general methodology for setting up bond models, Morton4 looked at solutions of the stochastic differential equations above with the hope of obtaining a basic description of forward rates. Equation (1) is a special case of term structure models known as Heath, Jarrow, Morton5 models. Morton considered the very useful risk-neutral version and what he found was surprising.
Volatility Models of the Yield Curve 239
A risk-neutral asset model is characterized by the property that each asset has the same rate of return as a money market account (see Chapter 13, page 309 of Musiela and RutkowskP). This leads t o a formula for a in any risk-neutral model of the form
df = adt + adW(t) The formula (see Eq. (13.15) of Musiela and RutkowskP) is
4
T
a
= a(t,T )
a(t,u)du
In our case, the risk-neutral model defined by Eq. (1) satisfies
An unfortunate feature of Eq. (2) is illustrated by the simpler SDE
+
d F ( t ) = C2F(t)2dt C F ( t ) d W ( t )
(3)
The two equations have a quadratic drift and a linear diffusion coeficient. This feature allows the possibility that their solutions explode. To illustrate this, Let M ( t ) denote the martingale solution of the SDE
d M ( t )= CM(t)dW
(4)
with M ( 0 ) = F ( 0 ) . Then one may easily check that the solution of Eq. (3) is given by
One can see that the denominator, with positive probability, may reach zero in a finite amount of time. This defect of the model defined by Eq. (2) is noted in Heath, Jarrow, M ~ r t o n They . ~ propose some elementary modifications of the model. Others have proposed more sophisticated alternative models. An important class involving discrete sets of maturity dates, termed m a r k e t models, is investigated in Brace, Gaterek, M u ~ i e l a . ~ The defect of the one-factor model is serious since it allows arbitrage within the model. This makes the model useless for computing hedging quantities.
240
V. Goodman
The well known modifications of the one-factor model, although arbitrage-free, do not extend t o a three-factor model which incorporates the volatilities given in Eq. (2). That is, none of alternatives addresses the possibility of incorporating linear and quadratic volatility coefficients to model forward interest rates. The explosive feature of Eq. (2) is an obstacle to producing an arbitrage-free three-factor model. We now describe a modeling method which avoids this explosive feature. This method leads to an explicit model, so that some financial derivative prices may be evaluated using this approach (Goodman and Kims).
3. Forward Measure and the One-Factor Model
It is well known that a useful martingale model can be formed from a risk-neutral HJM model by requiring all assets t o have the same return rate as some specifically chosen bond, whose maturity date is, say, T*. The mechanism for this transition is t o change measure so that a pathtranslated Brownian motion becomes a zero-drift Brownian motion under the new measure. Features of this change of measure are described in Chapter 13 of Musiela and Rutkowski,' Section 13.2.2. For our purpose, we record the well-known SDE for forward interest rates using a T*-forward Brownian motion, W . The formula for a in any T*-martingale model of the form
df = adt + crdW(t) is CY =
-a(t, T)
LT*
a ( t ,u)du
For the model defined by Eq. ( l ) ,the SDE becomes
df ( t ,T) = - C 2 f ( t ,T)
LT'
f ( t ,u)du. d t + C f ( t ,T ) d W ( t )
(1)
Notice that the diffusion coefficient is the same as in Eq. (1) but now the drift term is negative (at least for positive rates). Solutions of Eq. (1) remain finite provided that one uses only this system of equations to define rates driven by the noise W . As hinted earlier, Eq. (1) has an explicit solution, and we may express forward interest rates as simple functions of an exponential Brownian motion and its time integral.
Volatility Models of the Yield Curve
4.
241
The New One-Factor Model
Assume for the moment that Eq. (1) has been solved and its solutions are sufficiently regular so that the processes
Y ( t ,T') =
ly
f ( t ,T)dT
are well-behaved. For convenience, we write W for its integrated form:
dY(t,T')= -C2
w in Eq. (1)and consider
L: ST* f(t,T)
f(t,u)du.dT.dt+CY(t,T')dW(t) (2)
T
Notice that the drift term in Eq. (2) can also be expressed in terms of Y . The drift is C2
--Y(t,T')2dt
2 and the equation reduces to the single SDE
C2 dY = --Y2dt 2
+ CYdW
whose solution is
Y ( t , T ' )= 1
+
M(t) M(u)du'
(3)
In this solution, M ( t ) denotes the simple martingale defined in Eq. (4) whose initial value agrees with Y ( 0 , T ' ) . Equation (3) forms the starting point for the model developed in Goodman and Kim.' We prove that positive bond prices defined through the relation
form an arbitrage-free system for any family of maturity dates T which do not exceed the long bond's maturity date, T* (Theorem 3.6, Goodman and Kims). Therefore, Eq. (1) does define a viable martingale model for forward interest rates which avoids the serious price collapses in the riskneutral version.
242
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One cannot perform a change of measure with this new model to recover the (original) risk-neutral model. Although the measure for the new model is absolutely continuous with respect to the risk-neutral measure, the two measures are not equivalent. The new measure is concentrated on the event that the T*-bond price remains positive and reaches its face value at maturity. This new model represents a conditioning of the risk-neutral measure on the T*-bond price surviving until maturity.
5. Multi-factor Models The modeling approach explained in the previous section for the one-factor case generalizes to the three-factor case, Eq. (2).
Theorem 5.1. Suppose that C(u),L ( u ) ,and Q(u),are continuous functions on [O,T*]. Let V ( u ) denote the function
V ( u >= (C(u),Uu),&(.>I and let W ( t ) denote %dimensional Brownian motion. The stochastic differential system
df ( t ,T ) = - f ( t ,T )
/
T’
V ( T - t ) . V ( u- t )f ( t ,u)du * dt
T
+f(t,T)V(T
-
(1)
t ).dW(t)
has positive, finite solutions, f ( t ,T ) , for 0 5 t 5 T 5 T * . In addition, the family P ( t , T ) of bond prices defined as
has the feature that each process
is a local martingale on the interval [0,TI. Proof. Let M ( t ,T ) denote the family of martingale processes defined as t
A4 = exp ( /V ( T - s ) . d W ( s )- 1/2 0
I”
V 2 ( T- s ) d s )
Volatility Models of the Yield Curve
243
Notice that M satisfies an equation similar t o (1) but the drift term is zero: d M = MV(T - t) * d W ( t ) We pose the existence question for f in terms of a representation of the form
We seek sufficient conditions for the process g(t, T ,w) t o be smooth in t. If some process g has smooth sample paths, then df = exp(g)dM
+ exp(g)Mgdt
which can be written as df
= fV(T - t ) . d W
+ fjldt
But, Eq. (1) would require this expression to equal T*
fV.dW- f /
V-Vfdudt T
Therefore, we obtain the necessary condition that g satisfies the 0. D. E.
g(t,T) = -
f’
V . Vexp(g(t, u ) ) M ( t u)du ,
(3)
for each Brownian sample path. Now the solution to this O.D.E. exists. The integral weight function
v . VM(t,u) is defined for almost every path as a continuous function of t and u.A Picard-type iteration argument, applied to the family of continuous functions over the triangular region 0 5 t 5 u 5 T * , shows the existence of processes g(t, u,u ) which satisfy Eq. ( 3 ) . Therefore, we may reverse the steps in deriving Eq. ( 3 ) to show the existence of solutions f ( t , T). 0 The result in Theorem 5.1 is not limited to three dimensions. The same argument shows that a T*-forward bond model exists for any given ndimensional continuous vector-valued function V (t,T ) which might serve as a parameter in the n-dimensional version of Eq. (1).The conclusion concerning local martingales implies that the bond model is arbitrage-free.
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V. Goodman
Acknowledgment
I have enjoyed my long-term friendship with Professor Hui-Hsiung Kuo, and I am pleased to contribute to this volume in his honor. We share a common perspective in using Brownian motion t o define analytical concepts in infinite-dimensional spaces, and I respect and admire his extensive work on the structure of white noise mathematical models. References 1. R. A . Carmona and M. R.Tehranchi, Interest Rate Models: an Infinite Dimensional Analysis Perspective (Springer-Verlag, Berlin, 2006). 2. R. Rebonato: Interest-Rate Option Models., 2nd edition ( John Wiley & Sons, New York, 1998). 3. T. Wilson: Debunking the myths; in Risk vol. 7,67-73 (1994) 4. A.J. Morton: Arbitrage and martingles (Doctoral dissertation, Cornell University, Ithaca, U.S.A., 1989) 5. D. Heath, R. Jarrow, and A. Morton: Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation, in Econometrica 60,77-105 (1992). 6. M. Musiela and M. Rutkowski: Martingale Methods in Financial Modelling (Springer-Verlag, NewYork, 1997). 7. A . Brace, D. Gatarek, and M. Musiela: The market model of interest rate dynamics, Math. Fin. 7,127-147 (1997) 8. V. Goodman and K. Kim: One-factor term structure without forward rates, in ArXiw math. (PR/0612035, 2006).
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AUTHOR INDEX Barhourni, A., 135 Becnel, J., 24
Potthoff, J., 53
Chow, P.-L., 53
Redfern, M., 13 Riahi, A., 135 Rornanelli, S., 226
Goldstein, J. A., 226 Goodman, V., 236 Goswami, D., 175 Hahn, A,, 201 Hall, B. C., 161 Hida, T., 1 Manna, U., 90 Menaldi, J. L., 90 Mininni, R. M., 226 Ouerdiane, H., 135
Saito, K., 149 Sakabe, S., 149 Sengupta, A. N., 24 Sinha, K. B., 175 Sritharan, S. S., 90 Stan, A., 42 Sundar, P., 114 Yin, H., 114
