Infinite Dimensional Stochastic Analysis: In Honor of Hui-Hsiung Kuo

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Author: Ambar N. Sengupta | P Sundar

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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:


VOl. 7:


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

-

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

U K oflce: 57 Shelton Sweet, Covent Garden, London WC2H 9HE

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

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright 0 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by World Scientific Printers

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

Ito , 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


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


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,)O1 - 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


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


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


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


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,),


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


(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


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


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


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


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


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


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:


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


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


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)


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


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


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 .-


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
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
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


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 .


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).

88


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.


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


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


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)


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


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 ,


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)), ,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


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)


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


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)


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


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


L2-norm, we deduce

Rearranging the terms we have

111

112


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.


113

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

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 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 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)(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 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 , )

Quantum Stochastic Dilation

189

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

190

D. Goswami and K. B. Sinha

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

+

+


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,,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

~ ~ ) +

192

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

+

+


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


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

-


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


197

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,

198


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))


199

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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A . Hahn

(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

204

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 )

206

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


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 )

208

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


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 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(-;

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


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’.


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) 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 >xHE)dv = exp(-$ 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(-;

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:


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 >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


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 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= (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.

244

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).

245

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

Infinite Dimensional Stochastic Analysis: In Honor of Hui-Hsiung Kuo

Infinite Dimensional Stochastic Analysis: In Honor of Hui-Hsiung Kuo

Infinite Dimensional Stochastic Analysis: In Honor of Hui-Hsiung Kuo

Infinite dimensional analysis

Infinite-dimensional analysis

Infinite dimensional harmonic analysis III

Interest Rate Models: an Infinite Dimensional Stochastic Analysis Perspective