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Remarks on the existence and approximation forsemilinear stochastic differential equations in HilbertspacesY. El Boukfaoui a & M. Erraoui aa Département de Mathématiques , Faculté des Sciences et Techniques Guéliz , BP 618,Marrakech, Moroccob Département de Mathématiques , Faculté des Sciences Semlalia , BP 2390, Marrakech,MoroccoPublished online: 15 Feb 2007.

To cite this article: Y. El Boukfaoui & M. Erraoui (2002) Remarks on the existence and approximation for semilinear stochasticdifferential equations in Hilbert spaces, Stochastic Analysis and Applications, 20:3, 495-518

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REMARKS ON THE EXISTENCE ANDAPPROXIMATION FOR SEMILINEAR

STOCHASTIC DIFFERENTIAL EQUATIONSIN HILBERT SPACES

Y. El Boukfaoui1,* and M. Erraoui2,†

1Departement de Mathematiques, Faculte des Sciences et

Techniques Gueliz BP 618, Marrakech, Morocco2Departement de Mathematiques, Faculte des Sciences

Semlalia BP 2390, Marrakech, Morocco

ABSTRACT

In this paper we first shall establish an existence and uniqueness

result for the semilinear stochastic differential equations in Hilbert

space dX ¼ (AX þ f(X ))dt þ g(X )dW under weaker conditions

than the Lipschitz one by investigating the convergence of the

successive approximations. Secondly we show, under the

assumption of so-called pathwise uniqueness (PU ), the conver-

gence of the Euler and Lie-Trotter schemes in L p, p . 2 and the

continuous dependence of the solutions on the initial data and on

the coefficients for such equation. Finally we study the existence

of the solutions when the coefficients f and g are only defined on a

subset of the state Hilbert space.

Key Words: Stochastic evolution equations; Pathwise uniqueness

495

Copyright q 2002 by Marcel Dekker, Inc. www.dekker.com

*E-mail: yelboukfaoui@ucam.ac.ma†Corresponding author. E-mail: erraoui@ucam.ac.ma

STOCHASTIC ANALYSIS AND APPLICATIONS, 20(3), 495–518 (2002)

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1. INTRODUCTION

It is well-known that a wide class of stochastic partial differential equations

and stochastic delay equations can be described by the stochastic differential

equation

dXðtÞ ¼ ðAXðtÞ þ f ðXðtÞÞÞdt þ gðXðtÞÞdWðtÞ

Xð0Þ ¼ x:

(ð1Þ

Here A is the generator of the C0-semigroup e tA, t $ 0 on a real Hilbert space H

and W is a Wiener process on U with a covariance operator Q defined on a

complete probability space (V, F, (Ft)t$0, P ), where U is another Hilbert space.

For simplicity of presentation we can assume without any restriction that H ¼ U.

Let HT be the Banach space of all continuous Ft-adapted H-valued processes X

defined on the interval [0, T ] such that

jXjT ¼t[½0;T�sup EjXðtÞj

2

!12

, 1:

By a mild solution of Eq. (1) on [0, T ], we mean a stochastic process

X(·, x ) [ HT such that for t [ [0, T ] we have

Xðt; xÞ ¼ e tAx þ

Z t

0

e ðt2sÞAf ðXðs; xÞds þ

Z t

0

e ðt2sÞAgðXðs; xÞdWðsÞ: ð2Þ

It is well known that the existence and the uniqueness of mild solution of Eq. (1)

is guaranteed under the condition that f and g are Lipschitz using the convergence

of successive approximations. Recently, many results on the existence and

uniqueness of a mild solution for Eq. (1) have been obtained without assuming a

Lipschitz condition see [8,9]. However, in many case the existence and the

uniqueness of mild solutions are proved by methods different from the successive

approximations. So, it is important to find some weaker conditions than the

lipschitz one under which the Eq. (1) has a unique mild solution by investigating

the convergence of the sequence of stochastic processes defined by the successive

approximations. The first aim of this paper is to show the convergence of the

successive approximations scheme when f and g satisfy some Caratheodory-type

conditions. In particular we can see that the Lipschitz condition is a special case

of our proposed conditions. On the other hand we obtain the existence and

uniqueness result of the mild solution of Eq. (1) under weaker assumptions than

those proposed by other authors [8,18]. Recall that in the finite dimensional case

such problem has been studied for different equations recently by several authors

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see [12,25,27,28,30]. This paper can be regarded as an extension of some of the above

results to the Hilbert space.

As we know, in general, the existence of a mild solution of Eq. (1) on a

fixed probability space is rather strong. So, we look for a weaker concept, that is,

the existence of a weak solution which we state briefly as follows:

For a given data H, U, Q, A, f, g, x0 we look for a probability space

ðV; F; PÞ; filtration {Ft}, cylindrical Ft-adapted Wiener W process and Ft-

adapted stochastic process X which is a mild solution of Eq. (1). The sequence

ðV; F; P; {F t}; W ; XÞ is called a weak solution of Eq. (1). For such formulation

the reader can see [2,13,14,20,24,29].

In particular, D. Gatarek and B. Gołdys in [14] prove the existence of a weak

solution of Eq. (1) when A generates a compact semigroup and f, g are continuous

and satisfy the linear growth condition. Using the factorization method introduced

in [7], they show the tightness of the laws (L(Xn)) in C([0, 1] ; H ), where {Xn,

n $ 0} are constructed by the usual Euler scheme. This allows them, by the

Skorohod embedding theorem and the representation theorem for martingales, to

construct a weak solution ð ~V; ~F; ~P; { ~Ft}; ~W; ~XÞ such that the process X has the

distribution m which is the weak limit of a subsequence of (L(Xn)). Unfortunately,

under these circumstances, we can not ensure that the sequence Xn converge in L p,

for every p . 2: The difficulty here is that we are not able to prove the almost sure

convergence of the whole sequence Xn. Now the question is: Under which

conditions the sequence Xn converge in L p, for every p . 2? Another interesting

question: Does the solution depend continuously, in an appropriate sense, on the

initial data and on the coefficients under the above assumptions?

A second aim of the present paper is to give a positive answers. We shall

propose the pathwise uniqueness assumption (PU ) (see definition below) for Eq.

(1). First, we prove the convergence for the Euler and Lie-Trotter schemes when

the assumption (PU ) holds. It should be noted that such problem has been studied

by Kaneko and Nakao [19], Erraoui and Ouknine [10,11] and Gyongy and Krylov[16] for finite dimensional SDEs. Recently, using such idea Nualart and Gyongy[17] prove, under the assumption (PU ), the convergence in probability of an

implicit scheme for stochastic parabolic partial differential equation. In a recent

paper [23], Liu has proved the convergence of Caratheodory approximation of

stochastic delay evolution equation in Hilbert space under the pathwise

uniqueness assumption. The result presented here can be regarded as an extension

of some of the above results to the Hilbert space. Secondly, we discuss the

problem of dependence of solutions on initial data under assumption (PU ). We

also note that this problem has been studied under Lipschitz conditions see [8].

Recently it is shown in Seidler [26] that, under the weak uniqueness of Eq. (1), the

weak solution depends continuously on the coefficients in the sense of weak

convergence of laws. Here we assume more, that is the assumption (PU ), but in

return we can get more, that is the continuous dependence in the L p sense, p . 2:

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At the end of this paper we consider the Eq. (1) with the coefficients f and g

are defined only on a subset E of H. Our aim is to prove the existence result of

such equation. It should be pointed that the obtained existence result may be

applicable to some stochastic PDE’s in domains of Rd and with discontinuous

coefficients as well. The method we apply to prove the existence of the solution

may be viewed as a combination of a version of Liapunov’s method and a

procedure proposed in [14] for the existence of weak solution. The difficulty here

is that the Liapunov function used is only defined on a subset E. Moreover the

terms which appear in the Liapunov operator is not defined on the whole space H.

This difficulty is overcome, using the version of Liapunov’s method, by showing

that the solution will never leaves E, so the values of the coefficients outside E are

irrelevant. This enables us to show the existence of solution in E using the same

procedure proposed in [14]. We also note that the above mentioned version of

Liapunov’s method has been used in [21] and [22] in order to study the stability and

behavior of diffusion processes in Hilbert space.

The paper is organized as follows. In the next section we prove the

existence and uniqueness result under some Caratheodory-type conditions. In

Section 3 we prove the convergence in L p, p . 2 of the Euler and Lie-Trotter

schemes under the assumption (PU ). In Section 4 we study the variation of the

solution with respect to initial data and the coefficients. In Section 5 we prove the

existence of the solution when the coefficients are only defined on a subset E of

H.

Notations: Let us first introduce some notations which are useful in the

sequel. We write k·; ·l the inner product and j·j the norm. We denote by L(·) the

space of bounded linear operators. For each x [ H we denote by S(x, r ) the set

X [ HT ; jXð·Þ2 e ·AxjT ¼t[½0;T�sup EjXðtÞ2 e tAxj

2# r

( ):

Finally k·k2 denotes the Hilbert–Schmidt norm of an operator.

2. EXISTENCE AND UNIQUENESS RESULT

In this section we shall discuss the existence of a local mild solution of the

stochastic Eq. (1). To state the main result, let us assume that f and g satisfy the

following Caratheodory-type conditions:

(K1) There exists a function K :Rþ £ Rþ ! Rþ such that

E j f ðt;XðtÞÞj2þ kgðt;XðtÞÞQ

12k

22

� �# Kðt; EjXðtÞj

2Þ

for all t [ [0, T ] and all X(·) [ S(x, r), where r is a fixed positive number.

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(K2) K(t, u ) is locally integrable in t for each fixed u [ Rþ and is

continuous monotone nondecreasing in u for each fixed t [ Rþ.

(G1) There exists a function G : [0, T ] £ [0, 4r ] ! Rþ continuous

monotone nondecreasing in u [ [0, 4r ] for each fixed t [ [0, T ] and is locally

integrable in t [ [0, T ] for each fixed u [ [0, 4r ] such that

Gðt; 0Þ ¼ 0

E j f ðt;XðtÞÞ2 f ðt; YðtÞÞj2þ kðgðt;XðtÞÞ2 gðt; YðtÞÞÞQ

12k

22

� �# Gðt; EjXðtÞ2 YðtÞj

2Þ;

for all t [ [0, T ] and all X(·), Y(·) [ S(x, r ).

(G2) If a nonnegative continuous function z(t ), t [ [.0, T ] satisfies

z(0) ¼ 0 and

zðtÞ # D

Z t

0

Gðs; zðsÞÞds for all t [ ½0; ~T�;

where D ¼ 2M 2ð1 þ TÞ; T [ (0, T ] and M ¼t[½0;T�sup je tAj then z(t ) ¼ 0 for all

t [ [0, T].

Remark 2.1. Let Gðt; uÞ ¼ uðtÞGðuÞ t [ ½0; T�; u [ [0, 4r ] where uðtÞ $ 0 is

locally integrable and G(u ) is a concave nondecreasing function from Rþ to Rþ

such that Gð0Þ ¼ 0; G(u ) . 0 for u . 0 andR

0þ1

GðuÞ¼ 1: It follows from Bihari

inequality (see [2]) that G satisfies assumption (G2).

Now let us give some examples of the function G. Let 1 . 0 be sufficiently

small. Define

G1ðuÞ ¼u log ðu21Þ; 0 # u # 1

1 log ð121Þ þ G01ð12Þðu 2 1Þ; u . 1

8<:

G2ðuÞ ¼u log ðu21Þlog log ðu21Þ; 0 # u # 1

1 log ð121Þlog log ð121Þ þ G02ð12Þ ðu 2 1Þ; u . 1

8<:

They are both concave nondecreasing function from Rþ to Rþ such that Gð0Þ ¼

0; GðuÞ . 0 for u . 0 andZ0þ

1

GiðuÞ¼ 1:

The main result of this section is given in the following theorem.

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Theorem 2.1. Assume that assumptions K1, K2, G3 and G4 hold. Then there

exists T2 [ (0, T ] such that Eq.(1) has a unique mild solution solution in HT2.

To prove this theorem, we need to prepare a number of lemmas. We first

note that from assumption K2 and Caratheodory’s Theorem [5] the ordinary

differential equation

u0ðtÞ ¼ 3M 2ð1 þ TÞKðt; Þ; ð3Þ

has a local solution u(·, u0) with any initial value (t0, u0), u0 $ 0: Now let us

define the successive approximations as follows:

X0ðtÞ ¼ e tAx;

XnðtÞ ¼ e tAx þR t

0e ðt2sÞAf ðXn21ðsÞÞds þ

R t

0e ðt2sÞAgðXn21ðsÞdWðsÞ; n ¼ 1; 2. . .

8<:

Lemma 2.1. Assume that assumptions K1 and K2 hold. Let x [ H and u0 a

positive number such that u0 . (3M 2 þ 1)jxj2. Then there exists T1 [ (0, T] and

u(·, u0) ¼ u(·) solution of Eq. (3) on [0, T1] such that

i)

n$1sup EjXnðtÞj

2# uðtÞ for all t [ ½0; T1�:

ii)

n$1sup

t[½0;T1�sup EjXnðtÞ2 e tAxj

2# r:

Proof. It follows from assumption K1, Schwartz inequality and the infinite

dimensional Burkholder inequality that

EjX1ðtÞj2# 3 je tAxj

2þ E

Z t

0

e ðt2sÞAf ðs;X0ðsÞÞds

��������2

"

þ E

Z t

0

e ðt2sÞAgðs;X0ðsÞdWðsÞ

��������2#

# 3M 2 jxj2þ TE

Z t

0

j f ðs;X0ðsÞÞj2ds þ E

Z t

0

kgðs;X0ðsÞQ12k

22ds

� �

# 3M 2jxj2þ 3M 2ð1 þ TÞ

Z t

0

Kðs; jXj2Þds:

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It follows from assumption K2 that there exists T0 [ (0, T] and a solution of

Eq. (3) on [0, T0] with u0 . (3M 2 þ 1) jx0j2 such that

EjX1ðtÞj2# uðtÞ; for all t [ ½0; T0�:

Using the same techniques as above we can see that

EjX1ðtÞ2 e tAxj2# 3M 2ð1 þ TÞ

Z t

0

Kðs; jx0j2Þds

# 3M 2ð1 þ TÞ

Z t

0

Kðs; uðsÞÞds;

for all t [ [0, T0]. It follows also from assumption K2 and the continuity of u(·)

that there exist T1 [ (0, T0] such that

EjX1ðtÞj2# uðtÞ; for all t [ ½0; T1�: ð4Þ

and

EjX1ðtÞ2 e tAxj2# r for all t [ ½0; T1�: ð5Þ

We next assume that the inequalities (4) and (5) hold for some n $ 1. Then using

the same inequalities as above we have, for all t [ [0, T1],

EjXnþ1ðtÞj2# 3M 2jxj

2þ 3M 2ð1 þ TÞ

Z t

0

Kðs; EjXnðsÞj2Þds # uðtÞ;

and

EjXnþ1ðtÞ2 e tAxj2# 3M 2ð1 þ TÞ

Z t

0

Kðs; EjXnðsÞj2Þds

# 3M 2ð1 þ TÞ

Z t

0

Kðs; uðsÞÞds # r: A

Lemma 2.2. Assume that assumptions K1, K2 and G1 hold. Then

EjXnþmðtÞ2 XnðtÞj2# D

Z t

0

Gðs; 4rÞds

for all t [ [0, T1] and n; m $ 1:

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Proof. By assumption G1 and Lemma 2.1 we have for all t [ [0, T1]

EjXnþmðtÞ2 XnðtÞj2

# 2 E

Z t

0

e ðt2sÞAðf ðs;Xnþm21ðsÞ2 f ðs;Xn21ðsÞÞÞds

��������2

"

þE

Z t

0

e ðt2sÞAðgðs;Xnþm21ðsÞ2 gðs;Xn21ðsÞÞÞdWðsÞ

��������2#

# D

Z t

0

Gðs; EjXnþm21ðsÞ2 Xn21ðsÞj2Þds # D

Z t

0

Gðs; 4rÞds: A

Now let us define two sequences of functions (am,n(t )) and (fn(t )) on [0, T1] as

follows:

f1ðtÞ ¼ DR t

0Gðs; 4rÞds

fnþ1ðtÞ ¼ DR t

0Gðs;fnðsÞÞds; n ¼ 1; 2. . .

am;n ðtÞ ¼ EjXnþmðtÞ2 XnðtÞj2; n ¼ 1; 2. . .

8>>><>>>:

It is easy to see that we can choose T2 [ (0, T1] such that

f1ðtÞ # 4r; for all t [ ½0; T2�: ð6Þ

Lemma 2.3. Assume that assumptions K1, K2 and G2 hold. Then for any

n, m $ 1, we have

am;n ðtÞ # fnðtÞ # fn21ðtÞ # · · · # f1ðtÞ; for all t [ ½0; T2�: ð7Þ

Proof. We prove this lemma by induction in n.

It follows from of Lemma 2.2 that, for all t [ [0, T2], we have

am;1 ðtÞ ¼ EjX1þmðtÞ2 X1ðtÞj2# D

Z t

0

Gðs; 4rÞds ¼ f1ðtÞ:

Now by assumption G1 we have

am;2 ðtÞ ¼ EjX2þmðtÞ2 X2ðtÞj2# D

Z t

0

Gðs; EjX1þmðsÞ2 X1ðsÞj2Þds

# D

Z t

0

Gðs;f1ðsÞÞds ¼ f2ðtÞ;

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for all t [ [0, T2]. Using again the monotonicity of G and inequality (6) we obtain

f2ðtÞ ¼ D

Z t

0

Gðs;f1ðsÞÞds # D

Z t

0

Gðs; 4rÞds ¼ f1ðtÞ for all t [ ½0; T2�:

Now assume that (7) hold for some n $ 2, then using the same inequalities as

above we get

am;nþ1 ðtÞ # D

Z t

0

Gðs; EjXnþmðsÞ2 XnðsÞj2Þds # D

Z t

0

Gðs;fnðsÞÞds

¼ fnþ1ðtÞ

for all t [ [0, T2]. On the other hand we have

fnþ1ðtÞ ¼ D

Z t

0

Gðs;fnðsÞÞds # D

Z t

0

Gðs;fn21ðsÞÞds ¼ fnðtÞ

for all t [ [0, T2]. A

Proof of Theorem 2.1. From the Lemma 2.3. it is easy to see that fn(·) are

decreasing in n. Note also that for each n $ 1; fnðtÞ is continuous and increasing

in t. Therefore we can define the function f(t ) by

fðtÞ ¼n$1inf fnðtÞ; t [ ½0; T2�:

It is easy to see that f(t ) is nonnegative, continuous, f(0) ¼ 0 and satisfies

fðtÞ # D

Z t

0

Gðs;fðsÞÞds

for all t [ [0, T2]. It follows from assumption G2 that fðtÞ ¼ 0 for all t [ [0, T2].

Now from Lemma 2.3 we have

t[½0;T2�sup am;n ðtÞ #

t[½0;T2�sup fnðtÞ # fnðT2Þ n!1

! 0:

That is Xn(·) is a Cauchy sequence in HT2. Therefore there exists a process X(·, x )

such that

t[½0;T2�sup EjXnðsÞ2 Xðs; xÞj

2

1! 0:

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Using assumption (G1) we can see that for all t [ [0, T2]

E

Z t

0

e ðt2sÞAðf ðs;XnðsÞÞ2 f ðs;Xðs; xÞÞÞds

��������2

1! 0;

E

Z t

0

e ðt2sÞAðgðs;XnðsÞÞ2 gðs;Xðs; xÞÞÞdWðsÞ

��������2

1! 0;

from which we obtain that X(·, x ) is a mild solution of Eq. (1) on [0, T2]. The

uniqueness is an easy consequence of assumption G2. A

Finally we end this section by the following remarks.

Remark 2.2. Clearly the existence result remains intact if we replace x with a

random variable independent of the Wiener process with Ejzj2, 1:

Remark 2.3. Another interesting case arises when we assume that

i) For any T . 0, u0ðtÞ ¼ Kðt; uÞ has a solution for any initial u0, u0 $ 0;on [0, T ].

ii) r ¼ 1:iii) Hypothesis (G2) holds for any T . 0:

Applying similar arguments as those given in the proof of Theorem 2.1, we can

easily show that Eq. (1) has a unique mild solution on [0, T ] for all T . 0.

3. THE CONVERGENCE OF THE APPROXIMATIONS

Throughout this section we assume that A, f and g satisfy the following.

ðH1Þ

iÞ A generates a compact C0 semigroup:

iiÞ The function x ! kf ðxÞ; yl is a continuous mapping on H for all y [ H:

iiiÞ The function x ! ky; gðxÞQg* ðxÞyl is a continuous mapping on H for

every y from a dense subspace of H:

8>>>>>>><>>>>>>>:

(H2) There exists K . 0 such that

j f ðxÞj # Kð1 þ jxjÞ;

kgðxÞQ12k2 # Kð1 þ jxjÞ;

for all x [ H.

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The following result on existence of a weak solution for Eq. (1) is due to [14].

Theorem 3.1. Assume that assumptions (H1) and (H2) hold. Then, for each

x [ H, there exists a weak solution X(·, x) of Eq. (1) on [0, T] for any T . 0.

Without loss of generality, we may assume that T ¼ 1: Next, we formulate

the definition of pathwise uniqueness (PU ).

Definition 3.1. There is pathwise uniqueness for Eq. (1) if whenever (X(·, x ), W )

and (X(·, x), W) are two mild solutions defined on the same filtered space

ðV; F; {F t}; PÞ with W ¼ W and x ¼ x, then Xðt; xÞ ¼ ~X ðt; ~xÞ almost surely for

all t [ [0, 1].

Our goal in this section is to show the convergence in L p, p . 2 of the

Euler and Lie-Trotter schemes under the assumption (PU ).

3.1. Convergence of Euler Scheme

We consider now the approximating equation

dXnðt; xÞ ¼ ðAXnðt; xÞ þ f nðXnðt; xÞÞÞdt þ gnðXnðt; xÞÞdWðtÞ

Xnð0Þ ¼ x:

(ð8Þ

where fn and gn are defined as follows:

f n : Cð½0; 1�; HÞ £ ½0; 1�! H; f nðY ; tÞ ¼ f ðYFnðtÞÞ;

and

gn : Cð½0; 1�; HÞ £ ½0; 1�! LðHÞ; gnðY; tÞ ¼ gðYFnðtÞÞ;

where FnðtÞ ¼ k22n; if k22n # t # ðk þ 1Þ22n:It is well known from [14] that, under assumptions H1 2 H2, Eq. (8) has a

unique mild solution Xn(·, x ) on [0, 1] given by

Xnðt; xÞ ¼ e tAx þ

Z t

0

e ðt2sÞAf nðXnðs; xÞÞds

þ

Z t

0

e ðt2sÞAgnðXnðs; xÞÞdWðsÞ: ð9Þ

We will show that the sequence Xn(·, x ) converge to X(·, x ) in L p, p . 2;uniformly in t [ ½0; 1� and x [ B where B is a compact subset of H.

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We start with the following a priori estimate of Xn and X which are a direct

consequence of the infinite dimensional Burkholder inequality (see [8]) and the

Gronwall’s Lemma.

Lemma 3.1. Assume that assumptions H1 and H2 hold. Then for every p . 2

there exists a constant Cp . 0 such that

ðiÞx[Bsup

n$1sup E

t[½0;1�sup jXnðt; xÞj

p# Cp:

ðiiÞx[Bsup E

t[½0;1�sup jXðt; xÞp # Cp:

Now we are in position to prove the desired result on convergence for the Euler

scheme.

Theorem 3.2. Assume that assumptions H1, H2 and (PU ) hold. Then for every

p . 2 we have

n!1lim

x[Bsup E

t[½0;1�sup jXnðt; xÞ2 Xðt; xÞj

p¼ 0:

Proof. Assume, in contradiction, that there exist d . 0; x [ H and a

subsequence (xn0)n$0 converging to x such that

d #n0

inf Et[½0;1�sup jXn0 ðt; xn0 Þ2 Xðt; xÞj

p:

Without loss of generality, we denote (Xn0) and (xn0)n0$0 by (Xn)n$0 and (xn)n$0

respectively. From the proof of Theorem 1 in [14] it follows that the law of the

process (Xn(·, xn), X(·, xn), W ) is tight on C([0, 1]; H ^3), where Xn(·, xn) and

X(·, xn) are the solutions of (8) and (1) with the initial data xn respectively. By the

Skorohod theorem there exists a probability space ð ~V; ~F; ~PÞ and a sequence

ð ~Xnð·; xnÞ; ~Ynð·; xÞ; ~WnÞ that have the same distributions as (Xn(·, xn), X(·, xn), W )

and converge to (X(·, x ), Y(·, x ), W) a.s in Cð½0; 1�; H ^3Þ: Moreover, for every

t [ ½0; 1�; :

~Xnðt; xnÞ ¼ e tAxn þ

Z t

0

e ðt2sÞAf nð ~Xnðs; xnÞÞds þ

Z t

0

e ðt2sÞAgnð ~Xnðs; xnÞd ~WnðsÞ

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and

~Ynðt; xnÞ ¼ e tAxn þ

Z t

0

e ðt2sÞAf ð ~Ynðs; xnÞds þ

Z t

0

e ðt2sÞAgð ~Ynðs; xnÞd ~Wn; ðsÞ

hold for P almost every w. Letting n tend to infinity we can easily see that X(·, x )

and Y( ·, x ) are mild solutions of Eq. (1) on the stochastic basis (V, ~F, P, Q ) with

the Wiener process W and the same initial data. On the other hand, using the

statement (i) of Lemma 3.1 we have

0 , d # Et[½0;1�sup j ~Xðt; xÞ2 ~Yðt; xÞj

p¼ lim

ninf E

t[½0;1�sup j ~Xnðt; xnÞ2 Y ~nðt; xnÞj

p:

We may notice finally that, by the pathwise uniqueness, we have X(·, x ) ¼ Y(·, x )

almost surely. That is a contradiction. A

3.2. Convergence of Lie-Trotter Scheme

In this subsection we assume the following

ðH02Þ There exists L . 0 such that

kf ðxÞk þ kgðxÞQ12k2 # L for all x [ H:

It is easy to see that assumptions H1 and H02 imply the existence of a weak

solution of Eq. (1).

Let us define for arbitrary n $ 1 and T . 0 the following

Ji ¼

½ih; ði þ 1Þh½; i ¼ 0; . . .n 2 1:

½nh; T�; i ¼ n:

8<:

where h ¼ Tnþ1

:We now introduce the Lie-Trotter approximation which is defined as

follows:

For every n $ 1; we define

Ynð02Þ ¼ Xnð0

2Þ ¼ x;

Xnðt; xÞ þR t

ihe ðt2sÞAf ðXnðs; xÞÞds ¼ e tAYnðih

20; xÞ

Ynðt; xÞ ¼ Xnðt; xÞ þR t

ihe ðt2sÞAgðXnðs; xÞÞdWðsÞ t [ ½ih; ði þ 1Þh½:

8>>><>>>:

ð10Þ

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Following [3], we introduce

dðn; tÞ ¼

nþ1T

t� �

Tnþ1

; t [ ½0; T�

nnþ1

T; t ¼ T:

8<:

where [·] denotes the integer part. Then Eq. (10) can be written as follows:

Xnðt; xÞ þR t

0e ðt2sÞAf ðXnðs; xÞÞds ¼ e tAx þ

R dðn;tÞ

0e ðt2sÞAgðYnðs; xÞÞdWðsÞ;

Ynðt; xÞ þR t

0e ðt2sÞAf ðXnðs; xÞÞds ¼ e tAx þ

R t

0e ðt2sÞAgðYnðs; xÞÞdWðsÞ:

8<: ð11Þ

Before considering the question of convergence it is necessary to make some

preparation, as suggested by the following results.

The following Lemma shows that Xn and Yn tend to the same limit as n goes

to infinity.

Lemma 3.2. Assume that assumptions (H1) and (H02) hold. Then for every p . 2

we have

n!1lim

x[Bsup E

t[½0;1�sup jXnðt; xÞ2 Ynðt; xÞj

p¼ 0:

Proof. For n [ N we have

Ynðt; xÞ2 Xnðt; xÞ ¼

Z t

dðn;tÞ

e ðt2sÞAgðYnðs; xÞÞdWðsÞ:

Using the infinite dimensional Burkholder inequality, there exists a positive

constant C which is independent of n such that

Et[½0;1�sup

Z t

dðn;tÞ

e ðt2sÞAgðYnðs; xÞÞdWðsÞ

��������p

#Xnþ1

i¼1

Eti21#t#ti

sup

Z t

ti21

e ðt2sÞAgðYns; xÞÞdWðsÞ

��������p

# Cp

Xnþ1

i¼1ti21#t#ti

sup E

Z t

ti21

kgðYnðs; xÞÞQ12k2ds

� �p2

# Cp

Xnþ1

i¼1

E

Z ti

ti21

kgðYnðs; xÞÞQ12k2ds

� �p2

:

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It follows that

x[Bsup E

t[½0;1�sup jXnðt; xÞ2 Ynðt; xÞj

p# C

1

n þ 1

� �p221

:

Letting n tend to infinity we get the result. A

Lemma 3.3. Assume that assumptions (H1) and (H02) hold. Then for every p . 2

there exists a constant Cp . 0 such that

ðiÞx[Bsup

n$1sup E

t[½0;1�sup jYnðt; xÞj

p# Cp:

ðiiÞx[Bsup

n$1sup E

t[½0;1�sup jXnðt; xÞj

p# Cp:

Proof. Statement (i) is an immediate consequence of the infinite dimensional

Burkholder inequality and assumption H02. Statement (ii) follows directly from (i)

and Lemma 3.2.

Now, we can prove the convergence result. A

Theorem 3.3. Assume that assumptions H1, H02 and (PU ) hold. Then for every

p . 2 we have

n!1lim

x[Bsup E

t[½0;1�sup jYnðt; xÞ2 Xðt; xÞj

p¼ 0;

and

n!1lim

x[Bsup E

t[½0;1�sup jXnðt; xÞ2 Xðt; xÞj

p¼ 0:

The theorem can be proved as Theorem 3.2.

Remark 3.1. The approach adopted in this section can be applied to other

schemes. For the successive approximations some complications appear, so it

remains an open problem.

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4. CONTINUOUS DEPENDENCE

4.1. Dependence on Initial Data

In this subsection we show that under assumption (PU ) the family X(·, x )

depends continuously on the initial data in the sense specified in the following

theorem.

Theorem 4.1. Assume that assumptions (H1), (H02) and (PU ) hold. Then for

every p . 2 we have

x!x0lim E

t[½0;1�sup jXðt; xÞ2 Xðt; x0Þj

p¼ 0:

Proof. Suppose that the conclusion of the theorem is false, then there exists a

positive number d and a sequence (xn) converging to x such that

ninf E

t[½0;1�sup jXðt; xnÞ2 Xðt; xÞj

p$ d:

First using Gronwall inequality we can see that there exists a constant C such

that

Et[½0;1�sup jXðt; xÞj

pþ

n$1sup E

t[½0;1�sup jXðt; xnÞj

p# C: ð12Þ

Now, from the proof of Theorem 1 in [14] it follows that the law of the process

ðXð·; xnÞ;Xð·; xÞ;WÞ is tight on Cð½0; 1�; H ^3: By the Skorohod theorem there

exists a probability space ð ~V; ~F; ~PÞ and a sequence ð ~Xnð·; xnÞ; ~Ynð·; xÞ; ~WnÞ such

that

(i) ð ~Xnð·; xnÞ; ~Ynð·; xÞ; ~WnÞ have the same distributions as ðXð·; xnÞ;Xð·; xÞ;WÞ for

every n.

(ii) ð ~Xnð·; xnÞ; Y ~nð·; xÞ; ~WnÞ converge to ð ~Xð·; xÞ; ~Yð·; xÞ; ~WÞ a.s in Cð½0; 1�; HÞ^3:Moreover, for every t [ ½0; 1�;

~Xnðt; xnÞ ¼ e tAxn þ

Z t

0

e ðt2sÞAf nð ~Xnðs; xnÞÞds

þ

Z t

0

e ðt2sÞAgnð ~Xnðs; xnÞÞd ~W~nðsÞ

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and

~Ynðt; xÞ ¼ e tAx þ

Z t

0

e ðt2sÞAf ð ~Ynðs; xÞds þ

Z t

0

e ðt2sÞAgð ~Ynðs; xÞd ~WnðsÞ

hold for P almost every w. Letting n tend to infinity we can easily see that X(·, x )

and Y(·, x ) are mild solutions of Eq. (1) on the stochastic basis ð ~V; ~F; ~P; ~QÞ with

the Wiener process W and x as initial data. On the other hand, using estimation

(12) and (i) we have

0 , d # Et[½0;1�sup j ~Xðt; xÞ2 ~Yðt; xÞj

p¼ lim

ninf E

t[½0;1�sup j ~Xnðt; xnÞ2 ~Ynðt; xÞj

p:

We may notice finally that, by the pathwise uniqueness, we have ~Xð·; xÞ ¼ ~Yð·; xÞ

almost surely. That is a contradiction. A

4.2. Dependence on the Coefficients

The aim of this subsection is to establish that under assumption (PU ) the

continuous dependence on the coefficients is obtained. We consider the

stochastic evolutions equations

dXnðt; xÞ ¼ ðAXnðt; xÞ þ f nðXnðt; xÞÞÞdt þ gnðXnðt; xÞÞdWnðtÞ

Xnð0Þ ¼ x:

8<: ð13Þ

where fn, gn and Wn satisfy the following

(H3) Let Wn be Qn-Wiener process in H, Qn [ LðHÞ and Qn $ 0:(H4) For each n $ 1 we have

i) The function x ! k fn(x ),yl is a continuous mapping on H for all

y [ H:ii) The function x ! ky; gnðxÞQg*

n ðxÞyl is a continuous mapping on

H for every y from a dense subspace of H.

(H5) There exists a constant K1 . 0 such that:

j f nðxÞj þ kgnðxÞQ12k2 # K1ð1 þ jxjÞ;

for all x [ H and n $ 1:(H6) For each x [ H and y [ DðA* Þ we have

n!1lim kf nðxÞ; yl ¼ kf ðxÞ; yl;

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and

n!1lim ky; gnðxÞQg*

n ðxÞyl ¼ ky; gðxÞQg* ðxÞyl:

It is clear that, under assumptions (H1 2 H6), for each n equation has a weak

solution denoted by Xnð·; xÞ: Now we give the continuous dependence result.

Theorem 4.2. Assume that assumptions (H1 2 H6) and (PU ) hold. Then we

have

n!1lim E

t[½0;1�sup jXnðt; xÞ2 Xðt; xÞj

p¼ 0:

The theorem can be proved as Theorem 3.2 and Theorem 2.1 in [26].

5. EXISTENCE RESULT WHERE THE COEFFICIENTS ARE

NOT EVERYWHERE DEFINED

In this section we assume that there exists a measurable open subset E , H

such that the mappings f : E ! H and g : E ! LðHÞ are measurable.

Definition 5.1. By a mild solution in E of Eq. (1) on [0, T ], we mean an

E-valued stochastic process X [ HT satisfying Eq. (2).

For convenience we define f ðxÞ ¼ 0; gðxÞ ¼ 0 for x � E: Let @(A ) be the

resolvent set of A and RðnÞ ¼ nRðn;AÞ: The sequence of infinitesimal generator

Ln associated with this equation defined next plays a key role in the sequel. For

f [ C1;2u ðRþ £ EÞ and n [ N* let

LnfðxÞ ¼›

›tfðRðnÞxÞ þ

›

›xfðRðnÞxÞ;ARðnÞðxÞ þ RðnÞf ðxÞ

* +

þ1

2tr

›2

›x2fðRðnÞxÞRðnÞgðxÞQðRðnÞgðxÞÞ*

� �;

where C1;2u ðRþ £ EÞ denotes the space of all functions f : Rþ £ E ! R

continuously differentiable in the first coordinate and twice continuously

Frechet-differentiable in the second coordinate such that ›2

›x 2 f is uniformly

continuous on bounded subset of E.

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We assume the following

ðE1Þ

iÞ A generates a compact C0 semigroup:

iiÞ The function x ! k f ðxÞ; yl is a continuous mapping on E for all y [ H:

iiiÞ The function x ! ky; gðxÞQg* ðxÞyl is a continuous mapping on E

for every y from a dense subspace of H:

8>>>>>>>><>>>>>>>>:

(E2) There exists an increasing sequence of bounded open measurable subset

ðEkÞk$1 such that

k$1<Ek ¼ E

and for all k we have

x[Ek

sup j f ðxÞj2# Mk;

x[Ek

sup kgðxÞQ12k

2

2# Mk;

where ðMkÞk$1 is a sequence of positive numbers.

(E3) There exists a sequence of nonnegative function vj [ C1;2u ðRþ £ EÞ such

that

jlim

nlimLnvjðt; xÞ # 0; ;t $ 0; x [ E;

(E4) Let the function v : Rþ £ E ! Rþ defining by v ¼ limj

inf vj and satisfying

vkðtÞ Ux[›Ek

inf vðt; xÞk!1! ; ;t $ 0

where ›Ek denotes the boundary of Ek.

Lemma 5.1. Let x [ E: Assume that E1 2 E4 hold. Then if X(·, x ) is a mild

solution in E of Eq. (1) and vð0; xÞ , 1 we have

tE ¼ 1 a:s;

where tE U inf{t : Xðt; xÞ � E}.

Proof. First, we denote X(·, x ) by X(·). Without loss of generality we can

assume that x, RðnÞx [ Ek for all n; k $ 1: Let tn;kE U inf{t : ðRðnÞXÞðtÞ � Ek}

and tkE U inf{t : Xðt; xÞ � Ek}: We can see easily that t

n;kE ! tk

E as n !1: Since

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AR(n ) is bounded it follows from Lemma 3.5 [22] that RðnÞXð· ^ tn;kE ; xÞ ¼

RðnÞXð· ^ tn;kE Þ satisfies

RðnÞXðt ^ tn;kE Þ ¼ RðnÞx þ

Z t

0

ARðnÞXnðs ^ tn;kE ; xÞ þ RðnÞf ðXðs ^ t

n;kE ; xÞÞ

� �ds

þ

Z t

0

RðnÞgðXðs ^ tn;kE ; xÞÞdWðsÞ:

So, applying Ito’s formula to vjðt ^ tn;kE ;RðnÞXðt ^ t

n;kE ÞÞ and taking expectation

we obtain

Evjðt ^ tn;kE ;RðnÞXðt ^ t

n;kE ÞÞ ¼ vjð0; xÞ þ

Z t

0

ELnvj s ^ tn;kE ;RðnÞXðs ^ t

n;kE Þ

� �ds:

Letting n !1 it follows from Lemma 4.2 [22] that

Evjðt ^ tn;kE ;Xðt ^ tk

EÞÞ # vjð0; xÞ þ E

Z t

0n

limLnvj s ^ tn;kE ;RðnÞXðs ^ t

n;kE Þ

� �ds:

Since vj $ 0; v ¼ limj

inf vj and vð0; xÞ , 1 we obtain from the above inequality

that

Evðt ^ tn;kE ;Xðt ^ tk

EÞ # vð0; xÞ þ E

Z t

0j

limn

limLnvj s ^ tn;kE ;RðnÞXðs ^ t

n;kE Þ

� �ds:

It follows from assumption (E3) that

Evððt ^ tkE;Xðt ^ tk

EÞÞ # vð0; xÞ:

Hence

P tkE # T

� �# P EvðT ^ tk

E;XðT ^ tkEÞÞ $ vkðTÞ

� �#

vð0; xÞ

vkðTÞ:

Letting k !1; we arrive at P tkE # T

� �¼ 0:

Let

0 ¼ tm0 , tm

1 , · · · , tmi , tm

iþ1 , · · ·

be a sequence of partition of Rþ such that every T . 0

dmðTÞ Ui:tiþ1#T

sup jtmiþ1 2 tm

i jm!1! 0:

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Since f and g vanishes outside E we can define the Euler approximation as follows

Xmðt; xÞ ¼ e tAx þ

Z t

0

e ðt2sÞAf ðXmðFmðsÞ; xÞds

þ

Z t

0

e ðt2sÞAgðXmðFmðsÞ; xÞdWðsÞ;

where FmðsÞ ¼ tmi if s [ tm

i ; tmiþ1

� �: A

Now we can proof the main result of this section.

Theorem 5.1. Let x [ E: Assume that E1 2 E4 and (PU ) hold. Then Eq. (1)

has a unique mild solution in E denoted X(·, x ) on [0, T ] for each T . 0.

Proof. As one can see the proof of the existence is virtually a modification of

the proof that can be found in [8] or [14]. First we show that the family Xm(t, x ) is

tight in Cð½0; T�; HÞ: Let tm;kE U inf{t : Xmðt; xÞ � Ek}: Under assumption

E1 2 E2 we can see from the proof of Theorem 1 of [14] that the family

Xt

m;kEð·; xÞ

� �is a tight in Cð½0; T�; HÞ for every k and T where

Xt

m;kEðt; xÞ U Xmðt ^ t

m;kE ; xÞ:

So, for a fixed k, there exists a probability space (V, ~F, ~Ft, ~Q ) and a sequence of

random variables Xm,k on V such that the distributions of are the same as those of

Xt

m;kE

and

~Xm;k m!1! ~Xk a:s in C ð½0; T�; HÞ ð14Þ

for certain ~Xk [ Cð½0; T�; HÞ: Using the standard limiting argument we get that

~Xkðt; xÞ ¼ e tAx þ

Z t

0

e ðt2sÞAf ð ~Xkððs; xÞÞds þ

Z t

0

e ðt2sÞAgð ~Xkðs; xÞÞdWðsÞ

for t , ~tkE U inf{t : ~Xkðt; xÞ � Ek}: Let ~t

m;kE U inf{t : ~Xm;kðt; xÞ � Ek}; then from

(14) we have

limm!1inf ~t

m;kE $ ~tk

E:

Since Xm,k and Xt

m;kE

have the same distributions then using the precedent

inequality and Lemma 5.1 we arrive at

k!1lim lim

m!1sup P t

m;kE # T

� �¼

k!1lim lim

m!1sup P ~t

m;kE # T

� �#

k!1lim P ~tk

E # T� �

¼ 0:

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This implies that the family Xm(t, x ) is tight in Cð½0; T�; HÞ: The proof of the

existence may be completed by using the same argument as employed in the

proof of Theorem 3.2 (see also the proof of the existence theorems in [14] and [15]

and Lemma 5.1 gives the existence for all T. A

Remark 5.1. Using the same argument as employed in the proof of Theorem 3.2

we can show that

n!1lim E

t[½0;T�sup jXmðt; xÞ2 Xðt; xÞj

p¼ 0;

where Xn(·, x ) (resp. X(·, x )) is the mild solution in E of Eq. (9) (resp. Eq. (1)).

ACKNOWLEDGMENTS

The authors are sincerely grateful to Professor Y. Ouknine for very helpful

discussions and remarks. Also the authors would like to thank Professor M. Dozzi

for drawing their attention to papers [22] and [26].

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