Stochastic Partial Differential Equations and Applications II: Proceedings of a Conference held in Trento, Italy, February 1-6, 1988 (Lecture Notes in Mathematics, 1390) 3540515100, 9783540515104

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1390 G. Da Prato

L. Tubaro (Eds.)

Stochastic Partial Differential Equations and Applications II Proceedings of a Conference held in Trento, Italy February 1-6, 1988

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1390 G. Da Prato

L. Tubaro (Eds.)

Stochastic Partial Differential Equations and Applications II Proceedings of a Conference held in Trento, Italy February 1-6, 1988

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong

Editors

Giuseppe Da Prato Scuola Normale Superiore 56100 Pisa, Italy Luciano Tubaro Dipartimento di Matematica, Universita di Irento 38050 Povo, Italy

Mathematics Subject Classification (1980): 60H 15, 60G35, 93E 11 ISBN 3-540-51510-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-51510-0 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprintinq, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper

PREFACE

These are the Proceedings of the second meeting on Stochastic Partial Differential Equations and Applications, the first having been in October 1985 (with Proceedings published in the Lecture

Notes in Mathematics n. 1236). It seems that our wishes three years ago -

that this occasion for a more

direct communication among the researchers in this area of Mathematics occur every two or three years -

have been satisfied. Evidently two successes do not imply that there will be a third one, but they give

us good hopes ... ! The range of applications of SPDE becomes ever wider: filtering theory, biological models, control theory, field theory in Physics. They offer new problems and give at the same time hints for the solution. This time the Applications are better represented, although some lectures do not appear in these Proceedings because they have been published elsewhere. We wish to thank all the participants because it is due to them that the meeting has been successful. Finally we wish to thank the CIRM (Centro Intemazionale per la Ricerca Matematica) for its financial support. Special thanks again go the secretary, Mr. A. Micheletti, for his help (more than help) before, during and after the meeting.

Giuseppe Da Prato (Scuola Normale Superiore, Pisa) Luciano Tubaro (University of Trento)

CONTENTS

S. Albeverio, R. Hoegh-Krohn, H. Holden, T. Kolsrud-A covariant Feynman-Kac formulafor

.

unitary bundles over euclidean space A Bensoussan

On the integrated formulation of Zakai and Kushner equations

13

V.S. Borkar, S.K. Mitter - Lattice approximation in the stochastic quantization of($4hfields

24

M. Chaleyat-Maurel, D. Michel- The support ofthe density ofafilter in the uncorrelated case

33

P.L. Chow, J.L. Menaldi - Variational inequalitiesfor the control ofstochastic partial differential

equations

42

D.A Dawson, L.G. Gorostiza - Generalized solutions of stochastic evolution equations

53

A Dembo, O. Zeitouni - On the relation ofanticipative Stratonovicb and symetric integrals: a

decomposition formula

66

A. Frigerio - Some applications ofquantum probability to stochastic differential equations in Hilbert

space

77

I. Gyongy - The stability ofstochastic partial differential equations and applications. Theorems on

supports

91

G. Kallianpur, V. Perez-Abreu - Weak convergence ofsolutions ofstochastic evolution equations

on nuclear spaces P. Kotelenez

A stochastic reaction-diffusion model

H.-H. Kuo - Stochastic partial differential equations of generalized Brownian functionals

119 132 138

P.L. Lions - Viscosity solutions offully nonlinear second order equations and optimal stochastic

control in infinite dimensions. Part II: optimal control ofZakai' s equation

147

S. Meleard, S. Roelly-Coppoletta - A generalized equationfor a continuous measure branching

process

171

D. Nualart, AS. Ustiinel- Mesures cylindriques et distributions sur l' espace de Wiener

186

D. Nualart, M. Zakai - A summary of some identities of the Malliavin calculus

192

D. Ocone, E. Pardoux -A Lie algebraic criterionfor non-existence offinlte dimensionally

computable filters

197

M. Piccioni - A generalization ofWahba's theorem on the equivalence between spline

smoothing and Bayesian estimation P. Protter - A connection between the expansion offiltrations and Girsanov's theorem

205 221

VI

K.-U. Schaumloffel- White noise in space and time as the time-derivative ofa cylindrical Wiener

process

225

W. Smolenski, R. Sztencel- Large deviations for non-linear radontflcations of white noise

230

J. Zabczyk - Symmetric solutions of semilinear stochastic equations

237

List of Participants

257

A COVARIANT FEYNMAN-KAC FORMULA FOR UNITARY BUNDLES OVER EUCLIDEAN SPACE

Introduction. The purpose of the present article is to derive a kind of covariant Feynman-Kac formula, and with some more specificity, this refers to a Euclidean Brownian motion lifted to a unitary bundle. To motivate it, let us first go back to the Schrodinger equation d

(1)

iolJl/ot=-11281J1+VIJI=-112 l.:oj 21J1+VIJI, IJI(',O) =lJIo' j"l

(Here and below, physical constants are all equal to one; also 011=%xl1.) Regarding the potential V, which expresses interaction, as the electrostatic potential Ao' this equation becomes a special case of (2)

i(%t+iAo)IJI=-1/2

d

1.: j=1

(oj+iAilJl, 1JI(·,O)=lJIo'

where A=(AI , •. ,Ad):lR d-+lRd is the magnetic potential. Let xo=t and (3)

usd.

011 =0 11 +iAI1,

Then the 011 are covariant derivatives on a (trivial) line bundle IR d+ 1 xC, associated with the bundle IR d+ 1 xS1, and (2) can be written d

(4)

iOolJl =-1/2

1.:

0llJl, 1JI(',O)=lJIo'

j=1

Replacing t by -it we obtain instead the diffusion equation d

(5)

o

where the AlJ, are g-valued functions. The corresponding covariant derivatives are then

where CllJ,=Cl!oxlJ, and the same letter p is used to denote the induced

3

representation of g.. The covariant energy (Dirichlet) form is, for a function ql:N ..... W, (10)

JIDqll2 dvol, N

whereas the classical energy form is (11)

Jldqll2dvol, N

with d denoting outer differentiation. If we have a potential, i.e, a real-valued, and, to avoid complications, non-negative and smooth function V on N, we obtain the total energy by adding the term (12)

J Vlqll2 dvol, N

to

uo. 11). 2.

Let N be an oriented manifold and denote by tN all curves in N, i.e. all equivalence classes up to reparametrisation of piecewise smooth and continuous maps c: [O,l]..... N. We write c- =c(O), c" =c(1), and denote by c- 1 the inverse of c, i.e, c with reversed orientation. Whenever c 1+= c2- we can compose c 1 and c2 (c 1 followed by c 2) and we write c 1c2 for this curve. When defined the composition is associative, so tN is a partial group. We will use the notation txN for the curves c in tN with c" =x. Let now G and g. be as in the introduction, and denote by E the trivial bundle over N with fibre G, i.e, E=NxG. With a connection 't" we understand a bltingof curves in N to curves in E, preserving the partial group structures. We therefore demand (13) -r: tN xG tE, (14) 't"(o',g): txN t(x,g)E, i.e,

Now for some hE:G we have 't"(c,g)+=(c+,h). We will write h-gmtc): (16)

't"(c,g)+ =(c+,gm(c)).

We want 't" to be a homomorphism under the partial group structures: (17)

4

and (18)

'T(c,g)-1 ='T(c-1.gmtcl).

From (16) and (17) follows

=('T(c1 ,gh(c2,gm(c1)))+ ='T(c2,gm(cl))+ =(C2+,gm(c 1 Similarly (13), (14) and (16) yield (c- .gmfclmlc-1))=«C-1)+,gm(c)m(c-1» ='T(c -1 .gmtcl)" =('T(c,g)-1)+ ='T(C,g)- =(C- ,g). Consequently m must be a homomorphism of eN into G, or - better - a representation of eN in G: (19)

m:eN-.G.

(20) m(c-1)=m(ct 1.

(21)

We call m a multiplicative curve integral (MCl). The lifting of c to a curve in E, given the point gEG, is now defined as

where cl

denotes the curve

It is not difficult to check that there is a one-to-one correspondence between connections 'T and MCIs m. We have already seen how 'T gives rise to m. Conversely, given rn, we simply define 'T byeqn (16). Then (13-18) hold. Suppose now that we have an MCI which is smooth in the sense that t-.m(cl ) is smooth whenever c is. Then we can define 1

(24)

J

a (c) = m(c l r 1 dmtc') E g..

o

Then a is in the obvious sense a (smooth) additive curve integral (ACI). Conversely, given an ACI we obtain an MCI by solving the differential equations

5

Hence we have a one-to-one correspondence between smooth ACls and MCIs. Now, given an ACI v iff ,

n

vn(x,t) -> v(x,t)

vx ,

a.e. tin (O,T».

We then state the:

Theorem 3.2 : We make the assumptions of Theorem 2.1 and 3.1. Then the solution of (2.1) in the functional space (3.14) is unigue.

PROOF: Let us consider in (3.1) if> to be real and v to be the unique solution of (3.1), then we shall

prove that for any solution of (2.1), (3.14) one has: (3.15)

E 8 p(T)(if» T

=

1l"o(v(O» .

22 Therefore if there are two solutions Pl' P2 we shall have E 0T [Pl(T)(q,) - P2(T)(¢)]

=

0

and from (1.21), it follows:

This suffices to simply Pl(T) = P2(T). Let us prove (3.15). Notice first that by linearity (2.1) holds with consider the solution vof (3.1) which satisfies the perperties : (3.16)

Iv(x,t)1 , IDv(x,t)1 , 8v

lat l

a.e.

C,

C(I + [xl )

Ivk(x,t)1 , I Dv 8v

k(x,t)1,

k

a.e. t, lat l

ID 2v k(X,t)1

C (I + Ixl)

complex. Now let us

t

vx.

We can construct a sequence vk(x,t) which is estimates uniformly in k, i.e. (3.17)

vx,

.p

(complex valued) and satisfies the same C, YX,t ,

v x.

and converges to v pointwise as well as its derivatives, BVk

at

Bv

->

at '

vx .

a.e. t,

Indeed extend v(x;s) by v(x,T) for s ?: T and by v(x,O) for s n 1

k + vk(x,t) = n:;:)

11'2

f

n R +

1 (exp - k 2 [l x - Yl2 + It

0, and let: S12]) v(y,s)dy ds

which satisfies the properties (3.17). We deduce from (2.1) that:

E 0T peT) (vk(T)) + iV h*:IR k

- fT0 0t pet) (at BVk

1I'o(vk (0» + E

A(t) v k

-I (3) dt .

Now:

From the properties of the functional space (3.14, the property (3.15) follows.

23 Remark 3.1 : It is easy to check that the operator pet) defined by (J.l7) satisfies the requirements of the functional space (3.14). 3.3. Uniqueness of the solution of the integrated Kushner equation. The functional space for ?ret) is defined in a similar way as (3.14), namely: (3.18)

Yt, ?ret) E Q;'(B

w;

L 1(O,Zt,P))

and we can state the :

Theorem 3.3 : We make the assumptions of Theorem 2.1 and 3.1. Then the solution of (2.3) in the functional space (3.18) is unique. PROOF: It is the same as that of Theorem 3.2.

o

REFERENCES. [I]

A. BENSOUSSAN, Stochastic Control of Partially observable systems, to be published, Cambridge University Press.

[2]

N.Y. KRYLOY - B. ROSOYSKII, On conditional distributions of diffusion processes, Math. USSR Izvestija, 12, 336-356 (1978).

[3]

D.W. STROOK - S.R.S. YARADHAN, Multi dimensional diffusion processes, Springer (1979).

[4]

M. ZAKAI, On the optimal filtering of diffusion processes, Z. Wahrschein. verw. Geb. II, 230-243 (1969).

LATTICE

IN THE STOCHASTIC QUANTIZATION OF FIELDS l 2

Vivek S. Borkar Tata Institute for Fundamental Research (TIFR) P. O. Box 1234, Bangalore, India Sanjoy K. Mitter Department of Electrical Engineering and Computer Science Laboratory for Information and Decision Systems (LIDS) Center for Intelligent Control Systems Massachusetts Institute of Technology Cambridge, MA 02139 U.S.A.

I.

INTRODUCTION The Parisi-Wu program of stochastic quantization [8] involves con-

struction of a stochastic process which has a prescribed Euclidean quantum field measure as its invariant measure. carried out for a finite volume P. K. Mitter in [6].

2

This program was rigorously

measure by G. Jona-Lasinio and

These results .,ere extended in [2], which also

proves a finite to infinite volume limit theorem.

The aim of this note

is to prove a related limit theorem, viz., that of the finite dimensional processes obtained by stochastic quantization of the lattice fields to their continuum limit, i.e., the

2

process of [2],

[6].

2

The proof imitates that of the limit theorem of [2] in broad terms, though the technical details differ.

Note that this limit theorem can

also be construed as an alternative construction of the in finite volume. The next section recalls the finite volume

2

2

process.

process Section

III summarizes the relevant facts about the lattice approximation to the

2

field from Sections 9.5 and 9.6 of [4].

Section IV proves

the limit theorem.

lThe research of the second author was supported in part by the U.S. Army Research Office, Contract NO. DAAL03-86-K-Ol7l (Center for Intelligent Control Systems, Massachusetts Institute of Technology) , and by the Air Force Office of Scientific Research, Contract No. AFOSR-85-0227.

25 II.

THE

2

PROCESS

Let AcR 2 be a finite rectangle which, for simplicity, we take to be the unit cube

x =

(x , x ) 1

Laplace operator on A.

10 -< x 1 ,

x

< 1

2-

Let /:::, denote the Dirichlet

It is diagonalized by the basis

(x) = 2

sin (k x ) , x = (x , x ) , k£B = {(k , k ) I k . = rin , n > 1, 22 12 1 2 a In fact, - /: :, e = k 2 e where k = k 2 + k 2. For ct£R, let H

sin (k x) i = 1,

2

11

K

K

2

1

denote the Hilbert space obtained by completing

2

D(A) with respect

to the inner product < f, g>

ct

L:

=

(k 2)ct

ct

is the L

scalar product. Topologize Q=UH by the countable ct family of seminorms 2 !I • II = ,» > and Q"'= UH- ct via duality. n ct Let e = (_/:::,+lj-l, ( . , . ) its integral kernel, ect its ct-th operator power, and the centered Gaussian measure on H- 1 with co-

where

variance

e

e

Let·· denote the Hick ordering with respect to

[2], [6].

a,

(see [4], Ch.

for a

defined by exp

int t

A 1 if k = t, = 0 otherwise o . • 0 Lemma 3.2 ([4], p , 222) The map lO' defines an isometric imbed-

Also,

ding of t

(int A,,) .... L (A).

2

u

2

Let IT(j be the projection operator on L (A) z ef1 , so that

Fourier series at k 11T = a. ITo =2: 0

L'1


2

ro(S)(C:f)dS) oC 0+0 (s) (C t) ds).

(4)1(0),

r

It follows that

=

4>2 (t) (f) Similarly E[

I

(t

21

0 (s):o (Co1 -

t E

£I:

3

(0): 0 analogous to those above. loLl.

o

0

+

Jt

0

s



IR'

Proof : In [1], an integration by parts formula is obtained for the unnormalized filter in a very general setting, using Malliavin calculus. In the uncorrelated we can ennounce it as follows:

37 for each multi-index processes C: ,0 •

there exist integers n l

1,11 . . , l

• i k

I

•...• i

k

k

and

, •••

.ke[l.

such that (i)

for a.e.y .•

Vq>l.VT>O. E;( sup

and,

processes C

(ii) all the (w.x o ) (iii)

V >O, E;( sup

) • E;

V

Ie:.

and D

are adapted

)L I) = E; ((x i

)

to the filtration of

L ) I

I

k where B:

I,

As,

0

+

L

k=l [" •. ,

0 is the discount factor. The functional

(2) means the average total accumulative cost up to and stopping at

T.

The control

Let u(x) denote the optimal

objective is to minimize the average cost functional cost defined by

(3)

u(x) =

: T stopping time }.

As in finite dimensions (see, e.g. Bensoussan and Lions [2]), formally, we expect that the optimal cost u(x) is the solution of a variational inequality (V.I.) in H:

(4)

(Lu - au

+ f)

1\

(tP - u) =

0

where L is a second-order differential operator associated with the diffusion process Yt in H, and (a 1\ b) = min{a,b}. By means of an appropriate Gaussian measure J.L on H, we will formulate the problem (4) in an L 2(H,J.L)-setting. This enables us to generalize the V.I. results in finite dimensions to that of infinite dimensions. Parabolic and elliptic equations in infinite dimensions have been studied by several authors, including Daletskii [3,4], Piech [5] and Chow [6]. The idea of an L 2-formulation was already discussed in [6]. The control of stochastic P.D.E.'s has been treated by Bensoussan and Viot [7] and DaPrato [8], among others.

2. Preliminaries. Let (V, H, V') be a Gelfand triple, where V and H are real separable Hilbert spaces and V'is the dual of V such that the inclusions V

'---+

H

'---+

VI

are Hilbert-Schmidt (H-S). Then (i,H, V') is an abstract Wiener space in which a standard Brownian motion

Wt

is defined so that

E

< w"v >= 0

and

E < w"u >< w"v >= (t 1\ s)(u, v), VU,v E V, where (".) and

< " . > denote the inner product in H and the duality between V' and

V, respectively. Consider the linear stochastic equation in V':

(5)

dYt

+

Yo =

A Ytdt

x,

= o dso,

44

where A is a closed linear operator in H with a dense domain D A C V, a : V'

-I>

H is

a bounded linear operator, and x E H. Proposition 2.1. Suppose that (-A) generates a strongly continuous semigroup Tt such that ds

< 00, "it > O.

Then the equation (5) has a mild solution given by

(6) which is a diffusion in H, where 11·112 denotes the H-S norm in H. The proposition is just a re-statement of Theorem 2.1 in [6], in which Theorem 2.2 implies that Proposition 2.2. Let A be self-adjoint so that Tt is a contraction. Then the solution process Yt has an invariant distribution p. which is a centered Gaussian measure on H with the covariance operator

(7) where S = oo", Let S(H) denote the set of smooth functionals on H, that is, Co -functions depending only on a finite number of coordinates. For u E S(H), let Dzu,

. . . , stand for

the first, second H -derivatives and so on. Define

(8)

1

Lou = 2 Tr

- (Ax,Dzu).

Then we have the following Green's formulas, (Lemma 3.3, [6]). Proposition 2.3. Let A be given as in Proposition 2.2 and let AS = SA on S(H). Then Vu,u E S(H), we have

The Green's formulas, as in R!', are important in studying infinite-dimensional differential equations in an L 2-setting. In fact it was also shown that L o can be extended

45

continuously to a self-adjoint operator L in a Hilbert space to be introduced later. For elliptic operators with variable coefficients, the formula (9) has been generalized by Daletskii [41 in some special situation (see §5). 3. Variational Inequality for Linear Stochastic Equation Consider the linear stochastic equation (5). We assume that

(11)

(i) A satisfies the conditions in Proposition 2.3. (ii) a : V'

-+

V is linear and continuous and its

restriction UH to H is H-S. Let

)I =

L 2(H,p,) and let V be a subspace of

(12)

)I

defined by

V = {v E)I : !)SDzv, Dzv)p,(dx) < co},

which is a Hilbert space with the inner product

((u, v)) = !H{SDzu, Dzv)

+ a(u, v)}p,(dx) , a> O.

Let V' denote the dual of V so that

V

c)l

c V'

form a Gelfand triple. Finally introduce

(13) Then we have (see [6]) Lemma 3.1 The operator Lo on S (H) has a self-adjoint extension L in D. Moreover the Green's formulas (9) and (10) hold for u, v ED. Given a positive functional

)I

with domain

1 on H, consider the expectational functional

(14) where Yt is the solution of (5) and E z (-) = E{·IYo = z}. Lemma 3.2. Let integer m

(15)

1:H

-+

R+ be continuous and let there exist constant c > 0 and

> 0 such that

I/(x)1 s c(1 + Ilxll"'), (ii) I/(x) - I(y) I::; c(1 + I xii'" + Ilyllm) Ilx - yll, z, y E H,

(i)

46

where

II . II

denotes the H -norm. Then the functional v defined above satisfies the

equation Lo - av + /

(16)

0 with v E [).

Proof. Choose a complete orthonormal system (C.O.N.S.) {en} C D A in H.

Pnx

Let

Ej=l(x,ei)ei be a projection of H onto H n = Pn(H). Consider the following finite-dimensional approximation =::

(17)

dy!n)

+

Any!n)dt = IT,.dwt,

en)

Yo

Xn ,

(18) Then it satisfies the following equation

L(n)vn =

(19) To show

V n -+

- (A,.x, Dzv n) = aV n - /(Pnx).

Tr.

v, we note that en)

(20)

Yt

en)

r

(n)

= T t Pnx + 10 Tt_.lTndw•.

where Tt(") is the semigroup generated by An. In view of (6) and (19), we get

s

(21)

E{loo I/(Ttx + Zt)

/(Tt(n)xn + z!n)) !e-Cddt}

+ E{loo I/(T/n)x n + Zt) where Zt =

f(T/n)x n + z!n»)!e-atdt},

Tt_.lTdw. and similarly for z!n). Since TtCn)

-+

Tt strongly and z.Cn)

-+

z.

in mean-square, the above difference (20) goes to zero, if / is bounded and continuous. Under the assumptions (15), by the dominated convergence theorem, the assertion is still true, that is, v(n)(x) -+ v(x)V'x E H. To verify that v satisfies the equation (16), it suffices to show LCn)vn

L

2(H,p,).

3.1. 0

-+

Lv in

But this is a consequence ofthe Green formula (10) which is valid by Lemma

47

Let ao (u, v) be a bilinear form on 11 defined by (22)

ao(u,v)

= !H(SDzu,Dzv)/l-(dx),u,v E 11.

For u E VeL}, by the Green formula (19), it yields (23)

ao(u,v) = - !a "Iv E 11, where h,) denotes the )I-inner product and < ',' >, the duality between 11 and 11'.

Theorem 1. If the conditions (11) hold and I E 11', then the variational problem has a unique solution. If, in addition, the conditions (15) for

I are valid, then the solution

has the representation (14).

Proof. Since, by definition, a(u,u) =

the bilinear from a is coercive. As in

the theory of partial differential equations [9J, the well-known Lax-Milgram theorem ensures that the variational problems (24) has a unique (weak) solution. If I satisfies the conditions in (15), then, by Lemma 3.2, the representation (14) is a (strong) solution to the equation (16) and, hence, is also a weak solution. 0

4. Linear Variational Inequality. Consider an associated optimal-stopping problem. Let (0,1, P) be the standard Wiener space and let

1t i 1

be an increasing a-field.

Corresponding to the state

equation (5), the control problem is to minimize the cost functional

(25) as given in (2), where {r

t} E 1t is a stopping time such that Er


t/J,

where an is the appropriate finite-dimensional approximation of a. Formally, letting n -+ 00

in (27) and (28), they give rise to the corresponding problems in infinite dimensions:

(29)

Given

I

E

V' and t/J

E

V, find

U

in V such that

(Lu - au + f) /\ (t/J - u) = 0 in V', and (30)

Given

I E V' and t/J E V, find U E V with U

a(u,v - u) 2':< [,» -

U

:::;

t/J such that

>

for every v E V, v :::; t/J.

Lemma 4.1. Suppose that I and t/J are positive and continuous on H and I satisfies the conditions in (15). Then, if u(n) converges to

U

in V, the limit u is a solution to the

V.I. (30).

Proof. Let Un,Vn E Vn , regarded as a subspace of V, such that Un -+ U and Vn -+ v in V. Clearly < I, Vn - Un >-+< I, v - U >. Let an(u, v) an(u n, Vn), where Un, Vn are the projections of u, v E V into Vn , respectively. Since a and a are continuous and coercive in V

o

X

V, it is easy to show that an(un,v n) = an(u,v) -+ a(u,v). The rest is obvious.

Let us call the solution of the V.I. (30) a weak solution to the optimal-stopping problem (5), (25) and (26). Then we have

Theorem 2. Let the assumptions in (11) hold. Then, for each

I

E V', t/J E V and

49

a> 0, the V.I. (30) has a unique solution. to

U

in 11, then

U

In particular, if Un defined by (27) converges

is the (weak) solution of the optimal-stopping problem.

Proof. In a general Hilbert space setting, the existence of a unique solution to the V.I. (30), where the bilinear form a is coercive, has been proved (see, e.g., Kinderlehrer and Stampacchia [10l). The fact that

U

= lim n --+ oo Un in 11 is the solution of the V.I. (30)

follows from Lemma 4.1. 0

Remarks. 1). The present treatment of variational inequality with infinite-many variables in a Gauss-Sobolev space 11 seems to be new. 2). In contrast with the solution v to the equation (24), under some reasonable assumptions, the convergence of Un to the optimal cost u(x) defined by (26) has not yet been proved.

Example. As an example, let G C Rd be a bounded domain with a smooth boundary Let A

= (-A + k

2

),

oG.

where A is the Laplacian and k > O. Then (5) is a parabolic

Ito's equation, subject to, say, the homogeneous Dirichlet boundary condition. Define

H == L 2(G), 11 r

= Hr(G)

and V'

= H-r(G), where

= Hd + 1) so that the imbedding i: H

Then o = A -r/2 : H

-+

-+

Hr(G) is the Sobolev space of order

11' is H-S [9J. For x E 11, IIxllll

= IIA r / 2 x IlH .

ao" is of trace class. Let {lIn} and {en} be

H is H-S, and S

the eigenvalues and eigenfunctions of S, respectively. With respect the C.O.N.S. {en}, we have

00

Sv

= E IIn Vn f n ,

00

V

E H,

n=l

= E >'nvnen,

(E lin < 00) n=l

00

Av

v E D A,

n=l

where D A = {v E H: < oo}. Let yIn) = (ye, en) and wIn) =< we,en >. Then the stochastic equation (5) yields

=

x n , n = 1,2, ... ,

which has an invariant measure JL with covariance operator I' given by

[I'u, v) =

1

2 E >.;lllnUnVn, u, v E H. 00

n=l

50

Recall)l = L 2(H,p,). For u E S(H),

The bilinear form a takes the form

Then V is the space of functions u such that a(u, u)
with variable coefficients of the form

(31) where S(x) : H - H is of trace class and A(x) : DA C V - H is self-adjoint "Ix E V. Let L o be defined by

(32)

1

Lou = '2 Tr {Dz[S(x)Dzu]} -

1

'2 (R(x)

x,Dzu),u E S(H), x E V,

where R(x) = S(x)r- 1 and r is the covariance operator for a Gaussian measure p, on H with domain of

r- 1 in V.

It follows from a theorem of Daletskii [41 that the following

Green's formula holds (33)

r (Lou)vp,(dx) = lHr u(Lov)p,(dx) =

lH

r (S(x)Dzu,Dzv)p,(dx)

21H

for u, v E S (H). Clearly, as before, L o can be extended to be a self-adjoint operator L in)l = L 2(H,p,). Let us define the space V with the norm

The domain of L can be defined as

D(L) = {v E V : Lv E )I},

51

or, equivalently, v in V such that the mapping

can be extended continuously to an element of }I'

S(x) : H

-+ H

=

}I.

Suppose that the operator

satisfies

(34)

\Ix E H : IIS(x)lb

M(1

+ II z II"") for some constants

M> O,m 2:: 0. Then the set of all functions cp on H such that \Ix E H : IIDcp(x)11 2

C (1 + IIxll"") for

some constants C > 0, m 2:: 0, is included in V. Consider the bilinear form

(35) Clearly a(.,.) : V

a(u,v) = ao(u, v) X

V

-+

+ o:(u,v), 0: > 0.

R is continuous. Hence, as before, we can pose the following

variational problem:

(36)

Given

I

E V' and t/; E V, find u E V, u

t/; such that

a(u,v - u) 2::< t,v - u > for every v E V, v

t/;.

Since the V.I. (36) is similar to the V.I. (30), the same kind of arguments will verify the following

Theorem 3. Let the assumption (34) hold. Then the V.I. (36) has a unique solution. However, though motivated by an optimal-stopping problem, its connection with the V.I. (36) has not yet been established. Also the variational inequality associated with

L 1 has not been studied. The operator L may be regarded as the "principal part" of L 1 • The present result is a first step in the study of V.I.'s with variable coefficients in

infinite dimensions. This and other related problems will not be discussed here.

52

REFERENCES 1. P.L. Chow, Stochastic differential equations in infinite dimensions, Mathematics and Physics, Vol. 3, ed. by L. Streit, World Scientific Pub. (to appear). 2. A. Bensoussanand J.L. Lions, Applications de Inequations Variationnelles en Controle Stochastique, Dunod, Paris, 1978. 3. Yu.

L. Daletskii, Infinite dimensional elliptic operators and parabolic equations

connected with them, Russian Math. Surveys, 4.

(1976), 1-53.

On the self-adjointness and maximal dissipativity of differential operators for functions of an infinite-dimensional argument, Soviet Math. Dokl. 17 (1976),498-502.

5. A.M. Piech, A fundamental solution of the parabolic equation on Hilbert space, J. Functional Analys.

a (1969), 85-114.

6. P.L. Chow, Expectational functionals associated with some stochastic equations, in Stochastic Partial Differential Equations and Applications, LNM # 1236, SpringerVerlag (1987),40-56. 7. A. Bensoussan and M. Viot, Optimal control of stochastic linear distributed parameter systems, SIAM J. Control and Optim., 13 (1975),904-926. 8. G. DaPrato, Some results on Bellman equation in Hilbert space, SIAM J. Control and Optim., 23 (1985),61-71. 9. J.L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I, Springer-Verlag, N.Y. 1972. 10. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Acad. Press, New York, 1980.

GENERALIZED SOLUTIONS OF STOCHASTIC EVOLUTION EQUATIONS Donald A. Dawson! Carleton University Ottawa, Canada K1S 5B6 Luis G. Gorostiza'' Centro de Investigaci6n y de Estudios Avanzados Mexico 07000 D.F., Mexico 1. INTRODUCTION

In applications there arise stochastic evolution equations of the form dyt = A*ytdt + dZ t ,

t E [0, T],

(1.1)

defined on the nuclear triple S(Rd) C L2(Rd) C S'(R d), where S(R d) and S'(R d) are the usual Schwartz spaces on Rd. The processes Y and Z take values in S'(R d), Z is a semimartingale, Zo = 0, and A* is the adjoint of a linear operator A on L2(Rd) which generates a semi group of bounded linear operators '{Tt , t

O} on L 2(Rd ) and whose

domain, Dom(A), is assumed to contain S(R d ) . When (1.1) has a unique solution, it is given in the form (1.2) known as the evolution (or variation of constants) solution. Expressions (1.1) and (1.2) are symbolic and they may be interpreted in various ways. In many cases the operators A and Tt map S(R d ) into itself and these expressions are interpreted in the following (weak) forms:

and

i Research supported by the Natural Sciences and Engineering Research Council of Canada. 2 Research supported in part by CONACyT grants ICEXCNA-40319 and 140102 G203006, Mexico.

54

respectively, where (".} denotes the duality between S(Rd) and S(R d). A typical example is A If A and T; do not map S(Rd ) into itself, then the meanings of (1.3) and (1.4) must

be clarified because A

0, do not map

S(Rd) into itself because the

condition of fast decay at infinity fails. Let C(Rd) denote the space of real continuous functions on R d, and Co(R d) the set of elements of C(R d ) which vanish at infinity. For p > 0 we define the function

where

I.I is the usual norm on

Rd. We define

55

and we introduce the spaces

and

These are Banach spaces for the norm

Hp •

The duals of these spaces are written C;(R d )

and C;,o(Rd), with dual norm H-p • We also consider Mp(R d), the space of positive Radon measures p, on R d such that (p" p) == f pdp, < 00, with the p-vague topology. The spaces Cp(Rd) and Mp(R d) are in duality, and St maps Cp(Rd) into itself. Then St acts on Mp(R d) by duality:

Proposition 2.1. [4] For 0 < 0: < 2 and d/2 < P < (d + 0:)/2, (1) S(R d) C Cp,o(R d) C L 2(Rd) C C;,o(Rd) c S'(Rd).

(2) S(R d) is densely and continuously embedded in Cp,o(R d). and St for each t are continuous linear mappings from S(R d) into Cp,o(R d).

(3)

(4) t

1-+

(5) t

1-+

3.

St is a continuous curve in Cp,o(Rd) for each E S(Rd). StP, is a p-vaguely continuous curve in Mp(Rd) for each p, E Mp(R d).

GENERALIZED SOLUTIONS We denote 1i

= S(Rd )@D ([-

8,T ]) , where 1'([-8, T]) is the space of infinitely differ-

entiable real functions with supports contained in (-8, T), where 8 > 0 is arbitrary and fixed. For a function il>(x, t), x E Rd, t E R we write il>t

== il>(-, t).

We will give definitions of the symbolic equations

dyt = A*ytdt + dZt , and

Yi

= TtYo

+

it

Tt_sdZs,

t E [O,T]

t E [0, T],

(3.1)

(3.2)

which are generalized versions of (1.3) and (1.4). In order to motivate these definitions we start with some formal calculations. Applying (3.1) to il> =

0 I,

E S(R d ) , j E 1'([-8, T), integrating by parts and

extending the result to elements of 1i we obtain

56

where a

== a/at.

Similarly, applying (3.2) to W = 1/J i8I g, 1/J E S(R d ) , g E D([-b, T]), integrating by parts, using Fubini's theorem and extending the result to elements of H we obtain

i

T

(yt , wt}dt =

(Yo, iT TtWtdt) -iT (z., iT Tt_.aWtdt)ds,

In (3.3) and (3.4) the terms

WE H.

(3.4)

JoT (yt, Aif!t}dt, (Yo, JoT T t Wtdt) and JoT (Z., J.TTt_.awtdt}ds

are not well defined because Y and Z take values in S' (Rd) and they are applied to things which are not necessarily in S(R d ) . Nevertheless, (3.3) and (3.4) give us the basis for our definitions. We designate by

.e°(n, F, P) the space of equivalence classes of real random variables

on a complete probability space with the topology of convergence in probability, and by D([O, Tj, S'(R d)) the space of functions from [0, Tj into S'(R d) which are right continuous and possess left limits, with a Skorohod-type topology [11]. Definition 3.1.

Let Y and Z be processes in D([O, T], S'(Rd )) defined on the same

probability space (n,F,p). Then Y is said to be a generalized solution of (3.1) if there exists a Banach space of real functions on Rd, denoted by V(R d), such that S(R d) c

V(R d) C L 2(Rd), S(R d) is densely and continuously embedded in V(R d), A is a continuous (yt, Aif!t}dt is a random variable linear mapping from S(R d) into V(Rd), the expression

JoT

on (n, F, P) for each if! E (Dom(A)

n

V(R d))0D([-b, T]),

and (3.3) holds for each such if!

(equality in .eO(n,F,p)). Definition 3.2.

Let Y and Z be processes in D([O, Tj,S'(R d )) defined on the same

probability space

(n, F, P). Then Y is said to be the generalized evolution solution of

(3.1) if there exists a Banach space of real functions on Rd, denoted by V(R d), such that S(R d) c V(R d) C L2(Rd), S(R d) is densely and continuously embedded in V(R d), t; is a continuous linear mapping from S(R d ) into V(R d ) for each t E [0, Tj, t - T t 1/J is a continuous curve in V(R d) for each 1/J E S(Rd), the right hand side of (3.4) is a random variable on (n,F,p) for each WE V(R d)0D([-b,T]), and (3.4) holds for each such W (equality in .e°(n,F,p)). Note that only one process Y can satisfy (3.4) (see [3], Prop. 4.1). The next result shows that Definitions 3.1 and 3.2 are equivalent. Proposition 3.3. Provided that the conditions of Definitions 3.1 and 3.2 are satisfied,

the stochastic evolution equation (3.1) has a unique generalized solution and it is given by the generalized evolution solution.

57

E (Dom(A)

Proof (outline): Suppose that (3.3) holds for

n V(R d»®'D([-8,T]). To

show that (3.4) is satisfied for W E V(Rd)®'D([-8, TJ) it suffices to verify it for W of the form W = t/J I8l g, t/J E S(R d), 9 E 'D([-8, TJ). Let

= Then

E (Dom(A)

iT Tt-st/J(x)g(t)dt,

xER

d, S [-8, T]. E

n V(R d»®'D([-8, T]),

= -t/Jg(s)

-iT Tt-sAt/Jg(t)dt

iT Tt-st/Jg'(t)dt =iT Tt_soWtdt, =

and

Substituting into (3.3) yields (3.4). Now suppose that (3.4) holds for lit E V(R d)®'D([-8, T]). To show that (3.3) holds for q> E (Dom(A) n V(R d»®'D([-8, T]) it suffices to verify it for q> of the form q> = I8l f, E S(R d), f E 'D([-8, TJ). Let

s) +

lIt(x,s) =

s),

x E Rd ,

S

E

[-0, T).

Then lit E V(Rd)®'D([-o,TJ),

and

iT

Tt_sowtdt =

iT

Tt_s!"(t)dt +

iT

Tt_sA!'(t)dt = /,(s)

=

Substituting into (3.4) yields (3.3). Remarks 3.4 (a) In order to use the previous formulation in a specific case it must be shown that the terms in (3.3) and (3.4) which are not well defined a priori are indeed random variables

58

on (n,:F, P) for each 4.> E (Dom(A)n V(R d»0V([-6, T]) and W E V(Rd)0V([-6, T]).

{JoT {yt, A4.>t)dt, 4.> E H} defines an H'- valued random variable, then by the Hahn-Banach theorem JoT (yt, A4.>t)dt can be extended to 4.> E (Dom(A) n T V(R d»0V([-6,T]). Proving that {Jo {Yt, A 4.>t )dt , 4.> E H} is an H'-valued random

If one can show that

variable involves the distribution of the process Y, the fact that S(R d ) is dense in Dom(A) n V(R d) and the regularization theorem; this approach will be followed in section 4. The other such terms can be dealt with in a similar way. The terms

J:{Yt, {)4.>t)dt, Jt {Zt, {)4.>t)dt and Jt {yt, Wt)dt determine H'-valued random variables in (n,:F, P) whatever are the distributions of Y and Z (see [3]), and hence they can be extended by the Hahn-Banach theorem. (b) The difference between our formulation and the analogous one in the deterministic theory is that in ours some of the terms in (3.3) and (3.4) are not required to be well defined a priori, and they are shown to be well defined in specific cases. with V(R d) = Cp,o(R d) (Proposition 2.1). (c) The previous results hold for A =

4. THE GAUSSIAN CASE In this section we consider Gaussian processes with values in S'(Rd). For such a process X we denote the covariance functional by

In the setting of section 3, the next result gives sufficient conditions for an S'(R d )valued Gaussian process to satisfy a stochastic evolution equation with an S'(Rd)-Wiener noise,

Theorem 4.1. Let Y

== {yt, t

E [0,T]} be a continuous centered Gaussian S'(Rd)-valued

process such that

(i) KY(5, 4>; 5, 'ljJ), 4>, 'ljJ E S(R d), can be extended continuously to 4>, 'ljJ E V(R d) uniformly in

(ii)

5,

5 I-t

KY(5, 4>; 5, 4» is continuously differentiable for each 4> E S(R d),

K y satisfies the relation Ky(s,4>;t,'ljJ) = KY(5,4>;5, Tt-s'ljJ),

5

t,

4>,'ljJ E S(R d).

Then Y is a generalized solution of the stochastic evolution equation dyt = A*ytdt + dW t ,

t E [0, T],

(4.1)

59

where W is an S'(Rd)-Wiener process such that Kw(s, 4>;t,t/J) =

r:

l«

(4.2)

(Q..4>,t/J}du,

with (Q..4>, t/J)

d = du Ky(u, 4>; u, t/J) -

Ky(u, A4>; u, t/J) - Ky(u, 4>; u,At/J),

4>, t/J E S(Rd).

(4.3)

Moreover, Y is the unique generalized solution of (4.1) and it is given by the generalized evolution solution. Remarks 4.2. (a) Theorem 4.1 was proved in [1] under the assumption that A maps S(R d ) into itself, equation (4.1) being understood in the form (1.3). This result was extended in [2] for a general nuclear Frechet space F instead of S(R d ) , and A being allowed to depend on t but mapping F into itself for each t. (b) Processes Y satisfying equations of the type (4.1) are called generalized Gaussian Ornstein- Uhlenbeck processes. Expressions (4.2) - (4.3) are an abstract version of a fluctuation-dissipation relation. In [4,5] we studied a generalized stable OrnsteinUhlenbeck process. When the noise process Z has independent increments, as in these cases, we call equation (1.1) a generalized Langevin equation.

Proof of Theorem 4.1 (outline): We will consider first the terms in (3.3) and (3.4) which must be shown to be random variables. Let

II> E V(Rd)®V([-S, T]).

We will define JoT (Yi, II>t}dt as an element of £0(11,:F, P).

Since S(R d) is dense in V(R d), then 'H == S(Rd)®V([-S, T]) is dense in V(Rd)®V([-S, T]). Let (II>n)n be a sequence in 'H which converges to

II>.

Then JoT(Yi,II>'t}dt is a random

variable for each n (see [3]). In the next calculation we use Holder's inequality, Fubini's theorem and assumption (I), and

I· [v

denotes the norm on V(R d ) .

T T Eli (Yi, II>f}dt - iT(Yi, 1I>'!'}d{ = Eli (Yi, II>f T

$ const·i

- 1I>'!'}d{

EIf

T

= const.i Ky(t,lI>f T

s const.i IlI>f $ const. sup

IlI>f -

II>'!'Wdt

1I>,!,;t,lI>f - 1I>,!,)dt

60

The last term tends to 0 as n, m -+

00.

Hence (JoT (yt, ;)dt) n is a Cauchy sequence in

[,O(O,:F, P) and therefore it has a limit in [,O(O,:F, P); this limit is independent of the particular sequence (n)n' We designate the limit by JoT (yt, }(Yt, ¢»f(O)1'(t) + E(}Q,¢>}(Yt, A¢»f(O)f(t)]dt

T

T

T

T

[E (Ys , ¢>}(Yt, ¢»1'(s )1'(t) + E(Ys, ¢>}(Yt, A¢>}1'(s)f(t)Jdtds [E (Ys, A¢>}(Yt, ¢>}f(s )1'(t)

+ E(Ys, A¢>}(Yt, A¢>}f(s )f(t)]dtds

62 =

T

+ 2i

[J(yeo, T

+ 2i + 2i

10r

-2i

T[J(y(s,

i

= J(yeo, -2

iT [J(y(s,

T

)f(O)f'(t) + J(yeo,

0,

)f'(t) + J(y(s,

s,

+

s,

0, )f(O? - 2J(yeO,

T

2 s

T

0,

)f(O)f(t)]dt s,

)f(t)]dtds s,

0, )f(0)2

f(s)2ds

= iT =

On the other hand,

Finally, Proposition 3.3 implies that Y is unique and is given by the generalized evolution solution.

5. EXAMPLES (1) Let Xk, k = 1,2,··· be independent processes on R d which are spherically symmetric stable with exponent 0

< a < 2, whose initial point is distributed according to a probability

measure fL on Rd. Consider the empirical distribution of the first n processes, n

Nn(t) =

2: bx.(t), k=l

and the normalized fluctuation

t

0,

63 The following result can be proved by standard methods. Proposition 5.1. Y n converges weakly in D([O, (0), $I (R d)) to Y as n

-+ 00,

where Y is a

continuous centered Gaussian S(Rd)-valued process whose covariance functional is given by

Ky(s,¢>;t,'t/J)

= JRd f ¢>(x)(St-s't/J(x»P,s(x)dx - f

f (St_s't/J(x))P,s(x)dx, JRd ¢>(x)P,s(x)dx JRd

s S; t,

¢>,'t/J E S(R d ) ,

We note that conditions (ii) and (iii) of Theorem 4.1 are satisfied. Since P,s(x )dx is a finite measure on Rd, it is easy to show that

uniformly in s. Hence assumption (i) of the Theorem holds also. Therefore, applying Theorem 4.1 we obtain: Proposition 5.2. The process Y

== {yt, t E [0,Tn given in Proposition 5.1 satisfies the

Langevin equation

dyt = D.aytdt + dWt,

t E [0, T],

in the generalized sense, where W is an $I (Rd)_Wiener process such that

Kw(s,¢>;t,'t/J)

r: (Qu¢>,'t/J}du,

J

o

with

The initial value YO has variance

Ito [7] considered the case where the Xk are Brownian motions (a the usual interpretation (1.3) of the Langevin equation. In this case

{Qu¢>,'t/J} = where

>

is the scalar product in Rd.

f

JRd

"V¢>(x)· "V't/J(x)P,u{x)dx,

2), which allows

64

(2) The result of Martin-Lof [10] on the high density fluctuation limit of a Poisson system of Markovian particles can be extended to the case A = b.". The fluctuation limit process

Y is a centered Gaussian S'(Rd)-valued process with covariance functional

(see [1], example (2)). By Theorem 4.1, Y satisfies the Langevin equation

dYt

= b."Yidt + dWt ,

t E [0, T],

in the generalized sense, with Yo = Gaussian white noise on R d, and W is the S'(R d)Wiener process such that

Kw(s,;t,7jJ) = (s At) ( [b.,,(7jJ)(x) - (x)b.,,7jJ(x) -7jJ(x)b.,,(x)]dx,

JRd

,7jJ E S(R d ) .

Only assumption (i) of Theorem 4.1 needs verification:

IKy(s,;s,7jJ)I::; {

JRd 1(x)II7jJ(x)ldx

::; { (1 +

JRd

Similar results can be obtained for several fluctuation limits of infinite systems of symmetric stable processes with branching and immigration (see [1,6]). In these examples A

= b." + a,

where a E R is the Malthusian parameter.

Acknowledgement.

L.G.G. expresses his thanks for the hospitality of the Mathematics

Section of the International Centre for Theoretical Physics at Miramare-Trieste, Italy, where part of this paper was written. References 1. Bojdecki, T. and Gorostiza, L.G. (1986). Langevin equations for S'-valued Gaussian processes and fluctuation limits of infinite particle systems. Probab. Th. Rel. Fields 73, 227-244. 2. Bojdecki, T. and Gorostiza, L.G. (1988). Inhomogeneous infinite dimensional Langevin equations. Stach. Analysis Appl. (to appear). 3. Bojdecki, T., Gorostiza, L.G. and Ramaswamy, S. (1986). Convergence of S'-valued processes and space-time random fields. J. Functional Analysis 66, 21-41. 4. Dawson, D.A. and Gorostiza, L.G. (1988). Generalized solutions of a class of nuclear space valued stochastic evolution equations. Tech. Rep. 225, Center for Stochastic Processes, University of North Carolina at Chapel Hill.

65

5. Dawson, D.A., Fleischmann, K. and Gorostiza, L.G. (1987). Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium. Tech. Rep. 105, Lab. Res. Stat. Prob., Carleton University, Ottawa. 6. Gorostiza, L.G. (1983). High density limit theorems for infinite systems of unsealed branching Brownian motions. Ann. Probab. 11,374-392. 7. Ito, K. (1983). Distribution valued processes arising from independent Brownian motions. Math. Z. 182, 17-33. 8. Ito, K. (1984). Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. S.I.A.M. Philadelphia. 9. Ito, K. and McKean, H.P. (1965). Springer-Verlag. Berlin.

Diffusion Processes and their Sample Paths.

10. Martin-Lof, A. (1976). Limit theorems for the motion of a Poisson system of independent Markovian particles with high density. Z. Wahrsch. Verw. Gebiete 34, 205-223. 11. Mitoma, I. (1983). Tightness of probabilities on C([O, 1],8') and D([O, 1]'8'). Ann. Probab. 11, 989-999.

ON THE RELATION OF ANTICIPATIVE STRATONOVICH AND SYMETRIC INTEGRALS: A DECO,MPOSmON FORMULA AmirDembo Information SystemsLaboratory StanfordUniversity Stanford.CA 94305 OferZeitouni Laboratoryfor Information and DecisionSystems Massachusetts Instituteof Technology Cambridge. MA 02139

1. Introduction Let Wto te [0.1]. be a standard.Ft-adaptedBrownianmotion. Let Xt be another. not necessarily adapted.stochasticprocess and assume that. in some senseto be specifiedlater

= -oo L->oo

-J (L)

Let 0

"I

< "2 < "3 < ...< "N = 1, and let t1N = sup I". - ".1 .

Define I(m) N

(A3)

i,j

=! s

(m)(w

ke l

k

'ex

w

"t-!

1

J

),

where Sk(m) is defined as in (1.1), i.e, IN(m) is the approximation to I(m) using the N-mesh tl ... "N·

68

-(L) prob -(L) IN(O) J(O)

('VL < 00)

lim

..

lim J L

lim

for any N-mesh

't 1 ••• 't N

when

O.

0, in probability

N(M)

(A4)

(A5)

The following easily proved lemma is the basis for our later conclusions: Lemma 1: Assume (AI) -(A5). Then Xt is Stratonovich integrable, and

(2.1)

frQQ!. By (A3),

J(O)(L) are a Cauchy sequence W.r.t. convergence in probability and therefore they converge in probability to a limit and the R.H.S. of (2.1) is well defmed as this limit is probability. By definition (1), we therefore have to show that

L N

lim prob

= lim prob

wk

(2.2)

Xk

k=l.

We will show that: (2.3a) N

lim

lim

lim

£..J

,-,wk

k=l

(L) ] x- k(M) - J-N(O)

-

0 m . pro b

(2.3b)

and also that

L N

lim lim

-

==

0 in prob

k=l

Clearly, (2.3)

(2.2). Note that (2.3a) is exactly (A4). To see (2.3b), note that

(2.3c)

69

and (2.3b) follows from (AS). Finally, to see (2.3c), we will show that

3q>O

S.t.

(2.4)

from which (2.3c), and therefore (2.2), will follow. w

Let

-w

I

max

't;..l

't k

ke l ,..N

I; then, for q = e/q', E as in (A2),

(2.5)

For fixed N, the first term in the R.H.S. of (2.5) is finite, being the moment of the maximum (over a finite number) of finite variance zero mean normal variables, which concludes the proof of the lemma. We specialize the results of lemma I to two important particular cases. The first allows one to obtain a sort of "Taylor expansion", similar to the one in lemma (4.2) of [2]. The second will allow us to make connections with the Ogawa integral.

Lemma 2

Let St(m) be Ft-adapated continuous semi-martingales, and let their Doob-Meyer decomposition be: (m)

St

(m)

= So

+

A(m) t

+

M(m) t

where M;m) is a continuous martingale and A;m) is a continuous bounded variation process. Let a

m

E

1f2I1A

2

(m)11 , where II II denotes the total variation norm.

E 1/4[

'" .(m)

-

(t-s)

rvfm)

4

s ) ]

2

70

f 1

C

E1I2

m

(S(m»2dt

o

I

and assume that:

2:

< 00,

me.I

2:

L

< 00,

rne I

E«X(ffi»l
0

we are given well-measurable random processes

taking values in

L(E

processes E (t) , b (t) , oi Oi k:=1 ,2 ••• d 2 ) , taking values in

L(E

esses

ao(t)

Fo(t),

values in

fAt) ,

EO'

and the duality be-

EO

(,).)

Goi(t) ,

,

1,

(t) ,

(t)

(i:=1 ,2 ••• d 1 ' and well-measurable proc-

EO) ,

1,

(k)

goi(t) ,

Go i u,vE E

such that for every

well-measurable random

(t) ,

(k)

goi (t) ,

the processes

taking (E o i (t)u,v)

1 (G (t) ,v) , (goi (t) ,v) have the stochastic differenoi (k) k tials (E Oi (t)u,v) = (E (t)u,v)dNo(t) , d(b (t)u,v) = oi oi (k) k (k) k (t) u,v) dno (t) , d (G (t) ,v) = (G (t) ,v) dNo (t) , d (gOi (t) ,v) = (b oi oi Oi (k) k 1 d1 (goi (t) ,v)dno(t) , where Mo(t) :=(Mo(t) , ••. ,Mo (t» and d1 1 d1 (We recall ••• ,mo (t» are Fo t -semimartingales in R that the processes indexed by o are considered on the stochastic (b

oi

(t) u,v) ,

basis 0 o:=(no,Fo'Po' (F o t» .) We are also given -adapted continuous increasing-processes U ij ik Ho(t) , holt) , continuous F -adapted processes Co (t) , r o (t) ot of bounded variation (i,j:=1,2 .•• d k:=1,2 ••• d 2) and continuous 1, Fo t

1

d1

= (m (t) , ••• ,m (t.) o o d «dho(t) , and

1

-semimartingales

d2

Mo(t) :=(Mo(t) , ••• ,Mo (t» ,

for every

0 > 0,

dlMol1 (t)« dH (t) , o (t)«dho(t)

and

,

such that

d

mo(t)= Po -almost surely

(t)« dh (t) , o

dlrikll (t)«dho(t)

d!moll (t)«

for every

0 >0

i,j:=1,2 .•• d For every

k:=1,2 ••• d 2 • 1, 0 >0 we consider the stochastic evolution equations t

uo(t) =UoO+J(Ao(S)Uo(S)+Fo(S»dHo(S)+

o

t

.

(3.1)

99 t

vo(t)=VOO+j(ao(s)vo(s)+fo(S»dho(s)+

o

t

.

1 t{ to .. (s) + +2b lboi,bojl (s)vo (sl +boi (s)goj (s) -b oj (s)goi (s)

+J(b:)

(3.2)

(s»drik(s)

in the normal triple E 1 c:. EO $ c:. Er, where "so and v60 are f -measurable random variables in E and E resP oO 2, 7ctively, 1 [b oi ,b 6 j] :=b6ib6j-b6jb6i' and (B 6 i (s) "e (s) +G6 i (s) , are short notations for i 1 i j (B6i(S)U6(S)+G6i(S»dMc;(S)+2(BOiBOjUAS) +B 6 iG o j (s»d(S)+ 1 (k) (k) i k i +2 (B6 i (s) "e (5) +G 6i (5» d It) and (b o i (s) v6 (s) +goi (s) dmo (s) +

1 ) i j 1 (k) i k ) +2 (bQib Oj (s "« (s) +boig o j (s ) )d (5) +2 (b o(k) i (5) v6(s) +goi (s»dL(E

(i:=l,Z, •.• ,d d

Z:=[O,T]xC([O,T], R l ) ,

l,

l,

EO)'

B

(k)

i

k:=l,Z, ..• ,d

: Z ->L(E

l(dl+l)+l)

such that for every

dimensional semimartingale

y(t) ,

random processes

l,

EO) ,

Gi: Z ->E Z '

definedon

and every d l l satisfying the condition (a l), the and

u,vEE

(Gi(t,y(.)) ,v)O

have the

stochastic differentials (k)

d(Bi(t,y(.))u,v)O=(Bi

-k

(k)

-k

(t,y(.))u,v)Ody (t), d(Gi(t,y(.)),v)O=(G (t,y(.)),v)Ody (t) i Ak c11+jdl+l on the interval [O,T], where y (t) =y(t) (k:=l,Z •.• d y (t)= l), . l d (d +1)+1 -=(t) (j,l:=l ,2 .•• d , 9 1 1 (t) =t. (The a-algebra of well-neasurable subl) d l):X(U)EB} d l) sets of [O,T]xC([O,T]; R is generated by the sets 5,t[X{X(.)EC([O,Tb R ,

103

where

0

u

s

st

T

d

B EB

and

1)

.i

H be a subset of the absolutely continuous functions

Let

R

W: [O,T]

d

1

, w(O)=O,

H contains the infinitely dif-

such that

ferentiable functions vanishing at

0 E [O,T] •

We consider the stochastic evolution equation t u(t) =UO+!A(S,y(.) l u t s ) +F(s,y(·) )ds+

o

t . +! E. (s, y ( • ) ) u (s) +G. (s, y ( • ) ) dy1. (s) + o 1. 1. 1t + 2'! E.E. (s,y o 1. )

E!k) (s,y(.»u(s)+G!k) (S,Y(.»d(s) 01.1.

+

and for every uW(t)

. . u (s) +E. (s,y (.) luIs)+

°

2

Klul

2

u EE

3, are constants.

for

u

u:=0,2

u:=0,2,

I (Au, BiU)O +(u, on

u:=0,2

I (B i(k) u,u)OI

i,j :=1 ,2 ... d

1

'

2

k:=1 ,2 •.• d1.+1

where

K

Let

C([O,T]; EO) nL 2([0,T]; E be a continuous embedding from 1)-+X the space C([O,T]; EO) nL E ) , equipped with the norm lul= 2([0,T]; l

t 2 =sup!u(t) 10+Jlu1 (t)dt, 1

° the metric

with

into a metrizable topological vectorspace

P. Then the solution

ered as a random element in

u

to eq.

X

(4.1) can be consid-

X. Its distribution on

X

is denoted by

and the topological support of use the notation

is denoted by supp . We for the clouser of U:={u w: wEH} in the space X.

OX

Theorem 4.1. Suppose that the assumptions are satisfied. Then

supp

Proof. First we note that the clouser in w

00

{u : wEC ([O,T]; R same, where

d1

), w(O)=O}

AO([O,T];

functions on

[O,T],

T>

(a

= OX

d 1)

X

1),

(a 2) and (SE 1)-(SE 3)

of the sets

d {u: WEAO([O,T]; R 1 )} w

and

is the

denotes the set of absolutely continuous

taking values in

d

E 1

and vanishing at

0. To

d

by see this we approximate the function wE A ( [0 ,T] ; R 1) O d1 [O,T] , ) such that wa(O)=O and walt) -+w(t) on w ECCX)([O,T]; R a and uniformly in t , as 6 -+ 0. Then we get sup I u w6 (t) -u wI -+ T w6 2 Jlu (t)-uw(t)l as 6-+0 by applying Theorem 3.1 in a very

°°

°special case.

ldt-+O

Therefore it is enough to prove the theorem when

00 d H:={wEC ([O,T]; R 1 ),w(O)=O} . -X In order to prove supp X cU

proximating the semimartingale random processes (t) = 6- 1

/ 6 where

y (s) : =0

Y6(t) ,

fyi (t-s) ° for

we use the standard method of ap-

y(t)

with infinitely differentiable

defined by smoothing as follows: (6- 1 (s)) ds

s 0,

in the normal triple

Ao(S,W):=A(S,yo(·'w»

E

1

, F 6(S,W):=F(S'Y6(·'w»

GQi(s,w):=Gi(s,yo(·'w»

c. EO

c. E_

1 '

where

, Boi(s,w):=Bi(S,yo(·'w»

, a(s,w):=A(s,y(·,w»

, bi(s,w):=Bi(S,y(.,w» (k)

f(s,W):=F(s,y(·,w», gi(s,w):=Gi(s,y(.,w», gi

(s,w):=G

(k)

i

(s,y(·,w»

Taking into account Remark 3.2, we can easily see that these equations satisfy the assumptions of Theorem 3.1. Applying Theorem 3.1 we get P(p(uo'u) e c) .... 0

that X

X

....

X

weakly, where

Xl-X) U

fore supp

. llmsup 0..,.0

Proposition 4.2. Let (y(t»tE[O,T]

c > 0 . Consequently X.

is the distribution of . Slnce obviously

uX

1) {po: 6 >O}

we adapt a device from [9].

WEC 1([0,T]; R

and (A on

There-

Hence

d1

)

such that

be a continuous semimartingale in

the conditions (A measures

for every

X -X ) = 1 ,

For proving

let

o .... 0

as

w(O)=O,

and

. f . R d 1 , satls ylng

Then there exists a family of probability 2). (O,F), such that the following assertions

hold: (i) For every respect to (ii) as

o > 0 the measure

P, and

y(t)

Po (sup IY (t) -w (t) t:>T

0 .... 0,

for every

I

Po

is absolutely continuous with

Po -semimartingale.

e) ....

0,

Iw-yl·iYoll(T)

.

.

Po (sup I (Wl_yl) t:>T

.

1"

(t)

I

c) ....

:;;1

denotes the bounded where 1, Po -semimartingale y (t) •

i,j:=1,2 .•• d

variation part of the (iii)

is a

is

po-bounded as

6 .... 0

This proposition is not formulated separately, but actually it is proved in [9]. By this proposition the processes

Mo(t) :=y(t) ,

0

106

1 d No(t):=(y (t), ••• ,y 1 (t),t),

mo(t):=w(t) , and

Ho(t) :=ho(t) :=t

tic basis (n,r'Po,(r t»

satisfy the assumptions (A

(n,r,po,(r t» ,

1 d no(t):=(w (t), ••• ,w 1 (t),t)

on the stochas1)-(A3) • It is not difficult to check that on

equations (4.1) and (4.2), playing the role of equa-

tions (3.1) and (3.2) respectively, satisfy the assumptions (E 1)-(E5) w) Po(p(u,u

of Theorem 3.1. Applying Theorem 3.1 we get that as

0'" 0,

0>0

c > 0,

for every

since

X -X supp IJ. =U

P

e > O. Consequently for every e > 0 there exists P(p(u,uw) 0 for every

Po(p(u,uw) 0. Hence

such that o

«

P • Thus we have

supp IJ.X :::lUX,

and consequently

is proved.

5. Stability of stochastic partial differential equations Let

rand

T li: 0

be real numbers and let

be an. integer. The

m li: 0

theorem below is an application of the results of Section 3 to the stochastic partial differential equations

(5.1) with initial condition (5.2) dv6(t,x) = Dp(ar(t,x)DqVo(t,x) +fo(t,x»dho (t) + (t,X)DpVo(t,x)+goi

. p(k) (k) L, 1 i k ik +{D"Qi (t,X)DpV (t,x) r.(2(t)+r (t» 6(t,x)+goi

(5.3)

with (5.4) given on a stochastic basis where

tE [O,T] ,

x=(x

1,

•••

,X

d

d) E R ,

a Dp:=axp

for every for

p:=1,2 ••• d,

0>0 DO

is

107

the identity,

ik P [b",1.',b"'J'] P (t,x) ::= b 6 (-"- b 6j u u k

(The variable

w6 EQ 6

f

"x

t' ) H unc a.ons , ere

"x

•

is omitted everywhere from the arguments of the

pq A6 '

pq a6

bP 6i'

r

P B6 i ,

BP{k) 6i

bP{k) 6i d [0 ,T] xQ5 x R

05xB (Ed) -measurable bounded functions on . .m

-"- b P ) (t ' x) k 6i

u

. .m+1

are

r

.m+1

and

. .m+2

(r), G E (r), 95i E L 2 w (r), F5 E L 2 w2 (r) , f 5 E L 2 w2 Oi 2 (k) . .m+1 (k) . .m+1 EL 2 w (r), g5i EL W (r) for every 5>0, p,q::=O,1,2 ••• d, G 5i 2 2 2 i::=1,2 ••• d 1,

k::=1,2 ••• d

(G5 i (t) , 0 the derivatives in x ER of the functions

tions

a

up to the order [O,T]xQ5xR

d

m+1

• Moreover

I

are bounded measurable functions on F

I G5 il m+1

:;;K,

61

W 2

(r)

:;;K,

:;;K, I f 5 1. .m+1 :;;K, Ig 5 i l. .m+2 :;;K, :;;K, w {r} w (r) w (r) w (r) 2 2 2 2 where K is a constant which does not depend on 5 {p,q::=O,1,2 ••. d, 1.

£:=1,2 ••• d, (P

2)

i:=1,2 ••• d

As

u

5 ... 0 n

1.'

k:=1,2 ••• d

2).

(If

'

k::=1,2 ••• d 2 } •

we have

CL2'

all multi-indices

1

13B P u D 5i

hl:;;m,

in u ol3 b P 5i 1131

Y:=(Y1"' Yd)

, an

CL2

L{dH

5)L2,

f or every

(p,q::=0,1,2 ••• d, is a multi-index then

u ELand 2 i::=1,2 ••• d 1

DYf denotes

r

108

where

a. n

Goi m

W (r) 2

I y I : =y 1 + ••. +Yd .) (k ) Goi

C·.rn (r) w,2+1

(k)

in

Moreover

F0

f 0 in

in

and

CW;(r) ,

•

(P 3)

2 ••• d)

For every

u EL

(and

2

i:=1,2 ..• d

1,

k:=1,2 .•• d

2,

p,q:=0,1,

the random processes

p(k) ( ) (uD b Oi t) tE[O,T) are tight in for all multi-indices Iy I s m, I y

sI

C( [O,T) iL2) , m+1 .

uniformly in

0> 0 ,

Moreover the random processes (k )

(goi(t»tE[O,T) and (goi (t»tE[O,T) are tight in m •. .rn+1 . .rn C([0,T)i W , C([O,T) ,w (r» and C([O,T) iw2(r» respectively, 2(r» 2 uniformly in 0, for every i:=1,2 ••. d k:=1,2 ••. d • 2 1, (fo(t) )tE[O,T) ,

(P 4) For each

0> 0

for

d

L

e p q

p,q=1

Alel

••• ,e Rd , d)E 1, which does not depend on O. for all

d (t, wo,X)E [O,T)xrloxR

dtxdPoxdx -almost all

e:=(e

2

where

and

A >0

is a constant

m+2 m+3 There exists an L -solution v (r) n L (d h ) W (r) 2 2(r) S) oECW2 o 2 of the problem (S.3)-(S.4), for every 0>0, such that (vo(t»tE[O,T) (P

. .rn+2 CW 2 (r),

is tight in

po-bounded as

uniformly in

0 >0

2 Iv 01 . .rn+3 • h o (T) w (r) 2

and

0 .... 0 •

is

Theorem S.1. Assume (A

from Section 3 and (P above. 1)-(PS) 1)-(A3) Then there exists a unique L 2(r) -solution (uo(t»tE[O,T) of the problem (S.1)-(S.2) and

for every

0> 0,

po[supluo(t)-Vo(t) I t:oT W;(r)

as

0'" 0,

for every

U nL )w 2(dHo 2 oECW2(r)

po[luo-voI2m+1 W 2

£ >0 •

This theorem can be obtained from Theorem 3.1. the assumptions (E

1)-(E S

Corollary S. 2. Let

0

(t,x) E [O,T) xR sup

D uo(t,x) , d

y

D vo(t,x)

sup (1+lxl) x ERd (P 0 a. s .) and

y

.... O

ID uo(t,x) 1 d/ 2 +n • Then,

po-almost every

o E rio

W

are jointly continuous in

for all multi-indices

2 r

'Ho(T)

(In order to verify

be an integer such that

under the assumptions of Theorem S. 1, for the functions

(r)

(r)

) of Theorem 3.1 see [4).)

n

y

• .m- 1

m

such that

lr l s n sup

• Horeover

2 r Dy vo(t,x) I '2+n • Then

in place of

are satisfied. Let

S)

-X

supp lJ. =U

n

holds with

X.

[2]). Suppose that assumptions (SFO)-(SF S ) are satisfied with every integer m • Then ((t))t [O,T] is a 0,00 X -X random element in the spaces cO,oo(r), C , and supp lJ. =U with

cO,oo (r) ,

[1],

cO,oo

°

in place of

X.

By virtue of Sobolev's theorem on embedding

n C

in

we get

Corollary 7.2 as a special case of Theorem 7.1. Theorem 7.1 and 7.3 follow immediately from Theorems 6.1 and 6.3, respectively. Assume now that

r>d/4 . Then

:=(w(t)'''0=(\w(t),\-1)O \=\(x) :=(1+lxI 2)r. Since p(t)=y

-1

(t)(t)

are continuous in y(t) >

°

also belongs to

speak about the distribution of we denote by

lJ.

X

p'

y(t):=((t),"O

for all

m

-1 w

:w

the process Thus we can

on the space

X, which

m

uX

I}

p

in

is a

denote the clouser of

X

Theorem 7.4. Assume (SFO)-(SF S ) . If x -X then supp lJ. =U

P

(r).

m

inf Iy (t) t [O,T] w

H,

[O,T] ,

where

C ([0 ,T] ;W (r)) n L 2 ([0 ,T] ;H (r)) .... X 2 2

where

continuous embedding as above. Let Up:={Yw

t

yw(t):=

t [O,T],

. .m-s l

Cw (r ) n L W 2 2 2 (p(t))t [O,T]

and

r

> d/4

in Assumption (SF S)'

P

As a particular case of Theorem 7.3 we have Corollary 7.5. Suppose that assumptions (SFO)-(SF ) are satisfied with rn>d/2+n and r>d/4, where supp lJ.X=UX with CO,n (r) and cO,n

P

P

S

is an integer. Then in place of X.

Theorem 7.6. Suppose that assumptions (SFO)-(SF ) are satisfied

S

°

with every integer m and with a real number r > d/4 in AssumpX holds with cO,oo(r) and CO,oo in place tion (SF ) . Then supp lJ.X=U of

X.

p

S

p

Proof of Theorem 7.4. With the necessary changes we follow the way of the proof of Theorem 4.1. As in the proof of Theorem 4.1 we may assume that the elements of from [O',T]

in

d

:R 1

H are infinitely differentiable functions

vanishing at

t=O.

Together with the D-M-Z equa-

tion (7.2)-(7.3) let us consider the equation

116

1

d¢6(t,x)=(£(t'Y6(t»¢6(t,x)-2 NkNk(t'Y6(t»¢6(t,x»dt-

*

1

(k )

- 2(BB )ik(t'Y6(t»M i

(t,y 6(tȢ6(t,x)dt+

(7.6) with the same initial condition

(7.7) for every

6 >0 ,

where

Y6 (t)

is the smoot.h approximation for

y (t)

defined by (4.3). We can verify that the problems (7.2)-(7.3) and

(7.6)-(7.7)

satisfy the assumptions of Theorem 5.1. Applying Th 5.1 we

get that lim P ( I ¢ 6 -¢ 6->-0 for every for

I

E: >0,

UEC([0,T];W

-1 A E L 21

(7.8)

G E:) = 0

where

l u l r> sup lu(t,') tE [0 ,T]

m m+1 nL ( [ 0 , T ] ; W (r» 2 2(r» 2

lim P (sup 6->-0 t:;;;T

Iy 6 (t) -y (t) -1

y 6 (t) : = (¢ 6 (t) , 1) 0 = ( A¢ 6 (t) , A

t

GE:) = 0

)0

positive on

[O,T],

P (S6) -+ 1

as

0

•

Consequently, since

for every

>0 ,

I::

y(t)

wE H

6-+0,

where

inf (¢Q(t) ,1)0 >0 , O:;;;t:;;;T

such that

solution of the problem (7.4)-(7.5) with sequence of the fact

P(S6) -+1 ,

fine now a sto:hastic process and

W

A

Y6(t) :=(¢ (t) ,1)0

where

is continuous and strictly S6={wHt, inf

tE[O,T]

A

Let

where

w:=w.

such an element

¢w

Y6(t»0}.

is the

L (r ) 2

(As an obvious conA

w

exists.) We de-

Y6(t) :=Y6(t)=(¢6(t) ,1)0

w ES 6

if

w ltS6 • By virtue of (7.8) and

if

dt (r)

y(t):=(¢(t),1)0'

and

Hence, taking into account that

T 2 +flu(t,') I +1

t

P(S6)-+1

we have lim P( sup 6-+0 tE[O,T] for every E:

I::

ly6 (t ) - y (t

> 0 • Hence

> 0 • Therefore

lim P (sup IyA-1 6 (t) -y -1 (t) 6->-0 t:;;;[O,T]

lim P( Ip6-pl G E:) 6->-0

A-1

P6(t,x) :=Y6 (t)¢6(t,x) the distribution of G

) I GE:) = 0

P6

• Hence on

X

=0

for every X

116 -+ IIp

weakly as

X. Consequently

= 1

E:

I

;; 1::) = 0

>0

for every

by (7.8), where

6 -+ 0,

where

X

116

is

117

since

P(p

X cijX supp IIp _ P

Thus

oEUp)=l

we take an element orem 4.1 we show that 0

for every

£ > 0,

where the probability measures

P

are from proposi-

5

lim Po(sup I (¢(t) ,1)0-(¢w(t) ,1)0 1 z c ) =0 and conset;;:;T -1 w -1 lim P (¢(t) ,1)0 -(¢ (t) ,1)0 =0 follow for every 8(sup I 0"'0 t;;:;T

tion 4.2. Hence

£>0.

X => ijX supp IIp - P

and in the same way as in the proof of The-

0 ...0

quently

is proved. To get

0 ...0

Thus

5...0 w -1 w :=(¢ (t),1)O ¢ (t,x)

forevery . Therefore

since P « P for every o proof is completed.

£>0,

w p(lp-p 10

5 > 0 • Hence

X

s upp IIp

-X

Up

where

pW(t,x):=

for every

E

>0,

is obvious. The

We can prove Theorem 7.5 similarly, by using Corollary 5.3. References [1]

M. Chaleyat-Maurel and D. Michel, Une propriete de robustesse en filtrage non lineaire, Stochastics, 19 (1986), pp. 11-40.

[2]

M. Chaleyat-Maurel and D. Michel, A Stroock-Varadhan support theorem in nonlinear filtering theory, to appear in Stochastics.

[3]

I. Gyongy, On the approximation of stochastic partial differential equations I, to appear in Stochastics.

[4]

I. Gyongy, On the approximation of stochastic partial differential equations II, to appear in Stochastics.

[5]

I. Gyongy, On stochastic equations with respect to semimartingales III, Stochastics 6 (1982), 153-174.

[6]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., AmsterdamOxford-New York (1981).

[7]

N. V. Krylov and B. L. Rozovskii, Ito equations in Banach spaces, Itogi nauki, Teor. verojatn. 14. Moscow (1979), 72-147 (in Russian) •

[8]

N. V. Krylov and B. L. Rozovskii, On conditional distributions of diffusion processes, Math. USSR Izvestiya Vol. 12 (1978), 336-356.

[9]

V. Mackevicius, On the support of the solution of stochastic differential equation, Lietuvos Matematikos Rinkinys, XXVI. Nr. 1 (1986), 91-98.

[10] E. Pardoux, These, Univ. Paris-Sud, Orsay, 1975. [11] E. Pardoux, Stochastic Partial Differential Equations and Filtering of Diffusion processes, Stochastics, Vol. 3, 1979, pp. 127167. [12] B. L. Rozovskii, Stochastic evolution systems. The theory of linear equations with applications to the statistics of stochastic processes. Nauka, Moscow, 1983 (in Russian).

118

[13] D. w. Stroock and S. R. Varadhan, On the support of diffusion processes with applications to the strong maximum principle. Proc. 6th Berkeley Symp. Math. Statist. Prob. Berkeley, Univ. California Press, 1972. V. 3, p. 333-359.

WEAK CONVERGENCE OF SOLUTIONS OF STOCHASTIC EVOLUTION EQUATIONS ON NUCLEAR SPACES V. Perez-Abreu 2 G. Kallianpur 1 Center for Stochastic Processes Centro de lnvestlqaclon en Matem,hicas, A.C. University of North Carolina-Chapel Hill A.P. 402, 3600 Chapel Hill, NC, 27599-3260, U.S.A. Guanajuato, Gto., Mexico

INTRODUCTION In recent years there has been interest in the study of fluctuation limits of infinite particle systems of independent Brownian motions and different types of interacting diffusion systems. Such problems have been studied, among others, by Hitsuda and Mitoma [2], Ito [3], Tanaka and Hitsuda [s] and Mitoma [11]. In all cases the limit process is given by a generalized Langevin equation or a stochastic evolution equation driven by a Gaussian martingale on a nuclear space of distributions O satisfy the condition: For each T>O and q>O there exist M q and 0' q such that for IITn(s,t)tPll q

(3)

5

Mqe

0' q(t-s)

IItPll q for all tPEell and

(1.14)

{A(t)}t>O is a family of continuous linear operators on ell satisfying the assumptions of Theorem 1.1 and such that for each T>O and q>O there exists po-q satisfying sup

IIAn(t)-A(t)II1.(ell

[p]'

ell ) Iql

-+ 0

as n-+oo.

(1.15)

(4)

n M = (Mr)t>O, and M = (Mt)t>O are ell'-valued L 2-martingales vanishing at the origin and such that M n ::}-M in C([O,oo): ell').

(5)

n, n 1 and 1 are ell'-valued random variables such that 1 ::} 1 on ell' and for each 1 n and M n are independent.

For each d'7t

suppose that the stochastic evolution equation

+ dMr

=

=

has the unique solution d'7t

=

A'(t)'7t dt

=

n

(1.16)

(er) and that.

+ dM t

has the unique solution (

t>O, '70=1

(1.17)

t>O, '70=1.

(et ). Then

::} (

in C([O,oo);ell').

123

The proof of this theorem is given in the next section. 2. PROOF OF THE MAIN THEOREM In order to prove Theorem 1.2 we need the following lemmas. We will denote the space C([O,oo);iP') by C t:

iP

Lemma 1..1: For let Gn: C iPr-"C 4j' and G: C 4jr.... C4j' be such that Gn(x)-+ G(x) as n-+oo uniformly over compact sets of C t: Let P n nz:1 and P be probability measures on C "and Q n = P nG 1 nz:1 an!,Q = PG- 1. If Pn=>P in C and iP, G is continuous th*' Qn=>Q in C iP,.

n

Proof: Let C = C([O,oo);R) be the space of continuous function of [0,00) to R with the topology given in Whitt [13]. For ¢EiP denote by II¢ the mapping of C iP' to C defined by (II¢x).

=

x .[¢].

(2.1)

Since P n =>P in C , then => PIIt in C for all ¢EiP and therefore {P nII;£J n>1 is tight tn C since C is a Polish space. Then by (R.2.1) and Theorem 3.1 of Mitoma [6] {P n}n>1 is itself tight in C t: The remainder of the proof goes as we now show: Let 0' be a as in the case of complete separable metric bounded real valued continuous function on C , and let t'>0. Then there exists a compact set A in C iP' such that iP IIAC O'(Gn(a»dPn(a) -

I AC O'(G(a»dPn(a)!

21lO'lloo(P n(A c) where 110'1100

=

sup


0 and a neighborhood V t' of zero in C iP' such that Gn(a) -

G(a) E Vt' for all nZ:Nt', aEA.

Therefore since 0' is a bounded continuous function on C iP' to sup IO'(Gn(a»-O'(G(a))/ aEA

< t'/2 for

R (2.3)

all

Then for nZ:Nf IIA O'(Gn(a»dPn(a) -

I A O'(G(a»dPn(a) I S t'/2,

so that from (2.2), IIO'(Gn(a»dPn(a) -

I O'(G(a»dPn(a)1 S e

for

(2.4)

124

Since P n

=>

P, we have

f a(G(a»dPn(a)

-+

f a(G(a»dP(a)

(2.5)

since G is continuous. The assertion of the lemma follows from (2.4) and (2.5) which together imply

Ja(x)dQn(x)

-+

f a(x)dQ(x).

o

The proof of the above result holds without change if, in its statement, C (b' is replaced by CT,: = C([O,T];') where T0 and any bounded set B in there exists N>O such that {Xs: O$S$T} Taking e such that

¢EB

= 1 and B = {¢}, ¢

sup IXs[¢]1 O$s$T Define

C N{F E ': sup IF]¢]I < £}.

s N1¢
O

00.

(2.9)

125

(2.10) From a Baire category argument (see Lemma 1.2.3 in [10] or Lemma 2.2 in [4]) it follows that V T(¢) is a continuous function in and hence there exist (JT>O and QT>O such that

(2.11) Hence from (2.10) and (2.11) we have that X s E CT : =

sup IlXsll-QT O:s;s:s;T

O:s;s:S;T and

< 00.

(2.12)

Next if a family of linear operators {A(t)}t>O on Theorem 1.1, there exists rT>QT such that -

satisfies the conditions of

(2.13)

where M rT and UrT are stability constants and

(2.14) Then using (2.12) and (2.13) we have that for O:s;t:S;T Y t[¢]:

I

t

Xs[A(s)T(s,t)¢]dS

o

defines a continuous linear map on IYt[¢]I

l.e., Y t E

TUrT TCTKTMrTe 1l¢ llrT

for all

for all ¢ in

Also, if O:s;t:S;T,

(2.15)

It has been shown in Step 2 of Theorem 2.1 in [5] that there exists PT>rT such that Y:- := (Y t: map (2.8) sends CT, into CT,.

Then Y:-

for all T>O. Hence the

Let K be a compact set in CT,. By R.2.1 and Proposition 2.1 of Mitoma [6], there exists QT>O such that K is in Then if X E B,

and

DT(X) := sup IlXsll-QT O:s;s:s;T


-A(s)T(s,t)4>

= An(s)(Tn(s,t)4>-T(s,t)4»+(A n(s) -A(s»T(s,t)4> for all 4> in and O:s;s:s;t:S;T we have (2.17)

IIAn(s)Tn(s,t)4> -A(s)T(s, t)4>lIqT

IIAn(s)(Tn(s, t)4>- T(s,t»4>llqT + II(A n(s) -A(s»T(s,t)4>llqT' Now by (3) in Theorem 1.2 there exists PT>qT such that (2.18) where using (1.15), for

some nO>O (2.19)

Next since An(s) and A(s) generate the (C O,l)-reversed evolution systems T n(s,t) and T'(s.t) respectively, using the corresponding forward and backward equations we obtain T n(s,t)4> -

T(s,t)4>

=

t

Js T n(s,r)(An(r)-A(r»T(r,t)4>dr

(2.20)

s Jst IITn(s,r)(An(r)-A(r»T(r,t)4>lIm dr.

(2.21)

and therefore for each IIT n(s,t)4>-T(s,t)4>lIm

Next, using the equi-(C o,l)-evolution property (1.14) we obtain that for each IIT n(s,t)4>-T(s,t)4>lIm

t o (r-s) MnJ e m IIA n(r)-A(r»T(r,t)4>lIm dr

s

where M m and um are stability constants. Taking m Theorem 1.2 there exists PT>qT such that

= qT

(2.22)

and using again (3) in

Using the (C O,l)-evolution property of T'(r.t) we obtain from (1.5) of Theorem 1 (taking PT large enough)

127

II(An(r) -A(t»T(r, t)¢II QT

$

(2.23)

O'PT(t-r) MpTe

IPTI'

Thus by (2.23) and (2.22) max(MQT,M PT) by (1.15)

we

have that if 0'

IQTI

= max(O'QT'O'PT)

)' and

liTn(s, t)¢- T(s, t)¢IIQT

$

=

(2.24)

2 O'(t-s) TM II¢IIPT e sup

--+ 0

M

IPTI'

IQTI

)

as n--+oo for all ¢ in

Also using (1.15) from (2.23) we have that sup

II(A n(r)-A(r»T(r,t)¢IIQT --+ 0 as n--+oo for all ¢ in

(2.25)

and using (2.24) and (2.19) in (2.18) we obtain sup

IIAn(s)(Tn(s,t)¢-T(s,t»¢IIQT --+ 0 as n--+oo for all ¢ in

(2.26)

Then applying (2.25) and (2.26) in (2.17), from (2.16) it follows that if X E B, (2.27)

(Gn(X)-G(X»t[¢J --+ 0 as n--+oo for all ¢ in

--+ 0

as n--+oo.

Thus there exist PT>QT and F!}- such that for any X E B and (2.29) Moreover since B is a compact set in with norm sup XEB

and the latter is a metric space

sup IIXtll-QT' sup IIXtll-QT =: H

< 00,

i.e., I(Gn(X)-G(X»t[¢JI

$

HF!}-II¢IIPT

for all ¢ in

and X in B.

(2.30)

128

Finally let V be the collection of neighborhoods of zero defining the topology of ip', i.e., if v E V v := v(£,v) := {F E ip': sup IF[tPJ0. Let IIIXlllv

=

sup sup IXt[tPJI, O$t$T tPEv

v in V.

From Mitoma [6J, we have that C([O,T);ip') has the projective limit topology of {IIIXlllv: vEV}. Then if v E V and X E B IIIG n(X)-G(X)lIIv

=

sup sup I(Gn(X)-G(X»t[tPJI O$t$T tPEv

and by (2.30) IIIG n(X)-G(X)lIlv

HFq- sup IItPllpT -+ 0 tPEv

as n-+oo.

Hence Gn(X) converges to G(X) uniformly over compact sets of C The fact that G(X) is a continuous map from The proof of the lemma is complete.

to itself is easily shown.

0

From the proof of the above lemma (see (2.24» we obtain the following corollary. Corollarv 2.1: Let {An(t)}t>O' {Tn(s,t): O$S$tO. Proof Qf Theorem 1.2: M

n,

Let P n, n?:1, and P be the probability measures on C([O,OO);ip') induced by n?:l, and M respectively. By assumption Pn => P in C([O,oo);ip').

From (1.10) and (1.12) we have that for each n?:l the solution (1.16) can be written as

= and using (2.6)

er =

=

of

+l

(2.31)

+ Gn(Mn)t.

(2.32)

°

129

Similarly, (2.33) We will first prove that en,T => e T in CT, for each T>O. Here en,T(eT) is • •

• • the restriction of to [O,T). Define fn(X):

--+

by (2.34)

Then for X E

using Corollary 2.1 we have

Let B C CT, be a compact set. Then using the notation of the proof of Lemma 2.2 for v in V, IlIfn(X)-f(X)lIlv:=

sup O:5 t:5T H

sup O:5t:5T

--+ 0 Finally, for X E hn(X) = fn(X)

sup tPEv

Ixo [Tn(O,t)]tP-T(O,t)tPJI

sup I!Tn(o,t)tP-T(O,t)tPl!qT

tP Ev

as n--+oo for all X in B.

define

+ Gn(X)

and heX) = f(X)

+ G(X).

Then hn(X) --+ heX) uniformly over the compacts of C

pJ

implies n,T by M

=> PT where

and MT

(2.35)

pJ and

The assumption Pn => P

PT are the probability measures induced on

respectively where

= Mr (O:5t:5T) (similarly, M"[= M t,

n O:5t:5T). Since 'Y => 'Y and 'Y n and M n are independent, it follows from (2.31), (2.32)

and Lemmas 2.1 and 2.2 that Qn,T => Q T in CT,. For each tPE let II

. tP be the mapping introduced in Lemma 2.1, suitably modified. Then the relation

qn=>

c}

Yj implies that J.I.*j=>J.l.

tP j

where J.I.*j (and similarly

J.l.tPj)

is defined by J.I.*j=

Thus, for each j, {c}nII;p1} is tight for every tPE. Now, from the fact that C has the projective limit topology of {d}j>l it can be shown that the sequence of measures

J.I.*= QnIIt

is tight for each

By Theorem 3.1 of Mitom.a [6]

follows that {Qn} is tight. On the other hand, the weak convergence of

qn

to

q

130

for every j clearly implies finite dimensional convergence under Qn of (X Xtk[¢kJ) to its law under Q. Proposition 5.1 of [6] then implies Qn=>Q·

t1

[¢1].· ...

0

Remark: For (1.16) and (1.17) to have unique solutions. it is sufficient. according to Theorem 2.1 of [5] that for some rn>O. ro-o we have

2

EI'YI-r
0 is the standard scalar product on L2(8). Let denote the Laplacian on Ho' closed with respect to homogeneous Neumann boundary conditions, D > 0 a diffusion constant and R(x) m

Cixi a polynomial in x fIR such that Co reaction-diffusion equation:

(1.1)

0 and cm < 0 if m

+

x, (r)

2. Let us consider the following

R(X(t))

O.

It was shown in Kotelenez [7] that there is a unique global mild solution X of (1.1) if Xo (r) is

bounded and that X is "smooth" if Xo is "smooth". Moreover, X is bounded if Xo is and m 2. Now we introduce the stochastic (i.e., mesoscopic) model following Dittrich [2]. Let e > 0, EN : =

{.E

e bri' r i e

1=1

sJ'

where e bri are Dirac measures with weight e, and set

00

E: (Eo = {abstract point}). separable metric space.

EN

If d p is the Prohorov metric on E, then (E,d p ) is a locally compact

Denoting by C(EN) the continuous functions f from E into IR such that f( a) N

EM

we set

Co(E):

=

= 0 if r4 EN: =

00"

C(E N) and let C(E) be the continuous real valued functions

133

vanishing at infinity with norm

III fill

: = sup If (a) I. Clearly, Co(E) is densely imbedded into o2'fE

•

N "

C(E). Let f Nbe the canonical mapping from S onto EN and C2(E) be those f from C(E) such that fOfNis twice continuously differentiable and satisfies homogeneous Neumann boundary conditions on SN where SN is the Nth Cartesian product of S with itself.

Denote the corresponding closed

Laplacian (on C(SN, IR)) by t. N and the operator induced by f N

Now we define an

unbounded operator A on (;2(E) by

(1.2)

t. Nf) (a), if a:« EN.

(Af) (a) : =

Further, let G(t, x, y) be the fundamental solution (or Green's function) of D t. (t) (cf. Kotelenez [7]) and 0 < /'1 < /'2 < 00 some constants. For k to be symmetric and continuous functions such that J Qf (xl, ... , x k) dx>, ... , dxs =

Sk - 1

(1.3)

k

... ,xk)

(

=

2 we define

: Sk -l1R+

1 for allxlES, .

y). S

(x- , ... ,x k) for all (xi , ... ,x k) E Sk. N

For

E f0

02'=

i=1

i

X

we set

a: , ... ,. : = 'I

a-+ (sgn ck) f·

'k

E

0

jf{i , ... ,;X J 1 k

'

where it, ... , ik are (pairwise different) numbers from {I, ... ,N}. Now we define an operator on Co(E) by

(Rf) (a) (1.4)

(

m· N f k - l

:= E

ke 1

K

Ic k I

J[f(02'+foy)-f(a}]dy S

Finally we define on (;2(E) n Co(E) an operator A f by

(1.5)

f

•

A : = Dt.

+

•

R.

Theorem 1

A e generates a Markov process

x' on D([O ,00);E)

(the space of E-valued cadlag functions).

For both the terminology and basic facts on Markov processes we refer to Ethier and Kurtz [5]

a f

134

(i) Let fNEC(E N) (continuous functions from EN into IR) and IN E C(E) such that IN (aJ = fN( aJ, if a-E EN' and = 0 otherwise. If f E C(E) we write f = E IN' where IN (aJ = f (aJ if a-E EN' Denote by SN (t) the Markov semigroup (a positive contraction semigroup which generates a (cf. Dynkin [4]) and by SN(t) the operator family on

Markov process) on C(EN) generated by Co(E) defined by

S (t)f N

where O( aJ

==

M

= {SN( t ) fN if M = N 0 ifM f. N ,

0 for all .zoE E. We set for fE Co(E) S(t) f = E SN(t) IN N

and easily verify first that S(t) defines a semigroup of positive contractions on Co(E) and then since Co(E) is dense in C(E) by extension a Markov semigroupon C(E). (ii) Set

: = DLi + RN, where RN: = 1 - . Rand 1 - (aJ = 1 if a-E EN and = 0 otherwise. RNis a bounded operator on C(E) and EN EN

satisfies the positive maximum principle, whence

generates a positive contraction semigroup

;(t) on C(E) ([5]). Moreover, we have that the pair of functions (1,0) (with l(aJ

==

1) is in the

Therefore, ;(t) is a Markov semigroup.

bounded- point closure (bp - closure) of

(iii) We want to verify Th. 2.2 in Ch. 4 of [5] with respect to A f. Dom (A f) = 0)' Identify Illo with its strong dual Illi> and denote by Ill_a the strong dual of III 0" Then we have

(2.1) IHO' c 1H 0 = 1Hi> c IH_O' with dense continuous inclusions. Assume in what follows Cj

= 0 for all i except i = 1 and i = 2 and consider -Jt Z((t ) = (D.6. + Cl)Z{(t) + c2Z((t)B{(Z((t))

(2.2)

{

where B((rp)(r) : =

0

Z((O)

f

Qj (r,

rp (q)

q)dq.

The existence of a unique mild positive bounded solution

S

(if Z((O) is bounded) can be shown by the same techniques as the existence of a unique X for (1.1)

(cf. Kotelenez [7)). It turns out that Z( is "closer" to X( than X since the range of interaction is

N

e

for X{ which we also have in (2.2) in difference from (1.1). Let us make the following assumptions:

(H) (i)

X(O) is bounded, k times continuously differentiable with k

homogeneous Neumann boundary conditions:

+ ... + 1n

odd and q. E {O, I}, where 1j 1

(ii)

III X (0) -

Z(( 0)

III

$ const

alI

{)In

8qll

{)qIn

---r ... -----,

1

k.

X(O, q)

a and satisfies the repeated 0 if there is an i such that 1; is

n

2

(il .

(iii) X((O) is independent of M((t) for all small e > 0, where M( is the canonical martingale part of X( (from Dynkin's formula). (iv)

X((O) is Poisson distributed with intensity Z((O, y)dy.

Moreover, let M be an IH_O' -valued Gaussian martingale with covariance (23) E( < M(t), rp>2)

t

= Lds{< X(s), [2D

n

{)

-2 C2 < X2 (s), rp2 >o}, where

is the dual pairing extending

rp)2]

+ Clrp2 >0

p

< . , . >0 and rp di a .

In Kotelenez [9] the following theorem was proved (denoting by

I. I-a the Hilbert norm on IH-0')'

136

Theorem 2

(I) Assume (H) in addition to - xoL a -l 0 stochastically. Then !X((t) stochastically uniformly on any bounded time interval.

X(t)L a

-l

0

f'

1

(II)

Assume (H) in addition to (X((O) - X(O)) ::} Y(O) (converges weakly - where Y(O) is an lit_a - valued square integrable random variable) and assume n 3. Then Y(

(on the Skorohod space of

=}

Y on D ([0, tJ, "_a)

"_a-valued cadlag functions defined on [O,t]), where t satisfies tec1tsupe

III Z( (0) 111 < 1 and Y is a generalized Ornstein-Uhlenbeck process satisfying on lH_ the linear a stochastic partial differential equation:

(2.4)

dY(t)

(Dll

+ c1 + 2c2 X(t)) Y(t)dt + dM(t),

with X(t) acting as a multiplication operator. The proof depends on (i) the verification of propagation of chaos, (ii) generalization of estimates obtained by Dittrich [3J, (iii) and functional analytic techniques developed in Kotelenez [6], [8J for a different model of chemical reactions with diffusion.

o Remark It should be possible to obtain the same limit theorem for the general model defined in Section 1 if we can verify propagation of chaos for it. This will be investigated in a forthcoming paper.

Another problem is to compare the results presented in this paper to results obtained by Kotelenez [8J for a stochastic model defined on a grid as introduced by Arnold and Theodosopulu [I].

References [1] Arnold, 1. and Theodosopulu, M. (1980). Deterministic limit of the stochastic model of chemical reactions with diffusion. Adv. AIml. Prob. 12,367-379. [2] Dittrich, P. (1986). A stochastic model of a chemical reaction with diffusion. Preprint Akademie der Wissenschaften der DDR, Berlin [3J Dittrich, P. (1987). A stochastic particle system: Fluctuations around a nonlinear reaction-diffusion equation. To appear in Stochastic Processes Appl. [4J Dynkin, E.B. (1965). Markov Processes, 1, Springer Berlin.

137

[5] [6] [7] [8] [9]

Ethier, N.E. and Kurtz, T.G. (1986). Markov processes: Characterization and convergence. Wiley, NewYork, Kotelenez, P. (1986). Law of large numbers and central limit theorem for linear chemicalreactions with diffusion. Ann. Probab. 14, 173-193. Kotelenez, P. (1986). Report 146, Forschungsschwerpunkt Dynamische Systeme, Universitat Bremen. Kotelenez, P. (1988). High density limit theorems for nonlinear chemical reactions with diffusion. Probab. Th. ReI. Fields 78,11-37. Kotelenez, P. (1988) Fluctuations in a Nonlinear Reaction-Diffusion Model (preprint - submitted).

Stochastic Partial Differential Equations of Generalized Brownian Functionals

Hui-Hsiung Kuo* Department of Mathematics Louisiana State University Baton Rouge. LA

70803

USA

11.

Generalized Brownian functionals

Let

be the Schwartz space of rapidly decreasing functions on lR.

*

be the standard Gaussian measure on the dual space

of

Let

i. e. its

characteristic function is given by 2

exp[-nE;U /2].

r, B

where

2

is the

L (IR)-norm.

E;

J.

By Wiener-Ito theorem. the space

L2( u)

of

ordinary Brownian functionals has the followipg orthogonal decomposition

where

K consists of multiple Wiener integrals of order n the Brownian motion B(t.x) = ' x J*. For

n in

with respect to 2 L ( lJ) .

its

S-transform is defined by E; It is easy to see that if

In(f).

the multiple Wiener integral of

f

in

t2(lR n). then

=

f nP

Thus the S-transform of

f(uI,···.u) E;(uI).··E;(U ) duI···du • n

makes sense even when

n

n

f

is not in

t 2 ( lR n).

This

is the motivation for Hida [2] to define the generalized multiple Wiener integrals

I

n

(f)

for

f

in the Sobolev space

Research supported by NSF grant DMS-850I775

n).

For

n > 1, 'let

139

(-n) n

{I (f);

K

n

nI (f)ll (

K -n

n

Obviously,

(-n) K n

(L

2)-

n)}

)

n

is the dual space of

III

The space

f

n

(f) n

( ) Kn

= In!

n f n (

) /2

n

(IR n).

of generalized Brownian functionals and the space

(L

2)+

of

test Brownian functionals are defined by

L$

K(-n)

n=O

where

n

and

is the nonstandard real number system

(L 2)+

K(-n) 's n

L2( u )

are

(L 2) - .

c c regarded as vector spaces over IR. * Obviously, we have 2 2 The S-transform on L ( u ) can be extended to (L ) - . This extension is called

the U-functional map. often

expressed

{B(t); 0 ..

t .. I}

in

The analysis of generalized Brownian functionals is very terms

is

differentiation operator Laplacian [3,4,9,11]. derivative then

a tljl

tional

8 If

of

the

taken

as

at'

a

Gross Laplacian

F'(I;;t),

U-functional I;

F

8 of

V ljl

0 c t .. 1 such that

8

G, can in

system,

example, the

when

coordinate

Beltrami Laplacian be

8 B, defined as follows (L 2)has functional

F'(.;t)

is aU-functional,

is defined to be the generalized Brownian functional with U-func-

F'(.;t),

i.e.

For an ordinary Brownian functional respect to

coordinate

and Volterra Laplacian

L the

For

U-functional map.

t, the Gross Laplacian 8

Gljl

ljl

such that

at2ljl

[1] is defined by

is integrable with

140

The Beltrami Laplacian

6

is defined by

B

On the other hand, suppose

2 (L ) -

$

and its U-functional

has second

F

functional derivative given by F" (; j t , s)

where

a is the

If

[3,4,12]

a-function,

F " ( · jt ) dt 1

to

be

F1" ( ; j t ) dt,

=

is

the

a

F 1"(; ; t ) 0( t -s) + F 2" (; ; t ,s) ,

F

L

1"(;;.)

1(O,l)

U-functional,

generalized

we define

Brownian

U-functional,

2

L ( 0 , 1) 2) .

the Ltlvy Laplacian with

fl $ L U-functional

i.e.

fl can be expressed in terms of L point of view as follows:

F " (; jt , s ) 2

F " (; j . , . ) 2

functional

Actually,

If

and

at

is a trace class operator of we define

from the nonstandard analysis

2

L ( 0 , 1) 2)

the Volterra Laplacian

Brownian functional with U-functional

tr F

2"(;),

6 V$ i.e.

and

t

r F

2"(.)

is a

to be the generalized

In this paper, we consider the following stochastic partial differential equations associated with

(1)

(2)

Note functional for

a

6

G

and

6

V:

1 • 2' flGu(t,B) +

[

du(t,B)/dt

[

d$(t,B)/dt = 2' flV$(t,B)

=

S(t,:B)

lim U(t,B) = f(B) NO 1

•

lim $(t,in = a - 0

1 A (D>f(x+Ak, t ) -Dr(x) , h)

for some bounded self-adjoint operator

e H, t

x

for more details) that

,

lim

for all

is both a viscosity sub and

[1J

X , then one can show (see

eH, t A ->- 0+

v

A(x,t)

on

(A(x,t)h,k)

H, and

by the Lipschitz constant of

A(x,t)

is bounded

on any ball containing

x x t ; t Furthermore, n n the above limits are uniform on bounded sets of H' . Then, we denote by (x,t) ,and

A(xn,tn)

pointwise if

A(x,t)

2 D 'f (x , t ) = A(x, t )

2)

Observe that

2 a a Dt(x,t)(T x,T x)

view of remark 1) above and the fact that

is continuous uniformly in TO.

depends continuously on

a

in a

which

varies in a compact set. This explains the apparent difference between the definition above and Definition 3)

11.1 in [1J.

We do not know if it is possible to replace in the above definition is uniformly continuous on balls, DY' is continuous from

H into

H2}

154

4)

[34J, the inequalities (11)-(12) are

As in

one has formally

(ACtXo' B'(lxol) 1::1)

Finally, we recall that for any is bounded by nonnegative constant

o

A,

Ct

o

of

i

{l, ••. ,m}

the statement

in viscosity sense" (for some

r

Co ) means that for all X o

I)

- Ao ---,---:;:.x

C (l+lxJ)

maximum point (resp. maximum point)

natural since

somewhat;

B' (I

2

C (H;JR)

6

and for any local

u-)'> we have

Co (l+lxl)

resp.

- Co (l+lxl) • We may now state our results. Theorem 1.

1)

The value function

in

x

for all

t

0

and Lipschitz in

t

u

belongs to

C(H x

0 , positively homogeneous of degree x

for all

t

1

in

x

for all

0 with a Lipschitz

remains bounded on all finite time intervals weakly (sequentially) continuous on

is concave

[O,T]

Hx

which

• In addition,

u

is

0

and satisfies for all

>

0 ,

T


for some constant 2)

for all

for all

olx-yl + Co Ilx-yll_2

x,y t

Co.

e

Ct If, in addition, g H2 and (f)Ct A is bounded in H2 T < 00 , there exists a nonnegative constant Co such that

(13)

x

H,

then,

u(·,t) ,

w

H,

[O,T]

t

e

[O,T]

or equivalently for all

(13' )

e e

x,y t

and

C It-s I (1 + Ix1 )

(14)

lu(x,t)-u(x,s)1

s

(15)

2 Ct ex D w(x)(Tix,Tix)

is bounded by

for all Theorem 2. 2)

1) Let

ex

A,

i e {t , ••• ,m},

C0 (l+lxl) w

H , o

s,t

T

in viscosity sense,

= u(',t)

,

t 6 [O,T]

The value function is a viscosity solution of (10). v e C(H x [0,00»

be a viscosity subsolution (resp. supersolution)

of (10) such that ( 16)

= u(·,t)

[O,T] x

for all

0

H , w

Iv(x,t)1

C(l+lxl)

forall

x6H,t6[0,T]

155

and either

v on

(17)

is weakly (sequentially) uppersemicontinuous (resp. lower semiH x [0,00) , or

v

R < 00

satisfies for all

sup flv(x,t)-v(y,t) I I t

e [O,RJ,

Ixl

R , Iyl

s R, IIx-ylL2

.. °

Then we have ( 18)

u(x,t)-v(x,t)

as

s ....

sup u(x,O)-v(x,O) H

resp.

u(x,t)-v(x,t)

inf u(x,O)-v(x,O)

for all

x

H

Remark.

°

Since one has clearly

u(x,O)

(g,x)

eH, t

°.

H, 2) and 3) may be easily

on

combined to yield uniqueness results for viscosity solutions of (10) satisfying the same initial condition provided they satisfy (17) or that they are weakly (sequentially) continuous on

H x [0,00) •

III • Proofs of Theorems 1 and 2. Since most of the arguments needed for our proof of Theorems 1 and 2 are adaptations of the arguments introduced in details and we will only explain

[1J,

we will not give all

necessary modifications.

Proof of Theorem 1. First of all, several claims in part 1) are almost obvious since the cost function

J

and the state equation are linear in

homogeneous of degree

and concave in

the bound (7) immediately yields that concave in

x

for all

convex functions that

t u

u

x

x, hence

for all

t

u

is clearly

°. Furthermore,

satisfies (16). But since

u

is

°, we deduce from standard facts of the theory of x for all t ° with a bounded

is Lipschitz in

Lipschitz constant on bounded intervals We now prove the continuity in

t

[O,T] of

u

and in fact we begin by proving

(14) assuming thus that g H2 (as we will see (14) does not require that (fa) is bounded in H ). Indeed, since we have for all 0 s t T a e A 2 lu(x,t)-u(x,s)1

sup

a

IJ(x,t,a)-J(x,s,a)1

sup 1ft E(fa"X )dl1 + sup as' a Att AtS C (e - e ) + sup a

IE(g,Xt)-(g,XS)I

156

we just have to bound

d cit E(g.X t) • But. observe that in view of (6). we have

d dt E(g.X t)

hence

d Id t E(g,X t)I

for some constant

at

(A

E :;;

Ce

g,x

Alt

t) Ixl

° independent of

C

the control

a

and of

x • And (14)

follows easi ly. In order to prove that by

gn

HZ

such that

gn

e

u g

C(H

x

, we just approximate g e H un the corresponding value

[0,00»

and we denote by

functions. In view of what we just proved, it is clearly enough to show that u

converges uniformly on bounded sets of

n

easy since we deduce from the formulas defining lu n (x , t)

-

I

u(x, t.)

u

n

and

u

sup IE(gn'Xt) - E(g.Xt) I a Alt e Ixl Ign -gl

and we conclude. We now show that

u

u . But this is

to

H x [O,to)

is weakly sequentially continuous on

equivalently in view of what we already proved that if

H x [0,00)

(0.00)

t

or

is fixed

and

xn x weakly in H then u(xn,t) u(x,t) • And this will be the case if we show that J(xn,t,a) ri J(x,t,a) uniformly in a = (as)s • Since J

is linear in

x

we may always assume that

x

=

°, and we denote by

the

solution of (6) corresponding to

x Recalling that x are elements of H n n that is of LZQRN) , we begin with the slightly simpler situation when Supp x c: K where K is a given compact set of lRN and we will treat the n

general case afterwards. In that particular case,

°

in

H-ZQRN)

x converges strongly to n by the Rellich-Kondrakov theorem. Then. we use the following

auxiliary result whose proof we postpone. Lemma 3. only on

For each

T < 00 , there exists a nonnegative constant

T such that for all

x

e

H and for all controls for all

(19)

t

a

e

=

C depending (a ) s s

[O,T]

In particular, we deduce from this lemma that (ZO)

sup a s

E

n

°

uniformly on

[O,T]

• for all

T < 00 •

157

We next observe that since may find for all

E > 0

A is compact and

a

(fE)a

eA

fa

depends continuously on a, 2 2 H , gE e H such that

bounded in

(21)

Then, we remark that sup a s

IE

r

a (f s ,Xn)ds + (g,Xn) I s t

0

IE

sup a s

r 0

C sup E a S for some constants

C independent of

n

and

e: , and

This bound proves our claim letting first go to

n

0

Ce:

go to

a (f

s ,Xn )ds +(g ,X) n Est

sup s

t

E

cs

independent of +00

I + CE

and then

n .

e:

0

We now show the general case which is deduced from this case by a truncadon technique : let jz I any

1/2 x

,

e H

1;(z) :: 1

1;

e

Coo(IRN)

if

jzl ..!. n

,

and we conclude easily letting first

n

go to

+00

165

In the case when v

satisfies (17), we have to make some perturbation

argument in order to obtain a maximum point. We first choose un (x,t) - z(x,t) - v(x,t)

(40)

;;;

C R

=

such that

00

un (0,0) - z(O,O) - v(O,O) - 1

for all Then, we observe that on the set

R


° some

enough, the maximum, denoted by

C;

= {(x,t)

££

HZ

such that

achieves its maximum over (xo,t

CR'

II ££ liz

£

and

Because of (40), for

c

small

, must belong to

o)

H x [O,TJ / Ixl < R} . And we may now proceed as in the preceding

case noting of course that

(££,x)

adds in the equations a term which is

bounded by

a

sup

I (x,l\;) I

A

c 11££lIzlxl

and we conclude as before sending

c

to

0,

;; c£lxl n

to

+00

first and then

to 0.

0

The conclude the proof of Theorem Z, there just remains to prove part Z). Exactly as in

[lJ,

we just have to show that for any fixed

A

a

for all

x e

H , 0:;;

to :;; t

X

is the solution of (6) corresponding to the initial condition x t and the constant control at a. lJe first observe that by the preceding where

computation there exists

K>

° such that for all

0 > 0,

v0

=v-

2 oeKt Ix 1

is a viscosity subsolution of dW +

ot and, by letting

a

sup

A

0 go to

{ - -Z 1

m a a L D2w(T.x,T.x) + (x,ACL vw-f a} )

i=l

1

:;;

0

1

0, it is clearly enough to show for all

(41)

o and we fix

(4Z)

to

in

(0,00) • We then introduce for

y

sup

H{

U

0

0

to

e

H , t

£ > 0

1 (A-1 (x-y),x-y) } - --z £

x

for all

x

eH

166

x e H, t

for all

(43)

0

£ 2) on v In view of this inequality, H - C(1+lxI v6(x) 6(x,to) we deduce easily that the supremum in (42) may be restricted to y e H such

so that that

v£ 6

so that

Iyl :£ C(I+lxl)

satisfies in particular

(44)

where

C denotes various nonnegative constants independent of

v£ 6(x) since the supremum defining We next claim that

+

v

£.

as E + 0+ pointwise in H . Indeed, 6(x) may be restricted to a bounded set in H,

we see that it may further be restricted to a set of

y

-1

for which

(A (x-y),x-y) + 0 or in other words y x in H_ 1 and thus y x weakly o E in H. If v is weakly sequentially uppersemicontinuous, we conclude easily since in this case we have for such lim v

c

If

v

satisfies (17), then lim c

v

6(y)

v

:£

6(y)

6(x)

v(y) :£

y's

vex)

and we also conclude that

v (x ) 6

Having thus proven the pointwise convergence of

to

v

is proven if we show for all

(45)

x

we see that (41)

6,

eH,

At this point, we may now copy the proof given in

0 .

t

[IJ with analogous

modifications as the ones introduced in the proof of part 3) above cular, we observe that all the approximations used in the bounds (44), including the and P.L. Lions

inf-sup

a £,n W6 a t -"21

L

i=l

2 £,n

D w6

in parti-

preserve uniformly

convolutions introduced in J.M. Lasry

[46J. And we obtain, for all

n

a

a

1.

1.

eX z

1 , a function

satisfying in viscosity sense and pointwise for all

(46)

[1J

T


and

° and

x

e

H, t

e

[O,TJ ,

T. Then, exactly as in the n

large enough

(n

sup

ye for all

x

e H,

t

H

e [O,TJ ,hence :£

sup.£ (1+lyI2) _

y e Hn for all

and we conclude letting

n

go to

+00

go to

° and

x

e H , t e [O,TJ

thus recovering the

desired inequality (45).

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[2J

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[3J

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[4J

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[5J

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O. Hijab. Partially observed control of Markov processes. 1,11,111. Preprint. N. El Karoui, D. Huu Nguyen and M. Jeanblanc-Pique. Existence of an optimal markovian control for the control under partial observations. Preprint. W.H. Fleming and M. Nisio. On stochastic relaxed controls for partially observed diffusions. Nagoya Math. J., 2l (1984), p. 71-108.

[soJ

T.G. Kurtz and D.L. Ocone. A martingale problem for conditional distributions and uniqueness for the nonlinear filtering equations. Ann. Proba., (1988), p. 80-107.

A GENERALIZED EQUATION FOR A CONTINUOUS MEASURE BRANCHING PROCESS

svi vie MELEARD Facul

Sylvie ROELLY-COPPOLETTA.

du Kaine des Sciences

Laboratolre de Probablll

Ont ver-s t t.e Paris 6, Tour 56

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o INTRODUCTION Continuous measure branching processes (denoted for simplicity by M.B. processes) arise as limits of branching diffusion processes, i.e. processes which model a system of particles which diffuse and branch on a locally compact space E with some independence properties. They are Feller processes, whose semi group TIt satisfies the following fundamental additivity property with respect to the initial measure: TIt(m 1 , · ) • TIt (m2, .) = This property is also another way to characterize these processes (71. In this paper we give an approach of such measure processes as solutions of stochastic equations in a measure space. More precisely, we use techniques of stochastic calculus and the martingale properties explicited for continuous M.B. processes by S. Roelly-Coppoletta (14], and generalized for a bigger class of M.B. processes, including non continuous M.B. processes by N EI Karoui and S. Roelly-Coppoletta (7]. In particular, when the M.B. process X t is continuous, it is known that (Xt,f) is a semimartingale with increasing process

Jo (Xs,Cf t

2

)

ds , (c is a function) , for all functions f which are

Xt-integrable.

It always was a preoccupation to find a stochastic equation of which the M.B. process X is solution. N. Konno and T. Shiga (11] have studied the t particular case where the random measure has a density and They obtain a stochastic equation satisfied by the density. In dimension higher that one, X t is almost surely singular, and the covariance process associated to the semimartingale (Xt,f) is also degenerated, since the associated kernel is a Dirac measure. It implies that we can not find a classical equation satisfied by X in an Hilbert space with a White Noise term.(cf D. Dawson (1]). t

172

We define here a stochastic equation in a measure space thanks to the interpretation of the martingale term of the semimartingale (Xt,f) as the value in f of a continuous orthogonal martingale measure Mt(.) with intensity Xt(dx)dt, in the sense of J.B. Walsh [15], N. EI Karoui and S. Meleard [6] Ccf also E. Dynkin [4]). So X is solution of a stochastic equation in the space t 2 of vector measures with values in L CO) :

The first section of this paper is devoted to the investigation of the above problem when the M.B. process takes his values in the finite non negative measure space on a locally compact metric space E, denoted by HfCE). We recall the existence of such a process, its martingale properties, and we construct the associated martingale measure by an extension theorem . Then we prove an equivalence between the strong formulation of the equation satisfied by the M.B. process and a mild associated version (cf [15]). D. Dawson and L. Gorostiza proved a similar equivalence between a strong and a mild equation satisfied by the fluctuation limit of a branching particle system (of [3]). The second section contains the same results adapted to a M.B. process with values in the space H.p of cr-fini te measures suitably weighted by a fixed "good" function .p on IRd , By using the stability of a class of cumulant semigroups, we deduce the existence of a M.B. process with values in H.p from the existence of a M.B. process with values in Hf(1R d ) (cf [7]). We take here, as initial measure X the Lebesgue measure, for which all the computations o' are particularly simple. The method to obtain the stochastic equation is the same as in the first case. But the difficulty for the construction of the martingale measure is different from before: it specially consists to verify that the terms we have to integrate with respect to X are really X tt integrable.

I FINITE CONTINUOUS MEASURE BRANCHING PROCESS

1-1 NOTATIONS AND HYPOTHESES

- Let 21 denote the Borelian cr-field on IRd , and 21 sets of 21" d

b

the subspace of bounded

- Let m be a fixed element of the space HfC!R I, space of the non negative d, finite measures on !R m will always denote the deterministic initial d) value of the process X. HfC!R is endowed with the vague topology.

173

d d) - Let C (R denote the space of continuous functions on R vanishing at o d d, oo infinity, Cb(R ) denote the space of bounded continuous functions on R L d denote the space of bounded Borelian functions on R . On these spaces, we

consider the norm: IIfll",

='

If(x) I .

SUPd

xeR

For simplicity, since there is no ambiguity, we will denote all these spaces 00

by Hf , Co' Cb' L . d 2 - For Il belonging to Hf(R I, L (Il) will denote, as usually, the space of Borelian functions which are square integrable with respect to Il. The norm on L2(1l) will be denoted by 11.11

2,Il

.

- Let b and c be two functions of C with c non negative. b' - Let A denote the infinitesimal generator of a Feller semigroup Pt' Then is a family of contractive operators on Co' and the domain of A, Voo(A) , defined by : V (A)

'"

is dense in C . o A represents the spatial diffusion of the M.B. process. We suppose that Voo(A) is an algebra. b

The linear operator A+b is also associated to a semigroup denoted by P

t,

t

defined as usually by:

='

Ex(f(Yt)expJob(Ys)dS)

where Yt denotes the Markov diffusion process on Rd with generator A ; is then continuous and its norm is uniformly bounded on [O,Tl by the positive constant exp(Tllbll",). Let U be the non linear semi group solution of the following partial t differential equation : (0)

AUt + bUt Uf o

f , feV (A) 00

It is also the mild solution of the following integral equation t b b (1) Utf =' Ptf Pt-s (2C Usf) 2 ds, fec b. o The existence and uniqueness of a solution of (0) or (1) is proved in

J

Watanabe I16l, or Pazy [13l. Furthermore, the semigroup U is positive, and t this can be deduced from its interpretation as a cumulant semigroup (cf El Karoui [5]). - In all the paper, K will denote a positive constant, which can change from place to place .

174

1-2 EXISTENCE AND MARTINGALE PROPERTIES OF THE M.B. PROCESS

Let us recall the definition and the fundamental properties of a continuous M.B. process: THEOREM 1-1 : There exists a tmique probability measure Pm on t.he canonical space

of cadlag ftmct.ions from

canonical process X sat.isfies:

Xo=m

wit.h values in H

f,

Pm-a.s.

such t.hat t.he

and one of the following equivalent propert.ies : (i) for feC b,

and T>O,

is a Pm-mart.ingale t

(ii) for fe Vm(A), Mt(f) = (Xt,f) -(m,f) -J

o

(Xs,(A+b)f) ds

is a continuous Pm-mart.ingale with increasing process

t.

2

Jo (Xs,cf ) ds

Proof: It derives from the results of Roelly-Coppoletta [14] : (i) gives the form of the Laplace functional of X, which determines it completely; then the existence is obtained by approximation and convergence in the space C([O,T];H The uniqueness comes also from the property (i), which gives the f). uniqueness of the finite dimensional distributions of X. The equivalence between (i) and (ii) is proved in the theorem 1.3 of [7]. The increasing process associated to the martingale Mt(f) can be written as follows:

JJ t

o

f2(x) c(x) X (dx) ds = s

J

t 0

J

f2(x) I(dx,ds),

where we would like to interpret the measure I defined on

as the

intensity of a martingale measure. So, in what follows, we extend Mt(f) m, defined by (ii) only on Vm(A) to L and even to

where

is the

finite measure defined by : T

I(dx,ds) . o

It will prove that Mt(f) is really a martingale measure taken on f.

1-3 CONSTRUCTION OF THE MARTINGALE MEASURE

Let us recall quickly the definition and the fundamental properties of an orthogonal continuous martingale measure M (cf J.B. Walsh [15]).

175

where A is a o-fie1d on a

DEFINITION 1-2 : ", defined on

1usinian space E, is called an orthogonal continuous martingale measure if it satisfies: (1)

V A E A, V t>O,

( il)

V

A E A,

Uii) V A E

Uv) (v)


O, "t(') is a vector o-finite measure with values in L (O) V A,B E A, AnB=0, { } and { } are orthogonal.

Then there exists a random predictable real measure I on B([O,Tl)xA, called intensity of M, and defined by:

t

=

fof I(ds,dx) t

P- a.s., V t>O.

A

Moreover,one can construct a stochastic integral with respect to M, for all functions f defined on Ox[O,TlxE,

measurable,

is the predictable

o--field), and such that:

ff T

E [

f2(w, s , x) I(dw, ds , dx) ]

E This stochastic integral is denoted by o

PROPOSITION 1-3 :


A{2 is a continuous linear mapping.

2, Il f - > ".(r)

00

Then, we can extend it by continuity to L2(1l) , and, for fEL2(1l), the increasing process associated to "t(f) is also equal to

fo (X t

s

2 , Cf )

Furthermore, the extension of "t( ) on L2(1l) defines an orthogonal continuous finite martingale measure. Proof: The choice of the space L2(1l) is natural since thus, the mapping

(7)

00

(A), II

II

2, Il

)

-> L2(0)

is an isometry.

--> The continuity of the mapping is then clear: f

11M.

=

2

Mt (f)

l

:s 4

= 4 EU)lRdf2(Xl I(dS,dX»).

=4

2

(ll,f ) ,

by definition of Il.

ds

176

fo (m,p:(Cf» ds T

But

s T II mil eo exp(Tllbll eo) IIcll eo IIfll eo Then the measure V. is finite, with mass bounded by Tllmll eoexp(Tllbll eo)lIcll eo, which 2 implies that D (A), and also Leo, contained in L (v.) . Q) 2 2 By density of DQ)(A) in L (v.), we can define M.(f) for f in L (v.). Let us now verify that this extension

the

of the

of the martingale Mt(f).

Let feL2{v.) ; we have to

that, for OoSsoSt, - M:(f)

-f:

(Xu'Cf

each g bounded and

2)dU)g]=

0 ;

exists a sequence (fn)n of Deo(A) which conver-ges to f f'or- the nor-m II 11 2; satisfies, for each

f

E[t-:(fn '

n

-

":(fn '

-J:

0

Then

We decompose the right hand side in

:

oS II gileo E[M:(f-f n

1)

») + 211gllQ) E(IMt(fn)IIMt(f-fn>l) 1

oS II gileo [( v., (r-r n) 2) + 2E

/

2(M:

rr-r n» )

(fn»

oS 211gll_ (lIf-f 11 2 + IIf-f II IIf II ) n 2, V. n 2,11 n 2, V. which vanishes when n tends to infinity; ii)

also tends to zero by the the same

iii)

E[I

oS IIgll eo oS II gileo

dU] (v.,

If2

I)

oS IIgll_ rr-r II (lIfll + IIf II ) n 2, V. 2, V. n 2, V. which vanishes when n tends to infinity. Let us show the orthogonality

of M ; we have proved above that t the increasing process associated to Mt(f) is, for all feL 2 (v.) , t

1 et k E Z on represente par Dp,k le complete de l'ensemble des variables aleatoires polynomiales, par rapport it la norme

ou L est Ie generateur du processus d'Ornstein-Uhlenbeck it valeurs dans n. La limite projective de ces espaces est notee par Doc et son dual est represente par D- oc. On designera par D I'operateur de derivation, et par 81'0perateur de divergence dans l'espace de Wiener. Soit S = Cb(n) n Doc la classe des fonctionnelles F : n - t R continues et bornees, et qui appartiennent it I'espace D oo • On dira qu'une distribution T E D- oo est une mesure s'il existe une mesure de Radon fl sur 0, telle que T et fl coincident sur S. Nous allons montrer d'abord que dans la classe des probabilites de D- oc, la convergence dans D- oo entraine la convergence faible de mesures,

PROPOSITION 1: Soit Tn rule suite de distributions de D- oc qui sont des probabilites sur n. Supposons que Tn converge vers T dans D- oo • Alors Test une probabilite et Tn converge faiblement vers T.

187

Demonstration: Il suffit de montrer que la suite de probabilites Tn sur nest faiblement relativement compacte. En effet, comme Tn converge vers T sur S, qui est une partie dense dans Cb(n), toutes les limites faibles doivent coincider avec T. On considere la variable aleatoire definie par

Iw(t)

It

w(t')18 dtdt'. t'I 3

D'apres [1), G appartient a D= et pour tout a > 0, l'ensemble A a = {w En: G(w) :::; a} est compact pour la norme de la convergence uniforme de n, et de plus P(UaENAa) = 1. Soit 'P une fonction de C=(R) telle que 0 :::; 'P :::; 1, 'P(x) = 1 si x:2: 0 et 'P(x) = 0 si x:::; -1. Pour tout nombre reel a on pose 'Pa(x) = 'P(x - a). Pour tout n, on represente par /in la mesure sur n associee a la distribution Tn· Alors, on a

(1)

Nous allons montrer que

(2)

Pour tout

E

> 0 on ecrit

1111

Gt() W=o

0

Iw(t) - w(t')18 dtdt'. ( E+lt-t'1)3

Notons que 'Pa(G t) E S mais 'Pa(G t) rt S. Pour tout w on a limt!o 'Pa(Gt(w» = 'Pa(G(w», et en consequence, limdO In 'Pa(Gt)d/in = In 'Pa(G)d/i. Comme 'Pa(G t) E S, on a In 'Pa(Gt)d/in =< 'Pa( Gt), Tn > . Finalement, pour achever la demonstration de (2) il suffit de voir que limt.l.o < 'Pa(G t), Tn >=< 'Pa(G), Tn > et cela decoule du fait que 'Pa( Gt) converge vers 'Pa( G) dans la topologie de D=, quand E 1 o.

n existe un reel p > 1 et un entier k qu'on peut prendre negatif et pair, k = -2v avec v EN, tels que Tn converge vers T dans la norme de Dp,k' Par consequent, E['Pa(G)Tn]

= E[(I + L)"'Pa(G)

s s

(I + l)-IITn] II(I+L)"'Pa(G)lIq 1I(I+L)-IITn ll p Const. II (I +L)"'Pa(G)lIp,

(3)

188

ou q est le conjugue de p, Comme limatoo 11(1 + L)"!.pa(G)lIq = 0, les inegalites (1) et (3) entrainent que la suite P-n est faiblement relativement compacte, ce qu'il fallait prouver. 0 On dira qu'une distribution T de 0_ 00 est nonnegative si < T, F F E 0 00 tel que F 0 p.s.

°

pour tout

PROPOSITION 2: Soit T un element nonnegatif de 0_ 00 , Alors Test une mesure de Radon. Demonstration: On considere une base orthonormale {hi, i I} de I' espace de Hilbert 1, soit {)n la sous-e-algebre de H = L2(0, 1) telle que hi E n* pour tout i. Pour tout n :F engendree par Sh1 , ' •• , Shn. L'esperance conditionnelle E[TI{)n] est une distribution de 0_ 00 , determinee par < E[TI{)n],F >=< T,E[FI{)n] > pour tout F dans 0 00 (cf. [4]). Si F appartient Ii. 0 00 , I'esperance conditionnelle E[FI{)n] a une version de la forme !.pF,n(Sh 1 , · " , Shn) OU !.pF,n E coo(Rn). On sait (cf. [4]) que E[TI{)n] coincide avec la composition, au sense de Watanabe, d'une distribution temperee Tn E S'(Rn) avec le vecteur aleatoire Sh = (6hl,'" ,6h n), et on a < Tn,!.p >=< T,!.p(6h) > pour toute !.p dans S(Rn). Le fait que T soit nonnegative entraine que les Tn sont nonnegatives pour tout n. Par consequent, Tn est une mesure de Radon sur R", Cela implique que E[TI{)n] est associee Ii. une mesure de Radon P-n sur n. En effet, pour tout F dans Cb(n) n 0_ 00 on a

< E[TI{)n], F > =< T, E[FI{)n] >

=< T,!.pF,n(6h) >=< Tn,!.pF,n >=

in

Fdp-n,

n

OU P-n = Tn 0 1r n et 1r n est la projection de sur R" define par 1rn(w) = (Sh)(w). Les mesures P-n ont toutes la meme masse totale egale Ii. < T, 1 >. Comme la suite E[T\{)n] converge en 0_ 00 vers T, la Proposition 1 entraine que Test une mesure de Radon sur n. o Note: Nous avons appris par [1] qu'un resultat semblable Ii. celui de la Proposition 2 avait ete trouve par H. Sugita.

UNE APPLICATION AUX MESURES CYLINDRIQUES On considere une mesure de probabilite cylindrique {) sur n. On fixe une base orthonormale {hi, i I} C n* de H, et pour tout n 1 on designe par {)n l'image de {) par la projection 1r n ; n -I< R", 1rn (w) = (6h)(w) (cf. [2]).

189

D'apres les resultats precedents, si {Dn } est une suite bornee dans D oo , alors {Dn ; n E N} est faiblement relativement compact dans la topologie de Prokhorov, done D est de Radon sur n. On ne connait pas des conditions suffisantes simples et pratiques qui entrainent I'appartenance de {D n } a D- oo . D'apres Meyer[3], on sait qu'une variable Fn = 1 are probably new. Once an identity is "suspected", its proof or disproof is not difficult; the reason for presenting a collection of such identities is because they are very useful in many applications of the Malliavin calculus. NOTATION Let (T, be a finite atomless separable measure space. Consider a zero mean Gaussian process {W(B),B E '13 } with covariance function given by E(W(B I)W(Bi) = I nBi), defined in some complete probability space (n, :F,P). We assume that the a-field :F is generated by W and we denote by H the Hilbert space L 2(T, '13 ,IJ.). For p > I, a E R and X a real separable Hilbert space, we represent by IDp,ll(X) the Sobolev space of X -valued random variables which is the completion of the set of X -valued polynomial random variables with respect to the norm IIF IIp,ll = II (I + L )

aJ2F

II L'(Q;X)

,

where -L is the generator of the Ornstein-Uhlenbeck semigroup {PI' t 0 }. Namely, L is the number operator, it multiplies by n the elements of the n -th Wiener chaos and PI multiplies these elements by «», D denotes the differentiation operator and 0 is its adjoint. Recall that D is a continuous linear operator from IDp,ll into IDp,ll_I(H), and 0 is a continuous linear operator from IDp,ll(H) intoIDp,ll_I' SetID_= nIDp,llandID_= UIDp,ll' P,ll

p,ll

Before stating the identities, we remark that (a)

a will denote an arbitrary real number, and k will denote an arbitrary positive integer.

(b)

The domain of the identities is any Sobolev space IDp,ll(H 0n) (including ID_(H 0n» where H 0n denotes the n -th tensor power of H (which is isometric to L 2(r , '13 n , IJ.n» and n has to be chosen so that the identity will have a meaning, for example, if we deal with &i L Ok , then the domain is IDp,ll(H

(c)

Whenever L II is the first operator and a < 0, then it is assumed that the identity holds for zero mean elements of the domain.

THE IDENTITIES (la)

(I +L )llD = DL II

(lb)

(k/+L)llD k =DkL ll

(2a)

O(J+L )ll =L 110 For a = I and in the case where 0 coincides with an Ito integral this identity can be found, for instance, in Stroock's notes [6], or in [8].

193

instance, in Strooek's notes [6], or in [8].

(3b)

c," (kl +L )a =Lac," D c, commutes with c,D = L, and more generally with La. D" c," commutes with L a, and (D c,)" also commutes with La.

(4a)

If c,u

(4b)

If c," u

(Sa)

c,D =L

(5b)

c,"D" =P"(L), where P"(L) =L(L-I)(L-2I) ... (L-kl). (see Proposition 2.5 of [5])

(5c)

c,"(kl+L)aD" =P"(L)La=Lap"(L).

(00)

Set u 1 = DL -1c,u. Then

(2b) (3a)

=0, then ()L au =O. Conversely, if c,Lau = 0 for some a e R and Eu = 0, then c,u = O.

=0, then c," L au =O. Conversely, if c," L au = 0 for some a e Rand Eu

=0, then c," u =O.

(i)

u E ID:z,a(H) implies u 1 e Ifh,a(H).

(ii)

If u belongs to ID20 = L 2(0 X T), then Uo = u -

(iii)

D c,u (I+L)u 1. The process u 1 represents the gradient component of u . The explicit representation of u 1 as DL -1c,u was given by Ustunel in [7].

c,uo= 0, and u = u 1+ uo is the orthogonal decomposition given in Theorem 4.4 of [4]. U 1 verifies

=

Set u 1=D"P"(L)-Ic,"u. Then

(6b) (i)

u

(ii)

If u belongs to ID20{H 0") =L 2(0 x T"), then uo = u - U 1 verifies c," uo = 0, and u = u 1 + uo is the orthogonal decomposition given in Proposition 2.7 of [5]. D"c,"u =P"(kl +L)u 1.

(iii)

E 1f'L

(7a)

D ()u = ()Du

(7b)

D

(7c)

D"Oi u =

c, u ""

+u

(H 0") implies u 1 E ID2,a(H 0").

"[k.

=k!u + L

i=1

k' . .

J 7-c,'D'u l!

I

i!c,i-i D"-i u

Notes: (1) In formulas (7b) and (7c) u is supposed to be a symmetric function of its variables. (2) For all formulas 7, in the expressions c,"ID"'u the operator c, operates on the variables of u (and not on those introduced by the differentiations D"'). Consequently, in this interpretation c,D is not equivalent to L. (8a)

DP, =e-'P,D

(8b)

D" P,

=e-IaP,D".

These relations have been used by Meyer in [3]. Remark: We also have the relation

194

which is related directly to the general theory of semigroups. PROOFS:

(lb) Assume that F =I" (f,,), where I" denotes the multiple Ito-Wiener integral, and t; is a symmetric function of the space L 2(T"). Then I"_,,(f,, (s , . »

«kI+L)aD"F)s = (kI +L)a

nan! = (n-k)! I,,_,,(f,,(s,'» =na(D"F)s

where seT" and we assume n

= (D"LaF)s

,

k. If n < k both expressions are equal to zero.

Suppose that U e L 2(Q x T") is a measurable process of the form Us = I,,(f,,(s, . » where f" (s , r) is a function belonging to L 2(T"+") which is symmetric on t e T" for all s e T": Consequently (2b)

'O"(kI+L)au

='O"(k+n)au =(k+n)aI"+,,lf,,) =(kI +L )0.'0" u ,

where i" denotes the symmetrization of i" in all its variables. (3b)

Using (lb) and (2b) we have D"'O"L a = D"(L-kI)ari

and (D 'O)1c-1 L a(D '0) =L a(D '0)"

LaD"'O",

(D '0)" L a = (D 'O)Ic-1D (L+l)a'O

by a recursive argument. (4b)

=

Suppose that u e ID_(H 0") has the (possibly formal) representation Us = IJ,,(f,,(s,'

where I, e L 2(T"+") is a symmetric function of the last n variables. Then 'O"Lau

=

= LnaI"+,,lf,,)

,,=0

»,

,

,,=1

and the result follows easily. (5c)

is an immediate consequence of (2b) and (5b).

(6b)

(i) Let u e u/

=

o,,) be given by the representation Us = LI" (f" (r , .

=(D"P"(L)-I'O"u)s = i

,,=0

,,=0

n.

(n+krl(n+k-l)-I ... (n+l)-II"If,,(s ,

= LI,,(f:(s,'» , ,,=0

». Then .» (I)

195

and, therefore, II u III 2,a S II u II 2,a.

ou -

(ii) From (I) we get ouo = ou 1= O. (iii) Applying a slight extension of (lb) to a polynomial pk(kl+L) we get

pk(kl+L)u 1 = pk(kl+L)D k pk(L r10ku =D k pk(L )pk(L)-IOk U =DkokU . (7b) Let v e ID..(H 0k) be a symmetric process depending on k parameters. Applying the isometry property (cf. Proposition 2.3 of [5]) of the operator Ok, we obtain

E( )=E(otuotv) =k! < u , v >L2CD.xT·) +

i=l

J I. < o' a.o!» >L'(nxT"')

[k] k'

"

I:; =k!L2(nxT')+L . -.-'E«O'D'u,v » , i=l I l!

which implies the result. Assume first j = 1. Then, applying recursively the identity (7a) we get

(7c)

Dl:;ou =oDku +kDl:;-lu , for all k

1.

+[ J[ m kAm

k J(kAm)!(k-kAm)Om-kAmDk-I-kAm u kAm

=kA(m+l)[m+l] i iJ i !om+l-i o':' U which proves the result by induction.

,

196

(8b) Recall that for G =I,. (g,.), P,G =e- IIJ G. Suppose that f : (O,oo)-.+lR is a continuous function with polynomial growth. Denote by f (L) the operator which multiplies by f (n) every element of the n-th Wiener chaos. Note that the identities (lb) and (2b) can be generalized as follows (Ie) Dkf(L)=f(kI+L)D k , (2e) f(L)fi =f(kI+L)fJk . In particular, for f (x) =e-tx we have f (L) =PI and this proves (8b).

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[2]

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[3]

P.A. Meyer. Transformations de Riesz pour les lois Gaussiennes. Lecture Notes in Math. 1059, 179-193,1984.

[4]

D. Nualart and M. Zakai, Generalized Stochastic Integrals and the Malliavin Calculus. Probab. Th. ReI. Fields, 73, 255-280, 1986.

[5]

D. Nualart and M. Zakai, Generalized Multiple Stochastic Integrals and the Representation of Wiener Functionals. Stochastics, 23, 311-330,1988.

[6]

D. Stroock. Some applications of stochastic calculus to partial differential equations. Lecture Notes in Math. 976, 267-382, 1983.

[7]

A.S. Ustunel. Some comments on the filtering of diffusions and the Malliavin calculus. Proc. Silivri Conf. on Stochastic Analysis. Lecture Notes in Math. 1316,247-266, 1986.

[8]

D. Williams. "To begin at the beginning .....", Proc., LMS, Durham Symp; 1980. Lecture Notes in Math. 851, I-55, 1981. David Nualart Facultat de Matematiques Universitat de Barcelona Gran Via 585,08007 - BARCELONA SPAIN

Moshe Zakai Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000 ISRAEL

A LIE ALGEBRAIC CRITERION FOR NON-EXISTENCE OF FINITE DIMENSIONALLY COMPUTABLE FILTERS Daniel Ocone Mathematics Department, Rutgers University New Brunswick, NJ 08903 USA Etienne Pardoux Mathematiques, case H, Universite de Provence F13331 Marseille cedex 3, France

1. Introduction In this paper we shall prove a result about linear, stochastic partial differential equations and apply it to the question of exact, finite-dimensional recursive computation of optimal filters. Let {Yet), 0::5 t::5 T} be an lRP-valued Brownian motion. Throughout, we assume that Y is the canonical process on (0, F, P), where 0 == {I E G([O, T]j lRP ) , 1(0) == OJ, F is the u-algebra of Borel sets of 0 w. r, t. sup norm topology, and P is Wiener/measure. On lRd define the operator

A

82

L d

',j=l

and assume that a(:v) the stochastic p. d. e.

:VI :v1

.8

+ Lbo(:v)a-: + c(:v), d

i=l

:VI

== [ai,j(:v)h9,j$d is symmetric. We shall consider the solution p(x,t) to dp(:v, t)

= Ap(:v,t) dt +

L h.(:v)p(:v, t) dyi( t) P

(Ll)

i=l

phO) = ozo(')'

(1.2)

Sometimes we shall write p(.,tIY) to emphasize the dependence of p on Y. Suppose that a set of linearly independent functions {(Pt"",n} C L 2 (lRd ) is given, and form the random vector «l,p(·,tIY))"",(n,p(-,tIY))) on (O,F,P). Here (,1f;) = J(:v)1f;(x)dx. Our main result, Theorem 1.1, applies the stochastic calculus of variations, or Malliavin calculus, to (1.1) in the case that A is uniformly elliptic and all the coefficients are analytic functions. It states Lie algebraic conditions under which the probability distribution of admits a density with respect to Lebesgue measure on lRn for any n. This result is a refinement of a similar theorem proved in Ocone[18], in which the coefficients are assumed to be only infinitely differentiable, but the initial condition p(" 0) is assumed to be smooth. Introducing the analyticity condition not only allows non-smooth initial conditions, but also leads to a simpler Lie algebraic criterion that is easier to check. To state Theorem 1.1 we introduce the following notation. Let A denote the Lie algebra of operators generated by A A - 1/2 I:f hH:v) and (multiplication by) h i ( · ) , 1 ::5 i ::5 p, using the Lie bracket [B, G] = GoB BoG. The elements of A are all partial differential operators with variable coefficients. For:vo E lRd , let A( xo) denote the linear space of operators consisting of the operators of A with their coefficients frozen at :Vo; thus, for example, :v E A and :vo =/= 0 imply E A(xo). Also, given I E G=(lRd), {(Bf)(:vo), B E A} = ((Gf)(:vo), G E A(xo)}. Finally, let Gb'(lRd ) denote the space of real valued, bounded functions on lRd which are analytic at each point of lRd and whose derivatives of all orders are bounded. Also, let Hk(lR d ) be the Sobolev space of (integral) order k with norm = I:!"I$k 118"/lli.·

=

:z

:z

198

Theorem 1.1 . Assume that a( x) > eI

for some

t;

> 0 and all x E JRd ,

ai,i·), bi(·), c(.), h k(-) E Cb' for 1 S; i,i S; d, 1 S; k S; p,

aXl

alai

"'1

•••

ax

"'4

E A(x{l)

for every multi-index a.

(1.3)

(104) (1.5)

d

Then for any t > 0, for any n, for any linearly independent set {¢l, ... ,¢n} C L 2 (JR d ) , the probability distribution of admits a density with respect to Lebesgue measur;. Section 2 of this paper gives the proof of Theorem 1.1. Remark. Because of (1.3) and (104), equation (1.1)-(1.2) has a unique, adapted solution satisfying E

iT

dt < 00

for k

< -d/2

and T> O.

Moreover, p(·,t) E C([O,T];Hk(JR d)) a. s. for all k, and p(·,t) E COO(JRd) for all t > 0 a. s. These facts are proved in Pardoux[21]' see especially pp.227-228. For this reason, (¢, p(" t) is well-defined for any ¢ E Uk. ",¢n} C Hk(JRd) for any k S; O. We can apply Theorem 1.1 to the nonlinear filtering problem

b(X(t)) dt + u(X(t)) dW(t), X(O) h(X(t)) dt + dB(t), YeO) = 0

dX(t) dY(t)

(1.6) (1.7)

where Wand B are, respectively, JRI and JRP-valued Brownian motions, X(t) evolves in JRd, and Yet) evolves in JRP. Let a(x) uuT(x). In compliance with (1.3) and (104), we shall assume a(x»eI,

and

ai,j, bi, hk E Cb'.

(1.8)

Let p(., tlY) denote the solution to P

dp(x,t)

= Aop(x,t)dt + I>i(X)P(x,t)dyi(t)

(1.9)

I

(ai,j(x )u(x)) - L: k(bi(x )u(x)). p(', tlY) is an unnormalized conwhere Aou(x) = 1/2 L: ditional density of X(t) given the sigma algebra:Fr = u{Y(s), s S; t}. In (1.6)-(1.7) Y is not a Brownian motion, but the measure induced by Y on n is absolutely continuous with respect to Wiener measure. Hence the conclusion of Theorem 1.1 will not be affected when we apply it to (1.9). Recently there has been interest in determining when conditional statistics, such as (¢, p(" t)}, do or do not admit finite dimensional, recursive realizations, and Theorem 1.1 has implications for this question. We shall say that the collection of statistics {(¢i, p(" t)}, 1 S; i < oo} admits a finite dimensional, regular sufficient statistic a, if a : n -+ M is a measurable map into a finite dimensional, CI-manifold, such that, for each i, there is a Oi E CI(M;JR) with (¢i,P(-, t)} Let

7r(.,tIY)

Oi(a)

a.s.

p(" tlY)

= Jp(x,tIY)dx

199

F[.

denote the normalized conditional density of X( t) given

{(l/>i,1I"(·,t)), 1 S i
O}

admits a finite dimensional, regular, recursive, sufficient statistic if for each i there is a 0i E

Cl(M;JR) with (I/>i, 11"(', t)) = Oi(a(t)) where p

da(t)

(LlO)

f(a(t))dt+ Egi(a(t))dyi(t) 1

for some Cl-vector fields

f

and gi, 1 SiS p on M.

Corollary 1.2. Assume (1.8) and let A be the Lie algebra generated by A o - 2: and hI, ... ,h p • If A satisfies (1.5), there is no countably infinite, linearly independent set {I/>i, 1 S i < oo} C L 2(JRd) such that either {(I/>i,p(·,t)), 1 S i < oo} admits a finite dimensional, regular sufficient statistic for any t > 0, or {(I/>i, 11"("t)), 1 S i < 00, t > O} admits a finite dimensional, regular, recursive sufficient statistic.

Proof. The conclusion concerning {(I/>i, 11"("t)), 1 S i < {(I/>i,p(·,t)), 1 i < oo} by the identity

00,

t > O} follows from that about

where hies) == J h(x)1I"(x,t)dx. To prove the result about {(I/>i,p(·,t)), 1 S i < oo}, let us assume that a finite dimensional, regular statistic exists and derive a contradiction to Theorem 1.1. Existence of such an a implies that for any n, = ((I/>1>p(.,t)), ... ,(I/>n>p(·,t))) = (Ol(a), ... ,On(a)). However, because the Oi are differentiable, if n > dimM, the Lebesgue outer measure in JRn of ((Ol(m), ... ,O(m)) 1m E M} is zero. Thus can not admit a probability density in contradiction to Theorem 1.1. A simple example in which (1.3)-(1.5) hold is A

1)11"/2, n E Z}.

1 82

2" 8x 2 ' h(x) == cos z , and

Xo

¢; {n1l", (2n +

Recently, both Lie algebraic techniques and the Malliavin calculus have been increasingly used in nonlinear filtering theory, and we wish to compare these applications with Corollary 1.2. Brockett and Clark(4] and Mitter(16] introduced Lie algebraic and geometric methods into filtering with the insight that, in formal analogy to realization theory in differential geometric control, existence of finite dimensional, recursive filters should impose restrictions on the structure of A. This inspired a lot of work into the classification of the algebras A associated to filtering problems and into using algebraic properties to seek finite dimensionally computable optimal filters. A nice survey of this effort and related topics may be found in Marcus[13]. Also, Chaleyat-Maurel and Michel[6] and, independently, Hijab[10] rigorously developed the original suggestions of Brockett, Clark, and Mitter. For example, in [6] Chaleyat-Maurel and Michel introduce the following notion of universal finite dimensional computability, which we describe only roughly and in modified O} (or {11"(" t), t O}) is universally FDC with respect to a class of infinitely form. {p(" t), t differentiable test functions S if there is a system (1.10) with COO-vector fields such that for every I/> E S there is a E Coo(M) with O(a(t)) = (I/>,p(.,t)) (O(a(t)) = (1/>,1I"(·,t))). By comparing the Ito derivatives of O(a(t)) and (I/>,p(.,t)) at t 0, one can derive a relationship between A and the Lie algebra of vector fields on M generated by f, 91, ... , 9p, as long as S is large enough. For an appropriate choice of S, say all infinitely differentiable I/> so that (I/>, p(., t)) and all its Ito derivatives make sense, it is shown in [6] that dimA(xo) S dimM. By way of contrast, Corollary 1.2 draws inferences about finite dimensional computability from existence of probability densities for for any n. This is apriori a much stronger property and requires the stronger condition

°

200

(1.5) on A(zo). However, we are able to weaken the differentiability requirements on IjJ and in the definition of finite dimensional computability. Other applications of Malliavin calculus to filtering may be found in the work of Michel[14], Bismut and Michel[2], Ferreyra[8], and Kusuoka and Stroockjf l]. These authors study the existence and smoothness ofp(z, t) as a function of z. That is, they determine when the unnormalized conditional distribution, as a random measure on IRa, admits a smooth density p(e , t). In this paper, we are using Malliavin's calculus to study the measure induced on a function space by the solution of Zakai's equation. On the other hand, our work is related to that of Chaleyat-Maureljs], who studies continuity of nonlinear filters using Malliavin calculus. She gives conditions under which conditional statistics are in the Sobolev spaces on Wiener space, but does not analyze the Malliavin covariance matrix as we do here.

2. Proof of Theorem 1.1. Our proof relies heavily on the analysis of [18], which we shall use without repeating proofs. For simplicity of calculation, we assume throughout that p = l. We first need to define the gradient operator D on Wiener functionals. Let FE L 2(0, P). It admits an Ito-Wiener expansion F=

LIk 0y1:,

1:=0

where each 11: E £2([0, T]1:), which is the subspace of symmetric functions in L 2([0, T]1:), and where 11: 0 y1: is the multiple Wiener integral

I 1 T

11:0y1:

Let

JI)l,2

T

•••

!A:(tJ, ... ,t1:)dY(t1 ) ... dY(t1:).

denote the set of FE L 2(0,P) satisfying

L k(k!)II/1:111, < 00

(2.1)

00.

1

If F E JI)l,2, we may define

= L k/1:( ... ,8) 0 00

D.F(Y)

y1:--l,

0::; 8::;

T,

(2.2)

1

where 11:( . .. ,8) is the element of £2([0, T]1:-1) obtained by fixing the last variable at 8. Because of (2.1), the series on the right hand side converges in L 2(0 X [O,T],P x m), where m denotes Lebesgue measure on [0, T], and thus D.F(Y) is well defined up to sets of P x rn-measure zero. In fact, E[

Next, given F

= (Fh

l

... ,Fn )

T

o

E

L k(k!)1I/1:1I1.· 00

(D.F)2 d8] =

1

(JI)l,2)n,

we define the Malliavin covariance matrix of F: (2.3)

Let Po F- 1 denote the probability distribution of F; for a Borel set A C IR n , Po F-1(A) = P(F E A). The Malliavin covariance matrix is used to study the regularity properties of P 0 F-l. For example, Bouleau and Hirsch[3] prove the following result.

201

Proposition 2.1. Suppose that F E (ID 1,2)n and

VTFVF > Then P

0

°

(2.4)

c.s,

F-l is absolutely continuous with respect to Lebesgue measure on IRn.

Proposition 2.1 presents the simplest application of the Malliavin covariance matrix. The theory of the Malliavin calculus shows that moment bounds on the inverse of V T FV F imply smoothness properties of the density d(P 0 F- 1 )/dz. For an introduction to the complete theory and its applications, see Ocone[20] or Michel and Pardoux[15]. to prove Theorem 1.1. Notice that is We shall use Proposition 2.1 with F = adapted to u{Y(s), s t}. Therefore, we may replace T by tin (2.3) in discussing and so for the rest of the argument we assume T = t. Before continuing, we note that it suffices to consider operators A in (1.1) of the form that appear in Zakai's equation (1.9) modulo a potential term:

where O'(z) : IRd --+ IRdxd. This is possible because for any a(z) [ai,j(z)] satisfying (1.3) and (1.4) there is a u(z) E Ci:' satisfying a(z) = O'O'T(z). For example, following Friedman[9], pp. 128-129, we may take O'(z) = (1/271") Jr .JZ(a(z) - zI)-l dz, where r is a simple closed curve in z > containing all the eigenvalues of a(z) for all z E IRd • Thus, by suitably choosing band c we can transform any operator of the form in (1.1) to the form in (2.5). The advantage of (2.5) is that A + c is the forward generator of the diffusion associated to

°

dX(t)

= b(X(t)) dt + u(X(t)) dW(t),

(2.6)

and we can then represent the solution to (1.1) with the Kallianpur-Striebel formula from nonlinear filtering. Suppose that W, and hence X are defined on a second probability space (U ', P, Q). Extend W, X, and the canonical process Y on (U, P) to the product probability space (Ux U' ,.rx P,P X Q) by W(w,w')(t) = W(w')(t), Y(w,w')(t) = Y(w)(t), etc. Let Eq denote expectation with respect to Q on U', let X"o(t) be the solution of (2.6) with X"o(O) = Zo and set

L(Y,t)

= ex p

[l '

h(X"o(s»dY(s) -1/21' h(X"o(s»2 ds]

= exp[h(X"o(t))Y(t)

-1'

Y(s)h'(X"o(s))dX"o(s)

- 1/21' [h(X"o(s))2 + Y(s )tr(a(X"o(s ))h"(X"o{S)))] ds Let p(" tlY) solve (1.1)-(1.2) with A given by (2.5). Then the Kallianpur-Striebel formula, modified by the potential c gives

(¢, p(" tIY»)

= E q [¢(X"o(t))e- J: c(x. o('» d, L(Y, t)]

for P a. e. Y.

(2.7)

(2.7) is proved in Pardoux[21] with c = 0, and the method extends easily to non-zero c. The right hand side of (2.7) is well defined for every Y E U, and we always use this particular version of

(¢,P(., tIY»).

We are now in a position to calculate the gradient of (¢,p(', tIY») using some nonlinear filtering theory.

202 Lemma 2.2. Under the assumptions (1.4), if d, the duals of SOl := completion of S in the norm 14> 101="; < (-LH Ir 12t4>, 4> >H)' I don't know a general result on the extensibility of semigroups to the S_OI-SPaces but e.g. for the Laplacian this is a straightforward calculatiorr'. Since S(rn.d)* '-+ V(ffid)* (continuous dense imbedding), w. is a continuous V'-valued process and we take [ := V(ffid x 1R.t-) (the space of COO-functions 4> : rn.d x IR.t- -+ rn. with compact support). How can one associate an ['-valued random variable to a continuous E'-valued process?"

2 Let

f : IR.t- -+ V(ffid)* be continuous and 4> E E, Define F(4)):=

(where 4>(" t) : rn.d -+ rn.,r

1-+

faoo < f(t),4>(·,t) > dt

4>(r, t); < ',' > denotes the dual pairing of V(ffid)' and V(ffid)).

Lemma 1 4> 1-+ F( 4» is a continuous linear form on [. Proof: (0) First of all, the integrand is well defined: 4>(', t) is Coo for each t and since supp 4> is compact, we can find compact sets K C rn.d, t c: IR.t- s.t, supp 4> c K x I. Hence 4>(', t) E V(ffid) 2c.f. [Kotelenez,1985), p. 128. 3The same problem with & = S(IRd)@'D([O,Tj) was trea.ted in [Bojdecki/Gorostiza.,1986].

227

and Jet) e V(JRd)* by definition (so the integrand is a real valued function on identically outside I).

(1) Claim:

1I4, vanishing

1I4 3 t 1-+ ¢(', t) e V(JRd) is continuous.

Let U be an open neighborhood of zero in V(JRd). Since the topology of V(JRd) is the strict {'IjJ : IRd -+ IR is Coo, supp'IjJ C K n } , (K n ) any inductive limit of the topologies of V(Kn ) d sequence of compact sets with IR = Un K n , un V(K n ) is a zero-nhd. in V(Kn ) (each n). It therefore contains a set of the form

=

v=

{'IjJ: supp'IjJ C K and max sup I Dk'IjJ 1< s] Ikl:SmrEK

Each of the functions (r, t) 1-+ Dk¢(r, t) is uniformly continuous: VkVe > 0 36(k) > 0 s.t. 6(k), 1r - q 6(k)} < e, so It - 5 6(k) implies sup{1 Dk¢(r, t) - Dk¢(q, 5) I : It - 5 sUPrEK 1 Dk¢(r, t) - V¢(r,5) 1< e. Let 6 = minlkl:Sm 6(1 k I). Then 1 t - 5 6 implies ¢(.,t) - ¢(·,5) eVe u.

(2) Claim: IR+ 3 t 1-+


- < j(5), ¢(" 5) >1

5

el, 15

-

t 1< 6 :

1< Jet), ¢(', t) - ¢(" 5) >1 j(5),¢(., 5) >1·

+ 1< Jet) -

Since j is weakly continuous, the second term is small for 6 small enough (¢(', 5) is a fixed element of V(JRd». Since V(JRd) is a barrelled space, the Banach-Steinhaus theorem? applies: the family

{J(t) : t

e I} c V(JRd)*

is equicontinuous (it is pointwise bounded, since j is continuous and I is compact: I SUPIEI < Jet), 'IjJ >1< 00,V'IjJ). This implies: the first term will be small, if all ¢(" t) - ¢(-, 5) are contained in a small nhd. of zero in V(JRd), which follows from step (1) for 6 small enough. (3) By step (2), F( ¢) is well defined (an integral ofa continuous function over a compact interval). It is clearly linear in ¢. (4) We finally prove the continuity of F. F is continuous iffT I V(Kn ) is continuous (each n), where (K n ) is a sequence of compact sets with IRd x 1I4 = lfn":1 K n s . Since the V(K n ) are metrizable, this is equivalent to sequential continuity. It is therefore sufficient to show: every zero sequence in £ is mapped into a zero sequence in IR. If ¢i -+ 0 in £, there are compact sets K C IRd , J C IR, such that SUPP¢i C K x J and ¢, -+ 0 in V(K x J), that is: Ve> 0, n e IN 3i o e IN s.t. for i i o maxlkl:SnsuPIEJ sUPrEK I Dk¢i(r, t) 1< e. The Jet) are equicontinuous ee- 3 nhd. of zero U C V(JRd) s.t. 1< Jet), U 1 (say). U contains e} and for i i o ¢i(', t) e U implying a set of the form {'IjJ e V(K): maxlkl:SnsuPrEK I Dk'IjJ(r) 1< Jet), ¢i(', t) 1, all t e J, i i o. Hence the integrands are uniformly bounded by a constant. 4[Schaefer,1980]. cha.p III, thm 4.2 5[Scha.efer,1980], chap II, thm 6.1

228

By the continuity of f(t) we have lim,....oc

>

< f(t), i(" t) >= 0 (t E 1) and

JI < f(t), i(" t) > dt = 0

lim F(i) = lim ( 1-+00

$-+00

o

follows from the dominated convergence theorem.

3 By the lemma, each trajectory Wt(w) induces an element W(w) of eo. W( dsdt =

fooo fooo(t 1\ s) < (-,s),(·,t) >H dsdt

4 Theorem 1 Proof:

W =N

in distribution.

W is defined by W(w)( 0) Now take E

= na>O Dom({3-A)a/2 j E endowed with the topology induced by the norms I 'ljJ Im:=

..;< ({3 - A)m'ljJ,;P > is a nuclear Frechet space. The semigroup

etA

can be extended to each of

the spaces (Dom({3 - A)a/2)*.

The choice of E is more complicated. A possibility the following. Let En

= {¢ECOO(OxIR+)

: ;:'¢(.,t)EEeacht,jand ;:'¢(.,t)=:Ooutside[O,n]},

which is a nuclear Frechet space in the topology defined by the norms ()i II¢II:" := j=O Emfan0 I utJ

t)

I:"-j

dt).

Define E := strict inductive limit of the En. To prove the analog of the lemma, we have to impose one more condition • The norms

I. 1m are equivalent to the corresponding Sobolev norms.

See [Schaumloffel,1988] for details.

References [Bojdecki/Gorostiza,1986] Bojdecki, T. & Gorostiza, L.G. Langevin Equations for S'-Valued Gaussian Processes and Fluctuation Limits of Infinite Particle Systems. Probab. Th. ReI. Fields 73 (1986), 227-244 [Gel'fand/Vilenkin,1964] Gel'fand, LM. & Vilenkin, N.Ya. Generalized FUnctions. Vol. 4, New York, 1964 [Kotelenez,1985] Kotelenez, P. On the semigroup approach to stochastic evolution equations. In: Arnold, L. & Kotelenez, P. Stochastic Space-time Models and Limit Theorems. Dordrecht, 1985, pp. 95-139 [Schaefer,1980] Schaefer, H.H. Topological Vector Spaces. 4th printing. New York, 1980 [Schaumloffel,1986] Schaumloffel, K.-U. Verallgemeinerte zufii.llige Felder und lineare stochastische partielle Differentialgleichungen. Diplomarbeit, Bremen, 1986 [Schaumloffel,1988] Schaumloffel, K.-U. White noise in space and time and the cylindrical Wiener process. To appear in: Stochastic Anal. Appl., 1988 [Treves,1967] Treves, F. Topological vector spaces, distributions, and kernels. New York, 1967

LARGE DEVIATIONS FOR NON-LINEAR RADONIFICATIONS OF WHITE NOISE WLODZIMIERZ SMOLENSKI AND RAFAL SZTENCEL

Institute of Mathematics, Technical University of Warsaw 00-661 Warsaw, PI. Jednosci Robotniczej 1, Poland Institute of Mathematics, Warsaw University, 00-901 Warsaw, PKiN, Poland Abstract. Let (i, H, B) be an abstract Wiener space and let W. be the associated semigroup of gaussian measures on B; let f be a function mapping H into another Banach space E. Suppose that there exists an increasing sequence Pn of finite dimensional projections, converging strongly to identity, such that f(Pn ) converges in the measure W. to eWe give some sufficient conditions to ensure that the family of measures £(i.) is a large deviations system in the sense of Varadhan. We show that the Varadhan contraction principle works in this case, as if the cylinder gaussian measures on H were real measures.

i

l.Large deviations. Let P" be a family of Borel probability measures on a metric space X. We shall say that P" has large deviations property (LDP) with the rate functional I, if there exists I: X -+ [0,00] such that (LD1) I is lower semicontinuous, moreover {x E X: I(x) a} is compact for every aE

R.

(LD2) For every open set G

-inf{I(x):x E G}

liminfe 2logP,,(G) " .....0

(LD3) For every closed set F limsupe 2logP,,(F)

-inf{I(x):x E F}.

" .....0

If, for instance, P" is the family of gaussian measures on R d and P" N(O,cId), then LDP holds with the rate functional I(x) = !luI2, where '·1 denotes the euclidean roJ

norm. We have the following elementary proposition (" contraction principle", cf. [9]):

This research was partially supported by the AFOSR Grant 87-0136 while the second named author was visiting the University of Tennessee, Knoxville. The author wishes to thank Professors Balram S. Rajput and Jan Rosinski for their warm hospitality.

231

THEOREM 1. If f: X -+ Y is continuous and a family of measures P, has LDP witb rate functional I, then the family of measures Qe = P e 0 bas LDP witb the rate functional J, wbere J(y) = inf{l(x):f(x) = y}.

r:

PROOF: Upper bound: lim sup e2 log Qe(F) e-+O - inf{I(x): f(x) E F}

=-

= lim sup sf log Pe(f(F))

inf inf{I(x): f(x)

yEF

= y} = -

inf{J(y): y E F}.

For the lower bound the proof is analogous. It remains to prove that the set B = {y E a} is compact for every a. But B = F(A), where A = {x E X: lex) a}. Indeed, let Xo E A, then l(xo) a, hence J(f(xo)) = inf{I(x): f(x) = f(xo)} < a, i.e f(xo) E B. Conversely, let Yo E B, i.e. inf{l(x): f(x) = yo)} a. The set f-l(yO) n {x: lex) a + I} is nonempty and compact hence there is Xo such that l(xo) = inf{l(x): f(x) = Yo} a. Obviously, f(xo) = Yo, hence B C f(A). This completes the proof. The following modification also holds:

Y: J(y)

THEOREM 2. If P, has LDP witb the rate functional I and fe: X -+ Y are continuous, moreover i, -+ f uniformly on compact sets, then Qe = P e 0 f;1 has LDP with tbe rate functional J, where J(y) = inf{l(x): f(x) = y}. For the proof, see [9]. Our aim is to get suitable generalization of Theorem 1 to the case of gaussian cylinder measures on a Hilbert space. 2. Abstract Wiener spaces. Let H be a separable Hilbert space with the norm I. I, , - standard gaussian cylinder measure (white noise) on H with characteristic functional-j'[z ) = exp( Now, if a linear mapping i: H -+ B is an injection into a Banach space B, moreover i(H) is dense in B and ,oi- 1 extends to a Radon measure in E, then the triple (i, H, B) is called abstract Wiener space. In the sequel we will simply assume that H C B. If Pn : H -+ H is an n-dimensional orthogonal projection, it can be written as n

Pn(x) = L(x, e.}, i=1

where (e.) is a complete orthonormal system (CONS) in H. Observe now, that for every u E H the mapping x 1-+ (x, u), where x E H, can be considered as a gaussian random variable with law N(O, luI). In particular, it extends to a random variable (w, u), where wEB. For this reason, P also extends to a random variable P: B -+ H. 3. Radoniflcations. Let f: H -+ E be continuous. Consider an increasing sequence of finite-dimensional projections (Pn ) , converging strongly to identity.

232

DEFINITION. FUnction f is said to be radonifying with respect to the sequence (P n ) , if and only if the sequence of random variables f(P n ) converges in measure 'Y 0 i-I, or, equivalently, for every e > 0 we have lim 'Y(lIf(Pn )

m,n-+oo

-

f(Pm)1I > s)

= O.

The limit of the above sequence will be denoted by j. In the sequel we will also with respect to cylinder measure 'Y." which is the image of 'Y consider extensions under the mapping T." T.,(X) = ez; However, for the sake of simplicity, we will write f(£w) instead of l.,(£w). From now on we put W = 'Y 0 i-I, W., = 'Y., 0 i-I.

I.,

4. Abstract version of Freidlln- Wentzell inequalities. In this section we will deal with fixed abstract Wiener space (i, H, B) and function f: H -+ E, which is radonifying with respect to some sequence of projections (Pn ) and every measure 'Y., for £ :::; £0· P., will denote the law of Theorems 3 and 4 below are abstract versions of Freidlin-Wentzell inequalities (formulated originally for solutions of stochastic differential equations), which in turn give LD2 and LD3 under some additional assumption on f.

I.,.

THEOREM 3. H for every u E H,8

(*)

and f(h)

>0

lim sup W(lIf(e:w

.,-0

= x,

+ u) -

f(u)1I

> 8) = '1 < 1

then for every 8> 0 liminf e: 2 log P.,((B(x, 8))

--21IhI2

e-O

PROOF: Let 8 be such that 1 P.,(G)

(12 -

'1

> 0, G = B(x, 8), Whl e

= W(f(£w) E G) =

1

{w:f(ew)EG}

f

e

exp(

2

{w:f(ew)EG}

1 exp( --(h,w)

J{w:f(ew+h)EG}

W(f(e:w

1

= W

* 8hl e

dW dW dWhl., = hI.,

1

2

+ -22 1h l )dWhle = e

- -;'lhI2)dW

e

2£

1 1 + h) E G,8(h,w) < Ihl)exp(--Ihl-lhI 2) {}, 2£2

(1- () - W(f(£w

+ h) ¢ G)ex p (-

(1 - ()2 - '1) exp( - -.!.-Ihl {}£

1

1

£

2£2

-8 Ihl- -lhI 2)

J:...2). 2e: 2 1hI

•

We have

233

a _II.2 exp ( - - 12 (S 2 - as 2) 2e which in turn for a = 1]/2s2 can be estimated from above by

(2sV1]) exp( -

(s2 - 1]))

exp( -

and this, for e < e(n, 1], so) does not exceed 11 2 - exp( --(s - 1])) 2 2e2

Finally we get for M >

+ 1]:

12M p«K:n 5 1 2 exp (- 2e 2 (S2 - 1])) +exp(- 2e

2 )

2

exp(

- 1])) + exp(

2e2

2e2

- 1]))exp(

2e2

=

- 2M - 1]))

1

5 exp(2e2 (s 2 - 1])) for e 5 e(1], so) and s < So. This gives immediately the desired inequality. 2. If f is weakly sequentially continuous, then LDl and LD3 hold with the rate functional J(y) = infHluI2:f(u) = y}.

COROLLARY

PROOF: To get (LD1) it suffices to observe that A = {u E H: !lul2 5 a} is weakly compact and repeat the proof of Theorem 1 with obvious changes. i.e. infHluI2:f(u) E Now let FeE be closed. Suppose that inf{J(y):y E F} = F} = Take s < So. Then K,. n F = 0, moreover K,. is compact, as an image of a weakly compact set under f. Consequently, there is 8 > 0 such that K! n F = 0. Now 1 limsupe2logP (F ) 5limsupe2logP «K:Y) 5 --2 S2 e

e-O

e

e-O

for all s < so. This completes the proof. It turns out that the condition (*) from Theorem 4 implies the condition (**) from Theorem 1, namely there holds THEOREM

5. If lim limsupliminfe2log7(lIf(ePm )

n-+oo

m-+(X)

-

f(ePn )1I > 6)

=-00

then for every u E H lim sup W(lIf(ew + u) - f(u)1I > 8) = 1] < 1. e-O

PROOF:

Fix u

E

H and 8 > O. We have

W(lIf(ew + u) - f(u)1I > 38) 5 W(lIf(ew + u) - f(ePnw + P nu)lI > 6)+

234

which gives immediately desired conclusion. COROLLARY 1: If G is an arbitrary open set, the above reasoning gives immediately (LD2):

-inf{J(y):y E G},

.....o

where J(y) = infHluI2:f(u) == y}. Let now K. = {x E E: f(h) = x, lhl s} and let d(x, A) denote the distance between x E E and the set A. Put K: = {x E E: d(x, K.) S 6}.

s

THEOREM

(**)

4. !ffor some 6> 0 lim lim sup lim inf e2 log 'Y(II f(eP n )

n ........ oo

m--+oo

E-+O

-

f(ePm)\I > 8) = -00,

then

PROOF: Let (gi) be a sequence of independent, standard gaussian random variables on a probability space (n, F, P). We have

'Y(IePn l > s) + W(lIf(e1\) - f(ew)lI > 6).

+ "1.

Now fix "1 > 0 and M

>

hence taking m

we get for E: :::; eo (n )

-+ 00

Pick n so that for every m

> n and e S E:o(n)

hence

-

M

f(ew)\I > 8) S exp( -z)' for E: :::; eo(n). e Now we will estimate 'Y(lePn I > 8). We have W(lIf(ePn )

-

= (1 -

exp( _>'8 2 /e2 ) .

After substituting>' = (1 - a )/2 the above formula yields

235

By continuity of f , we can fix n such that 8 3 = O. Moreover, 8 2 = 1( {x E H: (1If(Pn(ex + u) - f(Pnu)1/ > 8}) and S2 < t for e < eo(n). Now we will estimate S1. W(lIf(ew

+ u) -

f(ePnw

+ Pnu)1/ > 8) =

{

1

_d::-W_ dW -hI" = dW_hl" 1

{wEB:llf("w+u)-f("Pnw+Pnu)lI>o}

r

J{wEB:IIJ("w)-f("Pnw)U>o}

exp( -(h,w) e

exp(

e

1

2

+ -21hl )dW_hl" =

-

2e

:::;

2e

1I/("w)- f("Pnw)lI>o}n{w:

IhF}

f(dnw)lI>o}n{w: ;{h,w)-

+ :::;

Ihl';:::

1

1 IhI2+ 1 1 2 W(lIf(ew)- f(ePnwl/ 2:: 8)'exp(-2 exp( -(h,w) - -21hl }dW = 2e {w:;{h,w);:::;\-lhI2 e 2£

1

W(I/f(ew) - f(ePnwll 2:: 8) -expf 2e21hl ) + 2' 21

Now, choose M > Ihl2, By assumption we have for e < eo(n):

hence

M

W(lIf(ew) - f(ePnw)1I > 8):::; exp(-2)' e

consequently S1 < i for sufficiently small e and finally 8 1 + S2 + S3 < This completes the proof.

for small e.

236 REFERENCES 1. R.G. Azencott, "Sur les grandes deviations.," Ecole d'Ete des Probabilites, Saint-Flour, LNiM 774,1978. 2. X.Fernique, IntegrabiliU des vecteurs gaussiens, C. R. Acad. Sc. Paris, Serie A 270 (1970), 1698-1699. 3. M.Freidlin and A.Wentzell, On small random perturbations of dynamical systems, Russian Math. Surveys 25 (1970), 1-55. 4. M.Freidlin and A.Wentzell, "Random perturbations of dynamical systems," Springer, New York, 1984. 5. G.Kallianpur and E.Oodaira, Freidlin- Wentzell type estimates for abstract Wiener spaces, Sankhya, Series A 40 (1978), 116-137. 6. H.H. Kuo, "Gaussian measures in Banach spaces," Springer LNiM 463, 1975. 7. W.Smolenski, R.Sztencel and J.Zabczyk, Large deviations estimates for Banach space valued Wiener process, Inst. Math. Polish Academy of Sciences, preprint 371 (1986). 8. W.Smolenski, R.Sztencel and J .Zabczyk, Large deviations estimates for semilinear stochastic equations, Lecture Notes in Control and Information Science 96 (1987), 218-231. 9. S.R.S.Varadhan, Large deviations, SIAM,. Keywords. large deviations, abstract Wiener spaces, non-linear radonifications 1980 Mathematics subject classifications: Primary 60FI0, secondary 60B11

SYMMETRIC SOLUTIONS OF SEMILINEAR STOCHASTIC EQUATIONS J. Zabczyk Institute of Mathematics Polish AQademy of Sciences Warsaw, Sniadeckich 8 Poland

1. Introduction

Let E be a metric space and B(E) the space of all Borel, bounded functions defined on E. A semigroup of linear operators Pt , t acting on B(E) such that: ({l e B(E), (i) PtC{' z, 0 for all 0 and nonnegative (ii) Pt1 = 1, t is called a Markovian or transition semigroup. If m is a probability measure on E and Pt , t a Markovian semigroup such that

°,

°

J'f(x)Pt'Y(x)m(dx)=Jr(x)ptO,

o

> 0 and all t 0, ISH(t)\ S Me-!)(t • Moreover the measure NCO, r) is the unique invariant measure for , t.?: and therefore m = N(O,r ). Denote the measure N(So(t)x, r t ) by Symmetricity of implies that: It(a, b)

= S[

°

ei 0

= 5[5 ei dr

>= S t

x,y Eo

Therefore, i f

t

Letting

t-l- 0

rx

Eo D(A

o)

for

x E

therefore

Taking into account (12): 2

and finally

0 •

one obtains that:

, Since

>

r

= - x

for all

x E

x,y Eo

.

+00.

242

r

Y

=-

1( *)-1 Ao

y e H

Y

r

Since is a self-adjoint, nuclear operator, the required properties follow now easily. To prove the converse implication denote by (- An) the sequence of strictly negative eigenvalues of the operator Ao' Then +00

)

Il:'1

n


a such that: provided

(18)

_xI[ ,liYIl

5

r.

To define the mild solution of the equation (5) it is convenient to assume that: There exists an aU-process z, E-continuous and Markov with respect to an increasing family of 0" -fields J="t ' t z a and to the transition semigroup Qt ' t a compare Section 2.2. If a process Z satisfYing (H3) is given then an E-continuous and adapted process X ,is said to be the mild solution of (5), if surely for all t a t

(19)

x(t)

= s(t)x + S so(t-s)U'(X(s))ds a

+ Z(t) ,

t

a

Thus the concept of mild solution depends on the version of the OU-process Z. If however the function U' is locally Lipschitz then, see Proposition 4, the finite dimensional distributions of X are independent of the version. Let Xx(.) denote the solution of (5) and let Pt , t a be the transition operators:

We have the following main result Theorem 2. Assume that (i)

the conditions (Hi), (H2) and (H3) are satisfied,

(ii)

the H-derivative of a bounded from above function

U exists

247

(iii)

and is locally Lipschitz, the mild solution of the equation (5) exists for arbitrary x Eo E •

Then the family semigroup where: (21 )

Pt , t

is a Markovian and M-symmetrie

M(dx) = ke 2U(x) m(dx) ,

k

=

m(dx»-1 E

After some preliminary results, the proof of Theorem 2 will be given in Section 3.4. Sufficient conditions implying global existence of the mild solutions can be found in (5) • Markovianity of the solutions is discussed in the next subsection. Example 2. Define E, H, A and nal U is of the form:

Ao

as in Example 1. If the functio-

7l'"

U(x) =

Jp(x(r) )dr

x

Eo

E

o

where p is a polynomial of even order, with the negative leading coefficient, then all the conditions of Theorem 2 are satisfied, see [5J. In this case the measure m is identical with the distribution of the Brownian bridge on (O,71). Under different conditions on function p , not covering however polynomial nonlinearities, the formula (21) was derived by Funaki [9J . 3.2. Semilinear stochastic equations - existence and Markovianity. To prove that the transition operators defined b,y (5) form a Markovian semigroup we consider more general equation: t

(22)

X(t)

= S(t)x + S So(t-s)F(X(s»dS o

+ Z(t),

t

[O,T] •

Here F is a transformation from E into H, x E: E, T > O. It will be convenient to introduce also a completely nonstochastic equation: t

(23)

u(t)

where

t'

= 1So(t-s)F(X(s»dS o

+ lj-'(t),

t

(o,T1 '

is an arbitrary E-continuous function. ec Let (-A o) be the fractional powers of the operator -Ao' with domains Rill, = I;J.. > 0, see [11, Section 1.4]. The sets H lit. with norms:

248

(24)

x E HOI

>

b(

>

are Hilbert spaces. Moreover, for arbitrar,y constant Lot. = L such that:

x

°

°

there exists a

Eo H

(26)

Proposition 4. Assume that Ao is a negative definite generator of a Co-semigroup So(t), t on a separable Hilbert space Hand E is a Banach space such that, for aome Eo (0,1) ,

°

HOI. C. E C H

with the continuous embeddings. If F is a locally Lipschitz transformation from E into H, then for arbitrary E-continuous function the equation (23) has a unique local solution. Proof. The proof uses Banach's contraction principle, compare [11, Section It follows from the assumptions, that for some positive numbers K, N and all x e H :

3.3J.

s

Klxl

For arbitrary

v

)'v(t) Then

Eo

I\x II 5

N1X\lI(

C(O,T;H) define: t So(t-S)V(S)dS ,

=J

°

t

= I (-Aof j

°

t

t

°L( 5 °

[o,T1

So(t-S)V(S)dS I

= I J(-Ao}So(t-S)V(S)dS I

s

Co

t

dz-)

1v(·)1

and by

t f:

[O,T

1.

249

ConsequentlYt for some

and -l- 0 Jrv is the space

as

c;

T.j. O. In a similar way one can show that the function continuous and thus also E-continuous. Consider next of all E-continuous functions v such that

h(t)- f(t). .:s: s , t e [O,Tl ' and let be the transformation:

:T'f"

' \f'

::

C(O,T;E] t

t

)v(t) =

't'

SSo(t-s)F(v(s»dS +

'Y(t) ,

t t- [O,T] •

0

Then:

provided T is sufficiently small, (F is locally bounded transforT mation). So we can assume that ./If': CT CT s • If v 1 tV2 6 Os s then 1\ J;(V1)(·) -

S

S: ¥ r Kv 1 ( · ) r > s +

and by (18) ,

IF(v1)(·) - F(v2 ) ( · ) \ - v 2 (· )K

provided

1i.I"(.) \

It is enough to choose T such that O

Denote by m£., the symmetri Gaussian measure on H with the co2 variance operator = +(-Ao ) - 1 • The proof of Theorem 2 gives also that the processes are symmetrizable with respect to invariant measures ME-: ME.(dX) = kc: e £: -

mE (dx) ,

Let us recall that a family of probability measures (Me) E>O on a Banach space E is said to satisfy a large deviation principle with the rate function J : E [0 1,+ (0 ) if the following three conditions hold: (i)

The function olE

(1i)

lO,+oo)

J is lower semicontinuous and for arbitrary the set {x G :E:; J(x) Cl( .5 is compact ,

For arbitrary open set

G c.:E; :

lim £2 In M (G) 2:::. - inf{J"(x); XfG 5 ' t-

ti» (iii)

For arbitrary closed set

K C E :

The follOWing large deviation theorem extends a similar result obtained for Example 1 by G.Jetschke Part

11 .

Theorem 5. Under the assumptions of Theorem 2, the family (M E.)' defined by satisfies a large deviation principle with the rate function:

J(x)

=

t

-2U(x) + 2 I(_A o)1/2x/2 +

00

,

if

x

E;

otherwise

D(-A )1/2 o

255

Proof. We will use the following extension, due to G.Jetschke [13, Part I, Prop. 4.2 of a classical result of Varadhan [191 :

J,

Proposition 6. Let (mE) f,>O be a familJ of probability measures on a complete, separable, metric space E, satisfying a large deviation principle with the rate function I . Let U be a continuous, bounded from above function on E. Then the family of measures (Me) e >0 of the form (36) satisfies a large deviation principle with the rate function J: J(x)

H1 / 2

= I(x)

- 2U(x) - inf[I(y)-2U(y); y

Eo E

As the reproducing kernel for the measure D(-Ao)1/2 , with the norm:

=

=

h

therefore, see [141 (mE) f ,>0 is exactly I(x)

=

x E E • m1 is the image

Eo

or [171, the rate function for the family

21(-Ao)1/2x/2 {

+00

Theorem 5 follows now from Proposition 6 •

References A.Antoniadis and R.Carmona, Eigenfunction expensions for infinIte dImensional OrnsteIn-Uhlenbeck processes, Probab. Th. Related Fields, 74(1987), 31-54. P.L.Butzer and H.Berens, Semigroups of Operators and ApproximatIon, SprInger Verlag, 1967. P.L. Chow , Expectation functionals associated with some stochastIc evolution equations, Proceedings of the Conference on Stochastic Partial Differential Equations and Applications, Trento 1985, ItalJ, Eds. G. Da Prato and L.Tubaro; LNM 1236, 40-56. G.Da Prato. S.Kwapien. J.Zabczyk, Regularity of solutions of lInear stochastIc equatIons in Hilbert spaces, Stochastics, vol.23, No.2, 1987, 1-23. G.Da Prato and A note on semilinear stochastic equatIons, Di£ferential--and Integral Equations, Vol.1, No.2, 1988, 143-155.

256

[81

[9J

R.Datko, Extending a theorem of A.M.Liapunov to Hilbert space, J.Math.Anal.Appl. 32(1970), 610-616. W.G.Faris and G.Jona-Lasinio, Large fluctuations for a nonlinear heat equation with noise, J.Pbys. A: Math.Gen., 15(1982), 3025-3055. M. Fukushima , Dirichlet Forms and Markov North-Holland, 1980. T.Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math.J., 89(1983), 129-193. R.Z.Hasminski, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, 1980. Geometric Theory of Semilinear Parabolic Equations, tNM 8 ,Springer-Verlag, 1981. N.lkeda and S.Watanabe, Stochastic Differential Equations and DIffusion Processes, North-Holland, 1981. G.Jetschke, Invariant distribution of a nonlinear stochastic partial differential equation and free energy of statistical physics. Forschungsergebnisse Friedrich-Schiller-Universitttt Jena, Part I (86/11), Part II (86/20), Part III (86/40), 1986. G.Kallianpur and E.Oodaira, type estimates for Abstract Wiener Spaces, Sankhya, 40(1978), Series A, Part 2, 116-137. R.Marcus, Parabolic Ito equations, TAMS, vol.198, 177-190. W.Smolenski, R.Sztencel and J.Zabczyk, Large deviations estimate for Banach space valued wiener processes, IMPAN Preprint 371, Warsaw, October 1986. W.Smolenski, R.Sztencel and J.Zabczyk, Large deviations estimates for semilinear stochastic equations, in Proceedings "Stochastic Differential Systems", Eisenach 1986, Eds. H.J. Engelbert and W.Schmidt, LNinCIS 96. Probability N.N.Vakhania, W.I.Tarieladze and Distributions on Banach Spaces, Nauka, Moscow, 1985 (in Russian). S.R.S.Varadhan, Large Deviations and Applications, CBMS - NSF Regional Conference Series in Applied Mathematics, SIAM, 1984. J.Zabczyk, Structural properties and limit behaviour of linear stochastic systems in Hilbert spaces. Banach Center Publications, vol.14, pp. 591-609 , Warsaw, 1985. J.Zabczyk, Symmetric solutions of semilinear stochastic equations, Pre print 416, Institute of Mathematics, PAS, Warsaw, March 1988. I.Shigekawa, Existence of invariant measures of diffusions on an Abstract Wiener Space, Osaka J. Math. 24(1987), 37-59.

LIST OF PARTICIPANTS

- Alain Bensoussan (INRIA, Le Chesnay, France) - Vivek S. Borkar (Tata Institute of Fundamental Research, Bangalore, India) - Piennarco Cannarsa (Dipartimento di Matematica, Universita di Roma IT, Italy) -Mireille Chaleyat-Maurel (Laboratoire de Probabilites.Universite Paris VI, France) .A.

Pao-Liu Chow (Department of Mathematics, Wayne State University, Detroit, USA)

- Paolo Daipra (Dipartimento di Matematica, Universita di Padova, Italy) - Giuseppe Da Prato (Scuola Normale Superiore, Pisa, Italy) - Donald A. Dawson (Department of Mathematics, Carleton Univ., Ottawa, Canada) - Markus Dozzi (Department of Math. Statistics, University of Berne, Switzerland) - Franco Fagnola (Dipartimento di Matematica, Universita di Trento, Povo, Italy) - Alberto Frigerio (Istituto di Matematica e Informatica, Universita di Udine, Italy) - Hisao Fujita-Yashima (Scuola Normale Superiore, Pisa, Italy) - Luis G. Gorostiza (CIEA-IPN, Mexico, Mexico) - Istvan Gyongy (Dept. of Number Theory, Eotvds University, Budapest, Hungary) - Ulrich G. Haussmann (Mathematics Dept., Univ. of British Columbia, Vancouver, Canada) - Torbjorn Kolsrud (Department of Mathematics, Kgl, Tekniska Hoegskolan, Stockholm, Sweden) - Peter Kotelenez (Mathematisch Instituut, Rijksuniversiteit te Utrecht, The Netherlands) - Hui-Hsiung Kuo (Department of Mathematics, Louisiana State University, Baton Rouge, USA) - Gianni Jona-Lasinio (Istituto di Fisica, Universita Roma I, Italy) - Fabio Martinelli (Dipartimento di Matemarica, Universita Roma I, Italy) - Jose Luis Menaldi (Mathematics Department, Wayne St. University, Detroit, USA) - Michel Metivier (Ecole Polytechnique, Palaiseau, France) - Sanjoy K. Mitter (MIT, Cambridge, USA) - David Nualart (c/o Sant Quinti, Barcelona, Spain) - Daniel L. Ocone (Department of Mathematics, Rutgers University, New Brunswick, USA) - Etienne Pardoux (Universite de Provence, Marseille, France) - Mauro Piccioni (Dipartirnento di Matematica, IT Universita di Roma, Italy)

258 - Maurizio Pratelli (Dipartimento di Matematica, Universita di Pisa, Italy) - Andrea Pugliese (Dipartimento di Matematica, Universita di Trento, Povo, Italy) - Sylvie Roelly-Coppoletta (Lab. de Probabilites, Universite Paris VI, France) - Edmundo Rofman (lNRIA, Le Chesnay, France) - Wolfgang Runggaldier (Dipartimento di Matematica, Universita di Padova, Italy) - Kay-Uwe Schaumloffel (Fachbereich Mathematik, Universitiit Bremen, Fed. Rep. of Germany) - Wlodzimier Smolenski (Institute of Mathematics, Warsaw Technical University, Poland) - Pierpaolo Soravia (Dipartimento di Matematica, Universita di Padova, Italy) - Rafal Sztencel (Institute of Mathematics, Warsaw University, Poland) - Luciano Tubaro (Dipartimento di Matematica, Universita di Trento, Povo, Italy) - Ali S. Ustunel (2, Bd. A. Blanqui, Paris, France) - Jerzy Zabczyk (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland) - Ofer Zeitouni (MIT, Cambridge, USA)