Stochastic Partial Differential Equations and Applications: Proceedings of a Conference held in Trento, Italy, September 30 - October 5, 1985 (Lecture Notes in Mathematics, 1236) 3540172114, 9783540172116

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

1236 Stochastic Partial Differential Equations and Applications Proceedings of a Conference held in Trento, Italy, Sept. 3D-Oct. 5, 1985

Edited by G. Da Prato and L. Tubaro

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Editors

Giuseppe Da Prato Scuola Normale Superiore Piazza dei Cavalieri 7, 56100 Pisa, Italy Luciano Tubaro Dipartimento di Matematica, Universita di Trento 38050 Povo, Italy

Mathematics Subject Classification (1980): 60H 15, 60G35, 93E 11, 60H 10

ISBN 3-540-17211-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17211-4 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, reprinting, 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 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

PRE F ACE The theory of stochastic differential equations has had a great development in many directions since the early works by Ito in which it was first stated rigorously. One direction is the study of stochastic partial differential equations and applications or, in more abstract terms, stochastic differential equations in Hilbert (Banach, when possible) spaces, a really interdisciplinary subject between Analysis and Probability. We took the opportunity offered by the CIRM (Centro Internazionale per la Ricerca Matematica) to organize a meeting on this very subject, with the ambitious intention to organize a similar meeting every two years as an occasion for a more direct communication among interested researchers. We wish to thank all the participants; it is due to them that the meeting was so interesting. Finally we wish to thank the CIRM for its support: most of the organization rested on its shoulders. Special thanks go to the secretary, Mr. A. Micheletti, for his invaluable organizational work.

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

LIST OF PARTICIPANTS - G.L. BLANKENSHIP (Elect. Engin. Dept., Univ. of Maryland, College Park, USA) - M. CHALEYAT-MAUREL (Lab. de Probabilites, Univ. Paris VI, France) - A. CHOJNOWSKA-MICHALIK {Inst. of Math., Lodz Univ., Poland) - P.-L. CHOW (Dept. of Math., Wayne State Univ., Detroit, USA) - G. DA PRATO (Scuola Normale Superiore, Pisa, Italy) - F. FLANDOLI (Dip. di Mat., Univ. Torino, Italy) - G. GORNI (Scuola Normale Superiore, Pisa, Italy) - B. GRIGELIONIS -

1.

(Academy of Sciences of the Lituanian SSR, Vilnius, USSR)

GYONGY (Math. Inst., Eotvos L.

Un.i.v ,. , Budapest, Hungary)

- W.E. HOPKINS, Jr (Dept. of Elect. Engin., Princeton Univ., USA) - D. KANNAN (Dept. of Math., The Univ. of Georgia, Athens, USA) - G. KOCH (Dip. d i, Mat., Univ.

"La Sapienza", Roma, Italy)

- H. KOREZLIOGLU (Dep. Systemes et Carom., ENST, Paris, France) - P. KOTELENEZ (Fachb. Math., Dniv. Bremen, W. Germany) - H.-H. KUO (Dept. of Math., Louisiana State Univ., Baton Rouge, USA) - M. IANNELLI

(Dip. di Mat., Unlv. Trento, Povo, Italy)

- A. ICHIKAWA (Faculty of Engin., Shizuoka Univ., Hamamatsu, Japan) - P. MARCATI (Dip. di Mat., Dniv. L'Aquila, Italy) - F. MARCHETTI (Dip. di Mat., Dniv. "La Sapienza", Roma , Italy) - M. METIVIER (Ecole polytechnique , Palaiseau, France) - R. MORKVENAS

(Academy of Sciences of the Lithuanian SSR, Vilnius, USSR)

- D. OCONE (Dept. of Math., Rutgers Univ., New Brunswick, DSA) - E.

PARDOUX (Univ. de Provence, Marseille, France)

- M. PICCIONI

(Dip. di Mat., II Dniv., Roma, Italy)

- G. PISTONE (1st. di Mat., Univ., Genova, Italy) - A. PUGLIESE (Dip. di Mat., Univ. Trento, Povo, Italy) - A. RASCANU (Fac. of Math., Univ. Iasi, Romania)

- J. REAL (Dep. de Ec. Funcion., Univ. Sevilla, Spain) - W. RUNGGALDIER (Seminario Mat., Univ. Padova, Italy) - L. TUBARO (Dip. di Mat., Univ. Trento, Povo, Italy) - A.S. DSTUNEL (2, Bd. A. Blanqui, Paris, France) - V. VESPRI

(Dip. di Mat., II Univ. Roma, Italy)

- J. ZABCZYK (Inst. of Math., Polish Academy of Sciences, Warsaw, Poland) - P.A. ZANZOTTO (Via S. Antonio 7, Pisa, Italy)

CONTENTS

P. Cannarsa, V. Vespri - Existence and uniqueness results for a non linear stochastic partial differential equation ..••.••...........•..•....• M. Chaleyat-Maurel - Continuity in non linear filtering. Some different approaches ....•..•...•..•...................•..•................•..

25

P.L. Chow - Expectation functionals associated with some stochastic evolution equations ..••...•.••...........••....•...............•...•.........

40

F. Flandoli - Dirichlet boundary value problem and optimal control for a stochastic distributed parameter system ....•..•..•................. 57 L. Hazareesingh, D. Kannan - Stochastic product integration and stochastic equations ...........•..•..•......•........•••.••.........•....•...• 72 W.E. Hopkins, Jr. - Some remarks on a problem in stochastic optimal control .. 121 H. Korezlioglu - Passage from two-parameters to infinite dimension ........•.. 131 H.-H. Kuo - The heat equation and Fourier transforms of generalized Brownian functionals . . . . . • • . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A. Ichikawa - The separation principle for stochastic differential equations with unbounded coefficients . . . . . . . . . . . . . . • . . . . . • . . . . . . . . . . . . . . . . . . • 164 M. Metivier - Weak convergence of measure valued processes using Sobolevimbedding techniques .. _ . . • . . • . . • . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . .. 1 72 D. Ocone - Probability distributions of solutions to some stochastic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . • . • • . . . . . . . . . . . . . . . 184 E. Pardoux - Two-sided stochastic calculus for SPDEs ...................•..••.

200

M. Piccioni - Convergence of implicit discretization schemes for linear differential equations with application to filtering

208

A.S. Ustunel - Some applications of the Malliavin calculus to stochastic analysis . . . . . . . • . . • . . . . . . • . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . 230 J. Zabczyk - Exit problem for infinite dimensional systems

EXISTENCE AND UNIQUENESS RESULTS FOR A NON LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATION ,

1)

Cannarsa Gruppo Insegnamento Matematiche, Accademia Navale 57100 Livorno (Italy) ,

.1) ,2) ,3)

Dipartimento di Matematica, II Universita di Roma Via O. Raimondo, Roma (Italy)

ABSTRACT We study the non linear stochastic partial differential equation n

du(t,x)

A(x,u,Du,D"u)dt +(

L G,(x)D,u(t,x)+h(x,u(t,x»dW(t) j=1 J J

where A is a convex functional and W(t) a real Wiener process. We study the corresponding non linear robust equation by linearization methods. We also prove some existence and uniqueness results for parabolic equations with unbounded coefficients in Holder spaces.

1. INTRODUCTION AND NOTATION n ... ,x a typical point in IR and by (,1,), 1' n) n, the scalar product and the norm in lR respectively. Define D, = Denote by x = (x

J

1·1 a

ax,

J and D., = D,D, for i,j=1, ... ,n. Set D = (D ... ,D ). It a is the r, J a a 1, n a multi­index (a , ... ,a ) , set D =D 1 ... Dnn and lal = a + a + .•. + a 1 n 1 2 n 1

1) The A.A. are members of G.N.A.F.A. (C.N.R.). This work is partially supported by the Research Funds of the Ministero della Pubblica Istru zione 2) The A. is presently on duty at Stato Maggiore Marina, Palazzo Marina, Rome 3) This A. held the present communication at Trento

2

Set D"=(D

a)

where a: !a!=2. We assume that all functions are real valued.

The aim of this paper is to prove the existence of solutions to the non-linear stochastic partial differential equation du(t,x)=A(x,u(t,x) ,Du(t,x) ,D"u(t,x» + h(x,U(t,x»)dW(t)

+ «G(x) IDu(t,x»

n (t,x) E[ 0 ,T] xm

+ (1.1)

with initial condition u(O,x) =uO(x),

xEm

n

(1 .2)

We assume that W(t) is a real Wiener process in (Q,F,P), A(x,n,v,p) is a functional and

h

hypotheses on A,G and

is a real valued function such that h(O)=O. Other h

will be stated later.

When A is a linear operator and h(x,u(t,x»=h(x) 'u(t,x), then equation (1.1) is related to non linear filtering problems for diffusion processes (Zakai equation). In this case several authors studied problem (1.1)

(1.2) under different conditions. It is sometimes assumed - see

e.g. [2], [16], [18]

- that the coefficients of (1.1) are bounded and

G=O. The case of unbounded coefficients,with G=O,is treated in [13 ],[3], [19]. In a recent work [12] Ferreyra considered the case of a degenerate Zakai equation. Under the hypothesis that G is not (necessarily) equal to 0, the problem has been analyzed in [9] work by Da Prato and Tubaro [10]

and in [6] • Only in a recent

has the non linear problem (1.1)

(1.2)

been studied in the one dimensional case and under some restrictive hypotheses on the coefficients. In section 2 of this paper we will formally deduce the robust equation associated with (1.1) and we will derive the linearized equation related to it. In order to treat the linearized equation, in the following section we will state a result about parabolic n) equations with unbounded coefficients in (m that will be proved in the appendix. Existence and uniqueness results for the equation (1.1)

(1.2)

will be derived in the last section from the previous ones. Furthermore we will improve the results on linearized equation already obtained in [ 7] •

We conclude this section introducing some notation. We denote by

3

L 2 , u (IRn ) ,

a




t->O

0

So, functionals are differentiable along H1• The spaces of differentiability are :

IDo (X)

1 (X)

£)pCX)

exists and is in

5\ (X

exists and is in

ill 0-1 (X J

J5

00

(X)

n

;ien ;(Jt)}

l.J

32

With suitable norms, these spaces are Frechet spaces.

analo90us to Sobolev

For the filterinp; problem, there are two V!iener spaces : A bi£ one

J):

(V)'x 0), Po on

there is the

7

f(xt)L t.

A little one:

0,O(= law of y under Po) on which there is the functional

One associates

£1

00

= {Coo func t i onal s on

{Coo

functionals on 0)

Let

f

b ; the

be in

C

(w,y)

11T x 0)

to

0

and

o.

to

flliictional

(w,y)

proved by S. Kusuoka-D.W. Stroock [11]) for by

(w,y)

7

f(xt)L t

(xt,L t)

is in

(result

is a gooe diffusion driven

with smooth coefficients.

In order to prove that Theorem 4

ptf

is in J)2

00'

it suffices to prove

(M. Chaleyat-Maurel [2])

(j'\

¢ be in

Let

then

EW o( ¢)

v

Ij\:.

v

Sketch of the proof One uses the following formula which indicates how 'l acts on stochastic integrals (see S. Kusuoka-D.W. Stroock [11] lemma 2.2). Let and e

a

be a matrix and

be a Brownian motion, then if w;(T)·h

with h

S a vector satisfying suitable conditions (cf. [11])

in

=

'la(t)·h deC!)

sCT) +

=

C

a(t)de(t)

Ca(t)h' Ct)

+

+

f:

S(t)dt

then,

'lS(t) ·h)dt

E1.

Let ¢ be a functional on 'Wx 0 belonging to Q)oo; in order to prove the result, it suffices to work with ¢ in each Wiener chaos, thanks to the Wiener-Ito expansion:

33

LZ(V 0"x st) The

nth

=

0 &tn. o

homogeneous Wiener chaos

ru)x

The Brownian motion on

st is

it is more convenient to denote it by the

nth

I

cf m-p

(6 , ... ,6

1

(w ..• ,wm'Y1' ... yp); 1,

+ , ... ,6 + ) ' Any element of m,6m 1 mp

. " ,1n

(S)J n no

/Z de. (s 1)"') 1n_1 n0

de. (s1)£' . (s1'''''s) 11 11,· .. ln n

is the set of arrangements which repetitions of the

{1,Z, ... d}

1 1"

de. 0 1n

.41

{i 1 , · .. ,in } E

f.

dimensional,

s

oo

bers

m-p

chaos is written as :

L

where

is defined as follows:

taken one by one,

=

n

: 0

{s E IR

s1

...

d sn}

entire numand

in

It is clear that if in a term of

1IF

rSn rSZ de. (s)) de. (s 1)"') de. (s1) fCs 1 , · .. ,sn)' o 1n n 0 1n_1 n0 11

J 00

A =

there exists

i

(1

the

(1

k k

k n)

n) are in

belonging to

{m+1 , ... .m-p l ,

It follows easily that, after expanding

where

'V

Z

is the gradient on

O. And if all

{1,Z, ... ,m},

EW(A) o

=

A.

¢ on the chaos, one can prove

st.

Finally, with an induction procedure, we deduce the result. And we get

Theorem 4 : (M. Chaleyat-Maurel [2])

One can also calculate the iterate gradients of

TItf

and get estimates for

their norms. From the continuity point of view, it means that errors made in are not too bad.

H1-directions

34

3. Continuity in Sussmann's sense.

There is a third notion of continuity which was proposed by P. Sussmann [19] and which, roughly speaking, is the fo Llowing t hinp : in general, the filter is not continuous for the sup-nann but, perhaps "the lack of continuity" is not too big ; and he suggested to study the behaviour of Ptf when the observation is near a function of H1. More precisely, we consider system (1) with coefficients indeuendent of

Yt'

and the following system (4) : dx

(4)

u t

u dLt

u u i '\, u·i Xo(xt)dt + \ (xt)odwt +Xt(Xt)u t dt 1 '\, u .i (hi(xt)ut + "2 Xi

1

"2

p L

hi

)

dt

(x , l ) . o

We have just replaced y by u

in the Stratonovitch form.

And we define (5)

We use (5) and not

equation because we want to work with very feneral

f.

Then, we say that the filter is continuous in Sussmann's sense if (6)

Tl

being fixed.

Let us this continuity is interesting only in the cases where continuity is not satisfied ; indeed, if the filter is continuous for the sup-norm, the probability written in (6) is exactly zero for small enough.

35

In a joint work with D. [3], vR proved that the filter ptf is continuous in Sussmarm's sense and we gave exn l ic i te estimates for the convergence under the following hypothesis 1) (H all the coefficients are h

and

C4. 'V

are bounded as well as the derivatives of X.1 "s

h until fourth order. i Xo and the first derivatives of Xi and X.X 1

Let

­.E

are of sub­linear provrrh.

be a bounded set of P 1 and K be a co:npact set of R n .

The estimate we get concerns the following class of If

B1 and BZ are two positive constants, we denote by

functions

f: m.n

(i)

f(O)

(ii)

va • 0, v(x,x')

+

the set of

lR such that :

=>

flR

Zn , [x] ;;; R,

[f(x) ­ f(x')1 s

Ix'l.,,; R, B

1

exo (BZR) Ix­x ' I .

The theorem is now stated as : Theorem 6

(M. Chaleyat-Maurel-D. Michel [3])

"IE, 0 < E ,

E,

K,J:), B1,

VfE 4>(B

1,B2)

,

:

This result is analogous to the estimations proved by D.W. Stroock e t S. Varadhan in their celebrated paper sions [16]. We encounter the same type of difficulties: s 0] where A is an event and y a tem as P[A / Bayes formula and a classical estimation on the Brownian

for diffusion processes on the support of diffuindeed, in order to handle a Wiener process, we use motion to obtain an esti­

36

mate of the type : C exp{+

+}

P[A n

;';

O

rt


the

"

Gaussian

exp

{ -

I

invariant

distribution. To show f

satisfying the equation (2.8), consider

=

(ftg,h)

I

t * 0 (Tt_sR T t_ s g,h) ds, g, h

E

V.

Therefore

Now, Remarks:

taking the limit t -> The

special cases.

covariance

f

may

In particular

(i) when A is self-adjoint,

we get the equation (2.8). # be

determined

from

(2.8)

explicitly

for

some

44 (ii) when A is a normal operator and A commutes with R, we get

r = (A+A*)-lR. (iii) The method of characteristic functionals has been used by Chow [5] and Zabczyk

(10]

to study

the solution measures of stochastic evolution equations.

Related problems about invariance distribution have been treated in (11]. Next we consider the nonlinear stochastic evolution equation (P2) where, in contrast with (PI), B is a nonlinear operator.

For monotone nonlinear

operators, the existence and uniqueness theorem has been proved by Bensoussan and Temam [12] and Pardoux (13]. (Bl) B: V

Let us assume

V' satisfies

/IB(u)/I*

Q

/Iu/l P -

1

for some a > 0, p > I, 'V u E V,

(B2) B is coercive: 3 constants a > 0, 2 +

IUl 2 + '6

t 3

a /Iu/l P + tr. R, 'V u E V,

(B4) B is monotone, that is, 2 +

A

/u_vI 2 > 0, 'V u, v

E

V.

Then we have Theorem

2.3

process in H.

The

problem

(P2)

has

a

unique

solution

which

is

a

diffusion

Furthermore we have

Remark: This is a special case of a general existence theorem proved by Pardoux ( 13]. 3. Infinite-Dimensional Diffusion Equations Let {u t} be a diffusion process in H.

For any Borel set A

transition probability (3.1 )

pt{v,A}

=

Prob. {u t

and the transition operator

E

AI

U

o •

v}, v

E

H,

c H, define the

45

(3.2) where ¢ is a bounded-measurable function on H.

For convenience, we set

(3.3) Consider

the

linear

self -adjoint and strictly (PI) L

has

2(H.IJ.)

the

invariant

stochastic

equation

posi t i v e ,

(PI),

By the Remark

distribution IJ. E N(O,r),

where (i)

A is

assumed

to

be

following Theorem 2.2,

where r

2

!

Let H =

A-1R.

be the space of real functionals on H that are IJ.-square integerable.

For

a smooth functional ¢ E H. denote by Dk ¢ (h) the k-th Frechet derivative of ¢ at h E H.

Let (e

k)

be a complete orthonormal system (C.O.N.5.) in H with ek's E V.

If f(x l.x 2 ••••• x k) is a real function in k variables, a smooth simple functional

= i=I.2 •••• k ,

(v,e n ),

We let 5 be the set of all smooth simple functionals on H.

i

which

grow at most like a polynomial. For ¢ E 5.

as

in lli!n. we expect

the average functional

F

t

given by

(3.3)

would satisfy the diffusion equation (3.4)

¢(v).vEH.

where (3.5) Note that, since ¢ is simple. LO¢(v) is well-defined. but. in general. it is just a formal expression. v E H.

As it turns out, say,

has no meaning for ¢ E

and

To overcome this difficulity. one may want to restrict v E V and D¢ E V

as a V-derivative.

L O'

In (3.5) the term "n be the eigenvalue corresponding to the eigenfunction en of A,

and

set Am "-n

(3.10)

=......k

j=l

,

nJoAJomj'

On S, we introduce a norm

!I • liZ defined by

(3.11)

and let HZ be the completion of 5 with respect to this norm. By invoking the above lemmas, we are able to prove the following theorem. Theorem 3.1 Under the hypotheses of Lemmas 3.1 and 3.2, the transition operators {P t}

defined

by

(3.2)

forms

infinitesimal generator LO of self-adjoint operator on H 2•

a

strongly

continuous

the semigroup defined on

semi group 5

on

H.

The

can be extended to a

Furthermore we have

(3.1Z) (Proof) •

By Lemma 3.Z,

it

suffices

to consider

indicate how the proof goes, let us take

the Hermite functionals.

To

48 Then, by the chain rule, we get

where 8 By a direct computation, it is easy to show that tr.

[R 0

2

(v))

On the other hand,

so that

[h n" (8) - 8 h'(8)]

which verifies the equation (3.12) in this special case. By the definition (3.2), we have

Since u 8t

t

is Markovian, the semigroup property is obvious.

X ), when D(T) is dense in H (similarly if TEL(X,Y), are Hilbert spaces). We shall set E(X) = {TEL(X) > 0 for all X(X}. adjoint operator in X, powers of A (cf.

when X and Y

; T=T*}, and

E+(X)

If A is a strictly positive self

then we shall denote by Aa,

for instance [15]). Given aO will be specified later, all

2

x

(f), where

n

L D.c .. i,j=l J lJ 1

°

-

version (3.4) of (1.1) condition aa'

b a EC

1

xr

A

1

2::

f

d

I a 1:s.1

and we define D as

=

Let us

a aD f

+ Af ,

for

is the solution of the elliptic problem in e,

on f.

Thus we consider the abstract

under this new notation.

Al'

We first note that the

y,

2

implies A B, B* EL(D(A'),H), where H=L (e) 1, statement for A* and B* follows from the Green formula); then, from 1 -Y, Y, remark 2, the extentions (A A and (A-'B) exist and are unique. Let us 1) verify (3.6), which is equivalent to (3.5) (from remark 2). Since: (8)

68 n

B*f

n

L

b,D,f 1

i=l

+

1

I:

(D, b , )f 1

i=l

b

+

1

o

f

if we replace B by B* in the proof of corollary I, follows that (3.6) hOlds for some D [O,l[

then from (2.19) it

and A>O (A appears in the above

definition of A). Then we may apply theorem 2, which yields the desired conclusion.

0

We study an optimal control problem for (3.4)

in the abstract set

ting of theorem 2. Under the hypotheses of corollary 2,

these results

can be applied to the concrete system (1.1). Thus, throughout this sec tion we assume that the hypotheses of theorem 2 hold. Consider the problem of minimizing (3.7)

J (u)

over all UE:M

+ E a and such t:hat:

and uniformly in

{An Cs , to )

w E Q.

}

converges in

Then t:he sequence

K point:wise in 2,

{b

t

A

(dB)

n(a), n

of product: int:egrals Is Cauchy in PROOF: p •

For any pair of integers

{a • So


, let us put

a

and any partition

l}

89

D(A ,A ) ., m

n

I 0

A

t

I

m(C1) -

t

o

(where remember that the tensor products are to be read off from right to left).

By Lemma 1.3,

k

k

9 Am(si _1)6(I i ) - 9 An(si _1)6(I i ) r r reI r r r al k E

j el

k 9

r ej+l

Am(si _1)6(I i )[Am(si._l)6(I i.) - An(si._l)6(I i.)] r r J J J J j-l 9 An(si _1)6(I i ) r-l r r

Therefore

j-l

n

r-l

CM

2k(Ii

r

k( I. ) 1

r

90

and hence

The hypothesis on the convergence of

2.10 A (s,oo) n

DEFINITION.

K 2

n--

Let

A(s,oo)

A's n

completes the proof now.

A: [O,t] x Q

into

in

and uniformly in

s E [O,t]

K 2

such that

2.9 justifies defining the seochaseic produce ineegral of

3

D

A

00

E

n.

Theorem

a

w.r.t

by

MULTIPLICATIVE OPERATOR FUNCTIONAL OF OF A SEMIMARTINGALE

We chose above the Brownian motion case to expound our method of defining a stochastic product integral.

With some extra work and assumptions, one can

define product integrals w.r.t

MOF's associated with certain semi-martingales.

Following the idea we set forth for a Brownian motion, we will introduce, in this section, the MOF

X

associated with a semi-martingale

analyzing the properties of this MOF representation of

X

and show that

X,

X

In

we first establish a Peano series

XE

This series representation

enables us to establish a stochastic version of Kato-Trotter formula; as a special case of our stochastic Kato-Trotter formula we derive the corresponding well-known deterministic theorem.

Of course, the wide

applicability of the standard Kato-Trotter formula is well-known.

However,

the full potential of our stochastic version is not clear to us now because we are just beginning to understand it and exploit it; these investigations will

91

Xe

appear elsewhere. Since and MOF

Y, X-I

X·y

the product

e

Finally, we shall construct an inverse

X.

of

The semi-martingale X· M + V.

form

X

' we can define, for two such MOF's

X

that we will be working with from now on is of the

The following basic assumptions on

M and

V

will be in

force throughout this and next sections. 3.1 ASSUMPTIONS.

(A)

M

is a K martingale such that 2-valued

EUM(t) - M(s)U

where

G

2 • G(t) - G(s) , K 2

(1)

is a right continuous function which is locally of bounded

variation. (B)

V

is a K - v a l u e d cadlag process that is locally of bounded 2

variation. (c)

(C')

The processes

M and

are independent.

The independence assumption between

v*(.).

provided we assume that

of bounded variation, where (D)

V

For each

0

t t

E{J o

IVI

M

and

V

can be relaxed

is a right continuous function is the total variation process.

and

dIVI(s)}2 exp[a

2

{J 0

t

dIVI(s)}]

2


0,

X E

B

= f(x), Let

is the Gross Laplacian defined by

denote the Wiener measure on tion

(H,B):

f

defined on

B

with variance

t.

For a bounded Lip-1 func-

B, the function

u(t,x)

satisfies Equation (5)

[1,8J.

Is it possible to obtain

u(t,x)

by the same

argument as in the finite dimensional case using the Fourier transform technique?

Obviously, there are several difficulties in generalizing the Fourier

transform in Equation (2) to the infinite dimensional case

1 00 (---) = /2; not

0, exp[-iJ

exist.

However,

it

k

= 00,

namely,

does not make a reasonably good sense, and is

shown

in

[9J

that

the

product of

dx

these

does three

factors 1 )00 exp[-i Jdx (---

/Z;

makes sense in the Hida calculus. Fourier transform.

Thus we can define the infinite dimensional

The purpose of this paper is to show that Equation (5) can

be solved by using this Fourier transform. be

used

to

solve

other

constant coefficients.

infinite

We remark that this technique can

dimensional

differential

equations

with

156

12.

The Hlda calculus

Let IR.

be the Schwartz space of rapidly decreasing smooth functions on

The dual space

,J

*

of

,)

consists of tempered distributions and has

the standard Gaussian measure 2

C(!;;) = exp[ -1I1;;11 / 2J , I;;

,)

B(t,w)

,

with the characteristic functional where

norm.

11·11

__ { dip cm a

Wom,p,a+r with

We recall also the following imbeddings (see Adams [1]). C) For every jp > d

II f II m.p,n +r

Km,p,a,r II f II m,oo,a

178

II f II m,oo,a s K II f II m-j.p,a

with

for some constant K

D) If j > dJ2 and r > dJ2 the linear imbedding of Hilbert spaces Wom+j,2,a '-7 W om,2,a+r is a Hilbert-Schmidt operator. As a consequence we can consider the following chain of imbeddings: for a > dJ2, m > dJ2, j > dJ2 and all r > 0 and I

O.

(3.1)

the first mapping being Hilbert-Schmidt (therefore compact). We have also the dual chain of imbeddings Ma+r y

W o-m,2,a+r y

Wo-(m+j),2,r.

IV - Tightness of a sequence of dynamical systems N,l - The sequence of systems considered As often done, when considering dynamical systems with a small parameter (here lIN), one considers a sequence of such systems indexed by N and see whether they have a "limit" when N

00,

under a

suitable "normalization". We assume therefore that we are given a sequence (11Nt)N 0 of positive measure-valued processes. For each N and each positive measure u , IJN( u ) is a linear mapping from:;; (the linear space of C"'-functions on Rd, with compact support) into :jj and a bilinear mapping aN( u ) from into

V x :r;

'1).

We assume the following properties for (fl N),(bN),(a N).

II bN( fl )u II s K II u II

[H.1]

CO

Cia

We call bN* ( u ) the transposed continuous linear operator from M a into ca-I c WO-(I +j),2,a [H.2] For every N, fl

E

Ma , u

E

cm a v

E

for j > dJ2.

C ma

s

K

II

fl lI a

II u IIm,oo,a

II v II m,oo,a

[H.3] The paths of the processes flN are right continuous, with left limits, as functions from R+ into the dual of CIa = C-I a Y

W-(I+d/2+ e),2,a.

We set 6 t flN = flN(t) - lim flN(s).

sit

179

[M] (i) For every u

E

'1J t

= + f ds + MtN,u ,

o

MtN,u being a real valued martingale such that t

iMN,u,MN,vi- =

(ii)

f

ds ,

o

denoting by {Nl,Wi- the Meyer-Process of two real martingales (Nl,W) (see for ex. P.A. Meyer [11]). Remark - Properties [H.l], [H.2] and [M] for the processes of example 1 and 2 in section II are clearly true as a consequence of the formulas (2.6), (2.7) and (2.10), (2.11). The property [H.3] in these (lx E C- 1a and the fact that the jumps of J.lN are

examples follows from the continuity of x isolated. IV.2 - Theorem 1.

Let us assume the hypotheses of IlL 1 and moreover: [H.4] For u a (x) := 1 +

I x Ia

some constant K and all J.l

E

MB

bN( J.l )u a (x) ::; K ua (x)

[H.5] The measures J.lN t are a.s. measures with compact support and sUPN Let us set v = (d/2 + I ) v (d + m) and write

PN for the law of the process

II J.lNO lI a
p} = 0, for every p > 0, then P is carried by

t::;T

C(O,T;MW b).

Proof - We start with a lemma. Lemma 1 - Under the above hypotheses there exists K such that for any N and any stopping time 'tN of the process J.lN: sup Ell (0) t::;T

N

II J.lN t 1\

1:

N

II-a::; II J.lN(O) II-a eKT

Proof - Using [H.3], the fact that J.lN t is with compact support and [M] (i) one obtains easily for every

-s r:

180

t

E«u Since IlNtJ\t

N

a' IlNtJ\

t

»

+ K J

E 0 and 'Il > 0 there exists 0, such that for every family ('tN) where 'tN is a stopping time of IlN, 'tN

T: - Il N II -v,2,0 > 'Il}

I { II Il N r

+8

"[

E

\oJ

But, by a trivial extension of a Rebolledo-result for the finite dimensional processes ([13], see also [8J for an exposition), condition c) is implied by the following condition: c') For every E > 0 and 'Il > 0 there exists 0, such that for every family ('tN)N>O where stopping time of IlN, 't N

T:

't N +8

(i) sup 8

P( III 'tN

s0

1JN*(lls)·ll sds II

\oJ

-vz > n } '

't N +8

(ii)

sup

P (f 't N

trace -v 2 aN( Il s)ds > 'Il} \oJ

'

E

E

't N

is a

181

We use then the fact that the embeddings (cf. TIl)

Ma y

W-v,2,O

for

s>d

is Hilbert-Schmidt (and therefore compact) to see that a) and b) derive immediately from Lemma 1. Now, making use of [H. 1] and the Markov Property of processes

IlN we can write for j > d./2

't N +8

EN II I IlN 't N

Il s)' Il s ds

11-(1 +j),2, a

o

8 E NEil 11 0

(III TN

Il s)' Il s ds

0

11-(1 +j),a) s E N (K 8 Ell N II Il s lI a) . 11 0

T

Using now Lemma 1 we get for all N 't N+8

Il s ds 1I-(l+j),2,a

EN N III

IIIlN(O) lI a eKT

't N

110

The hypotheses [HA] gives then immediately c')i).

If we observe that [H.2] and the imbedding (3.1.C) and (3.1.D) give for j > d II aN( 11 ) II

o (WO(ffi+j),2,O,WO-(ffi+j),2,O)

s

KlIlllia

(where O(W,W) denotes the space of number operators from W in W. We obtain c)ii by the reasonning already made to obtain c')i). Nk To fmish the proof let us consider any convergent subsequence (Il )Je() of

For any

T

h

E

C(T,R) and u

Coo, U

E

D(0,T;W-v,2) which is

0, h

0,

J h(s) < u, o

(s) > ds is a continuous function on

t k a.s. positive. Therefore if P is the limit of the laws pN kon D(O,T;W-v,2) T

o I

o

This implies for

v .

h(s) < u,

(s) > ds

P a.s.

(t) to be a positive distribution and therefore a measure P a.s. for almost all t.

Consequences

The Theorem 1 applies clearly to the sequences IlN considered in the examples 1 and 2 of section II. Example 1 - Case 1: v = (d./2 + 2) v(d + 1)

182

Any limit measure P of the sequence pN on D[0,T;W-v,2] is carried by C[0,T;W-v,2] and has the martingale property: t

0,

let us consider a finite grid G of equispaced points of h

distance h, which tends to cover the whole line as h + O. and define a Markov chain on G by the following non-zero intensities: h a\x.x-h)

h

a (x,x)

=

=-

for xEG h

_I_a(x) + -hI b-(x) 2 2h 1 - - a Cx) 2 2h 1

1

h

Ib Cx) I

ah(x.x+h) • - - a(x) +1.- b\x) h 2h 2

(6.2)

226 except for the first and the last point of the grid, which are made absorbent. Let

'b be

the inclusion of G into R, and let Ph be defined as in Section 4. h 2 It is clear that condition (2.6) is satisfied once we show that for f£C 1 o2f af sup 1"2 a Cx) + b Cx) a; xn ax

-::z-

1 1 1 1 (-2 a(x) + h b (x ) )f(x-h) + (-2 a(x) + h Ib(x)

-

Th

Th

I

l - (_l_a(x) +-h b+(x»f(x+h)I < O(h) 2 2h

where O(h) goes to zero as h

O.

+

f Cx)

(6.3)

The behavior at the boundary is controlled by

the boundedness assumptions on a and b and the fact that f£C

2•

The expression

of the r.h.s. of (6.3) can be rewritten as

2hL

(f(x+h) + f(x-h) - 2f(x) - f"(x)h

2»

+

+U& (f Cx-l-h ) - f(x) - f'(x)h) + h

- b-(x) (f(x-h) - f Cx ) - fl(x)h) h

which clearly shows uniform convergence (f" is in fact uniformly continuous). 2

Moreover, for any gee;, the boundedness condition (4.16) is verified, and Corollary 4.1 and Theorem 5.1 can be applied. Such method can be extended to the case RO, d tions on the coefficients l16]. is still straightforward.

>

1, with additional assump-

The verification of conditions (2.7) and (4.9)

It is clear that the method could take into account

boundary conditions, too. Example 2.

This rather artificial example serves only as a sample to show

that reasonable alternatives to the previous space discretization scheme exist, even in dimension one.

Of course, this is much more true in higher dimensions,

given that the complexity of the topology of a grid increases. a

=1

in (6.1), and write b

-av/ox.

Suppose that

Usually, it will be easier to compute the

"potential" V than its derivative so that it makes sense to define the following approximating chain, holding fixed the grid G as before: h

227

:z (t 1

--h

a (x,x-h)

{

1

;: _1 h2

ah(x,X+h) = { 1

2h h

2

+ Vex) -

otherwise,

(l.. 2

(6.4)

+ vex) - V(x+h»

, if V(x+h)

h

(including the boundary ones).

h

V(x-h)}

and the other terms to be zero

Condition (2.6) is reduced to checking

l(V(x) - V(x:!:h) )f(x±h) - (V(x) - V(x:!:h) )f(x) J +

'2

for each f £C , where 6(h) goes to zero as h

+ O.

f(x)

I'

6(h)

This is because, when the

V-terms in. (6.4) repeatedly disappear around x, as h ( 3V/ ax) (x ) = O.

< min!V(x),

otherwise

letting a (x,x) = -(a (x,x-h) + a (x,x+h»

sup x£R h

if Vex-h) , min[V(x), V(x+h)}

+

0, it is necessarily

2 This allows to prove the boundedness condition for any g£C b,

so the convergence property of the filtering algorithm derived from (6.4) is the aame as in the previous example.

The author wishes to thank Professor J. S. Baras for his valuable guidance during the realization of the present work.

REFERENCES 1.

J. S. Baras, A. La Vigna, "Expert systems and VLSI architectures for real-time non-Gaussian detectors and filters", in C. Byrnes and A. Lindquist ed s , , Proceedings of MTNS-85, North-Holland, to appear.

2.

Ya. I. Belopol'skaya, Z. I. Nagolkina, "On a class of stochastic partial differential equations," Th , of Pr ob , and its Ap pl , , 27, 592-599, 1982.

3.

P. Billingsley, Convergence of Probability Measures, John Wiley, New York, 1968.

228 4.

P. Billingsley, Weak Convergence of Measures: SIAM, Philadelphia, 1971.

Applications in Probability,

5.

J. M. C. Clark, "The design of robust approximations to the stochastic differential equations of nonlinear filtering," in J. K. Skwirzynski ed., Communication Systems and Random Process Theory, Sijthoff and Noordhoff, Aalpen aan den Rijn, 1978.

6.

E. B. Dynkin, Markov Processes ­ I, Springer­Verlag, Berlin, 1965.

7.

A. Cerman l , M. Piccioni, "A Galerkin approximation for the Zakai equation," in P. Thoft­Christensen, ed., Systems Modelling and Optimization, Springer­Verlag, Berlin, 1984.

8.

I. I. Gihman, A. V. Skorohod, The Theory of Stochastic Processes, III, Springer­Verlag, New York, 1979.

9.

E. Hille, Functional Analysis and Semigroups, AMS, New York, 1948.

10. J. Jacod, Calcul Stochastique et Problemes de Martingales, Springer­Verlag, Berlin, 1977. 11. G. Kallianpur, C. Striebel, "Estimation of stochastic systems: arbitrary system process with additive white noise observation errors," Ann. Math. St at ; , 39, 785­801, 1968. 12. T. Kato, Perturbation Theory for Linear Operators, Springer­Verlag, Berlin, 1976. 13. T. G. Kurtz, "Extensions of Trotter's operator s em.l g r oup approximation theorems," J. Func t , Ana l , , 3, 354­375, 1969. 14. T. G. Kurtz, "Semig r oups of conditional shifts and approximations of Markov p r oc es s es ;" Ann. Pr ob ; , 4, 618­642, 1975. 15. T. G. Kurtz, Approximation of Population Processes, SIAM, Philadelphia, 1981. 16. H. J. Kushner, Probability Methods for Approximation in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977. 17.

H. J. Kushner, "A robust discrete state approximation to the optimal nonlinear filter for a diffusion," Stochastics, 3, 75­83, 1979.

18.

H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, MIT, Cambridge, 1984.

19.

H. J. Kushner, H. Huang, "Approximate and limit results for nonlinear filters with wide bandwidth observation noise," Report LCDS 84­36, 1984.

20.

F. Legland, "Estimation de parametres dans les processus stochastiques en observation i ncompl ete," These, Uni ve r s I t e Paris IX, 1981.

21.

T. Lindvall, "Weak convergence of probability measures and random functions in the function space DlO,oo)," J. Appl. Prob., 10, 109­121, 1973.

22.

P.­A. Meyer, Martingales and Stochastic Integrals I, Springer­Verlag, Berlin, 1978.

23.

M. Metivier, Semimartingales, W. de Gruyter, Berlin, 1982.

24.

R. D. Rl c ht my e r , K. W. MOrtOll, Difference Methods for Initial­Value Problems, Interscience, New York, 1967.

229

25.

B. Stewart, "Generation of analytic semigroups by strongly elliptic operators," Trans. Amer. Math. Soc , , 199, 141-162, 1974.

26.

D. W. Stroock, S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979.

27.

H. Trotter, "Approximation of semigroups of operators," Pacific J. Ha t h , , 8, 887-919, 1958.

28.

H. Trotter, "On the product of semigroups of operators," Pr oc, Amer , Math. So c , , 10, 545-551, 1959.

29.

D. Williams, Diffusions, Markov Processes and Martingales, Vol. Wiley, Chichester, 1979.

30.

K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1968.

31.

M. Zakai, "On the optimal filtering of diffusion processes," Z. Wahr. ve rv , Geb •• 11, 230-243, 1969.

John

SOME APPLICATIONS OF THE MALLIAVIN CALCULUS TO STOCHASTIC ANALYSIS by A.S .Ustunel In this work we give some applications of the Malliavin calculus to some problems of stochastic analysis other than the study of the fundamental solutions of second order parabolic operators.ln the first part we extend the Ito Representation Theorem to the distributions on the Wiener space in the contexte of the Malliavin Calculus.The second part is devoted to the explicit calculation of the occupation density of the diffusion processes using two different methods: the first one is the application of Clark's formula for the Ito Representation Theorem (cf.[JOJ) which similar to that of

[15J

H

except that we need weaker differentiability

hypothesis; the second method is completely new and uses the theory of the stochastic partial differential equations,although we applied it to the one dimensional Brownian motion,it can be applied to much more general processes.ln the last part we solve the problem of time reversal of some stochastic processes with the help of the Malliavin Calculus. We need more differentiability conditions in the Sobolev sense on the Wiener space than

(12] for example,however it is not neccessary that the process is a semimartingale and the approach is valid also for infinite dimensional processes since we do not make any use of the Lebesgue measure on the state space . I.Notations and Preliminaries In the fo llowing we denote by.D, the Wiener measure on

and H

the Banach space

C( [0, I], [Rd),

.tt

is

is the Cameron-Martin space so that

,H,j:l) is the usual triple of the abstract Wiener space (cf.[4J) . A represents

the infinitesimal generator of the Ornstein-Uhlenbeck process

with values inA

(cf ,

[JI]).If

K is a separable Hilbert space, D

p,

I(K)

is the Sobolev space of K-valued Wiener functionals with the norm topology

I

f

II (I-A) 1/2 f II L!'(b K) ,if

K

IR then we write simply

D I D(K) p, and D' (K)

denotes the projective limit of { D I (K);p E. (I ,CO) ,IE p, is its continuous dual. On we use the canonical filtration of

tc4

Brownian paths and denote it with

{'rt; ;-tE.[o,i])

of

.

is the subspace

D k(H) whose elements have adapted Lebesgue densities to the above p, filtration with the induced topology. 2) is the projective limit of these spaces and

:J)'

is its continuous dual

231

II.An Extension of Stochastic Integration and ltD Representation Theorem 2

L (f ) with (F ,I) =0 ,the Ito Representation Theorem says that

If F

there is a unique

"W F

r

2 L (dtxd f

) ,adapted

such that

f

F Note

that

o

0WF

(aWF)(s) dW

s

is the adjoint of the Ito's i.s omec i y r or the stoc-

hastic integration. Key Lemma Suppose tha t

0 1 t

"2 !

o

6

to

u(·)

K 1

O.

be a bounded, closed set in-

>0

a number such that Then there exists

transferring

aEK

I u (s) I

B(O,r)

and all

L=B(K,6 ) 2

>0

1

such

to

b\tB(K,6 1 )

K

contains a

one has 2

ds

6 2.

Proof. Since (H1) holds therefore one can assume that ball

6

x E I,

for some

I ZX (t) I

r>

i.

o.

One can also assume that for some

t> 0

If the conclusion of the proposition is c, not true, then there exist elements a EK, b EL positive numbers


0,

O

2

>

0

such that for

n,

245

3)

y(n) cD

and distance

Proof. t ) Let

r

>r

3

x

set

>r, >0

2

K = {z ) z = z (t)

in

B(O,r 2 ) .

(y(n),aD)=O. be numbers such that

for some

x E B (0, r, )

Proposition 2 there exists

x y (T)CB(0,r ) .

xEB(O,r,),

0 >0

K=y (n)

o>

°

y(?O

Consequently appli-es to

exist xEK

K

s > 0,

n

T) > 0.

3

>r

T) < 0

2

>r, >0

and

and

o > 0,

T) < n ,

Assume first that

there exists

cB(y(r!l ,0)

and

f;


°

y (n) cD

T\

if

Then the

°

such that




°

and we see that (R2)

M>

°

and

holds true.

holds as well. Final­ 2 m L,j': L [0,+00; R ]­>E

and consider transformations

given by: T f

S(t)au(s)ds,

° 1'hen 'l't

+00

Y(n)=closure {x; X=LooU'

l' < +00.

1 +00

'2

f [u t s )

°

2

I ds .::.. n J .

in the operator norru and opera tors

L'l'

Since

L'l'­>L

as

are compact as uniform

limits of finite dimensional operators, therefore operator

L oo

is

compact as well. If

is a Hilbert space, an explicit formula for

E

y(n)

can be

given. It was announced in [10]. Here we give its detailed proof based on the following Lemma 1. For arbitrary

operators

where

and

'1'

= f S(t)aa*S*(t)dt,

Q

°

T

have the same range. )Vioreover if satisfying

Q'l'b=a

II

u(s) = a*S*('.i'­s)b,

transfers

to

0

rJ.'

1\

a

E

Range

and

b

with the minimal norm then control

a

is an element

"u :

s.::..1',

with the minimal norms

2

flu (s) I ds

°

-1

.

Proof. The proof that Range

uo].

Q1/2 = Range L can be found e.g. in T T /I By direct computations one checks that L'l'u=a and that for

arbitrary control

u(·)

transferring

0

to

a

in time

1':

248

T

fds

o

.

This implies that T

f !u(s)

o

I

2

'I'

ds

I

o

2

rl'

2

A

ds + f!u(s)-u(s)! d s , 0

so the results follows. Lemma 2. As

Tt+ oo,

QTtQ oo

and

Qoo

is the unique non-negative solution

of the following equation ( 18)

2 +

Moreover

Q

I o*x I 2 =0,

for

xED (A*) .

is a nuclear operator.

Proof. See [8J. Proposition 5. Under the assumptions of Theorem 3 and for Hilbert space

E, 1

y(n) = closure {xERange Qoo' where

Qoo

x,x> < 2n>},

is the unique solution of (18).

5. Exit theorems for delay equations.

E=C[-h,O; Rn],

As in the introduction and

F

for some

a transformation from

K> 0

and all

\P

implies that for arbitrary (20)

1

E

,tp2

1-

tp E E

into

L.

o

has a unique solution

for x(·)

h

is a positive constant such that:

Banach and

t x(t)=tp(O) + f F(Xs)ds + f(t),

x(e)=t:.>(e),

n R

f E Co

fixed point theorem easily the following equation

t>O

eE[-h,oJ which we denote by

It is

249

X satisfies conditions (i) and (ii) of the definition of the solution map. have also the following immediate that

Proposition 6. For arbitrary

(j)

1

,(j)

2

E E,

t.: 0,

and

f 1 , f2 E C a

Proof. If

x

i

(t)=(j)

i

(0)

t

+ f

o

i F(x Ids +

(t),

s

t.:O,

i=1,2,

then

t

x1 +Kfll o s

lids,

t> 0.

Therefore

t + K f

o

II

lids

and the Bellman-Gronwall lemma implies the result.

Z

Thus

is a solution map.

Proposition 7. If for sone y(n)

y (n)

n.: 0,

E,

then

is compact.

Proof. vie show that. functions from x(e)=o, for e E [-h,O], t

x(t) where

is a bounded set of

1

"2

bitrary

+00

f \ u (s)

°

f

o

I

2

t F(Xs)ds + f u(s)ds, 0

ds

n

and

are equi-continuous. If

y (n)

t.:O,

sup{!F( 0,

then

oEy(n)}

< M

then for ar-

250

IX(8)-x(t)

Therefore, for

I

tM + 2/t n

and arbitrary

(j) E y (n)

equi-continuity of elements of

+ 2

Mlt-8! 8

1

,8

2E

n.

(-h,O]:

follows now immediately.

yIn)

The following proposition is a consequence of Theorem 1, Theorem 2, Proposition 6 and Proposition 7 Proposition 8. If

F

and hypotheses (H1)

is a Lipschitz transformation from

into

n R

and (H2) are satisfied them formulae (12)-(14) are

r

valid. In the formula (14) sets hoods of

r

E

can be defined as

6

6-neighbour-

y(n)ndD.

Note that in the formulation of Proposition 8 set D is a subset of E and not Rn. We will derive now from Proposiiton 8 an exit theorem for delay systems in terms of the final dimensional reference n set GeR • To do so let us fix arbitrary positive numbers r 2>r 1 such that

Gr

Define moreover for arbitrary

G

r

r

0

an infinite dimensional set

{(j) E E;

=

I (j)( e)

r

=

{(j) E E;

I (j)(8) I

and either Lemma 3. Let n into R • If

r

II

h


0

u( 0)

n, R

whereas

Let

r >r2 >r1

be numbers such that (21) holds. If hypor D=G then

and IH2) are satisfied for

(n)

(iii)

as

ds:5. n l •

n

(ii)

As-

t E (0,T+0) ,

°

lim £2 l n JE(-r(j),£) < sup{n.::. O; r ( n l 5G} = £+0 lim £2 l n JE(-r(j),£) .::. sup{n.::.O; r In) cG} =21 £+0 r Define L n ac , For arbitrary 6 >

(i)

and

oc t-u.or ,

is a finite dimensional set, a subset of

E.

Since

In the latter case

imply that for some

r

r In) =closure{y ERn; y=yO,u l t) such

o E t-i.,o)

for all

as needed in the lemma - or

.pt(O)E G.

and

< r

for all

r (j) E G 1

for all

(j) E (G

°

and

r1

r (j) E (G 1)

0

) 0' :

lim lP(distance (X(j),£ (-r(j)'£),n < 6) =1£+0 Proof. Theorem 4 follows from Proposition 8 and Lemma 3. Specific conditions which imply (H1) and (H2) will be given in Now we apply Theorem 8 to systems for which mation ( 22)

and

°! A (de) (j)( e),

-h

is an

n x n

(j) E E,

matrix-valued measure on

It turns out that sets To see this let

144],

§5.

is a linear trans for-

F=L, L (j)

A

F

rio)

[-h,O]

of bounded

are ellipsoid, see also [9].

be the fundamental, matrix solution, see [5, p.

of the equation

252 (23)

X{t)=L{X

x{O)=\p{O),

t),

t,::O,

0 E: [-h,O].

Define also (24)

o

PO-)

AI -

p t x)

det P ( A),

f A{dO)e

1..0

-h

Ae

e,

Proposition 9. Assume that all zeros of

p{.)

have negative real

Q:

parts and define matrix

'.chen

r

Proof. Assumption about M > 0,

1

(T) ={y E: Range Q;

a > 0

p(.)

y, y> .:::. T)} •

is equivalent to the existence of

such that

see [5, p. 1 82]. I f +'"

q>{w)

then

eiwtq,{t)dt,

f

o

q,{w)=P{-iw),

WER

1•

1

+.00

°

f q,(t)aa*q,*(t)=2 'IT

Plancherel's identi.ty gives +co

f (P(-iw»-1 a a*(P*{iw»-1 d w•

An argument very similar to the one used in the proof of Lemma 1 implies the result. Example 1. Consider delay equation in R 1 dX(t) and let to:

a


O.

where

°

Hypothesis (H1) < C
0,

general definition operator

for all sufficiently small

I-A A

is onto

E.

According to the

is dissipative if and only if

A> 0

and all

l0,Vi E ]) ( A ).

Disspativity

together with the onto property is equivalent to the contraction property of the semigroup

T(t),

t>O

generated by

(23), see [1] and

[7]. Dissipativity condition (29) can be formulated equivalently in the following way. Lemma 4. Dissipativity condition (29) holds if and only if for all I$,ViE D(A)

such that

[1$(0)-Vi(0)\'>!1$(8)-Vi(8)!

for all

8E[-h,0),

one has

(30)

(F(I$)-F(Vi),

-h

1

>

-h

2

> ••• >

(Aoa,a) +

(ii)

n= 1

If >

r

>

r

j=1

then

F