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Results. Math. 63 (2013), 435449c 2011 Springer Basel AG1422-6383/13/010435-15published online September 21, 2011DOI 10.1007/s00025-011-0207-9 Results in Mathematics

Almost Periodic Solutions to a StochasticDifferential Equation in Hilbert Spaces

Yong-Kui Chang, Ruyun Ma and Zhi-Han Zhao

Abstract. In this paper, we prove the existence and uniqueness of qua-dratic mean almost periodic mild solutions for a class of stochastic dif-ferential equations in a real separable Hilbert space. The main techniqueis based upon an appropriate composition theorem combined with theBanach contraction mapping principle and an analytic semigroup of lin-ear operators.

Mathematics Subject Classification (2000). Primary 34K14;Secondary 60H10.

Keywords. Stochastic differential equations, quadratic mean almostperiodic, analytic semigroups of linear operators.

1. Introduction

Stochastic differential equation has attracted great interest due to its applica-tions in characterizing many problems in physics, biology, mechanics and so on.Qualitative properties such as existence, uniqueness, controllability and sta-bility for various stochastic differential systems have been extensively studiedby many researchers, see for instance [9,15,19,2123,2732,34] and the refer-ences therein. On the other hand, the existence of almost periodic solutions fordeterministic differential equations have been considerably investigated in lotsof publications because of its significance and applications in physics, mechan-ics and mathematical biology, see for example [1,2,13,14,16,18,25,35] and thereferences therein.

Recently, the concept of quadratic mean almost periodicity was intro-duced by Bezandry and Diagana [4]. In [4], such a concept was subsequently

436 Y.-K. Chang, R. Ma and Z.-H. Zhao Results. Math.

applied to proving the existence and uniqueness of a quadratic mean almostperiodic solution to the following stochastic differential equations

dx(t) = Ax(t)dt + F (t, x(t)) dt + G (t, x(t)) dw(t), t R,where A : D(A) L2(;H) L2(;H) for t R is a densely dened closedlinear operators, F : R L2(;H) L2(;H) and G : R L2(;H) L2(;L02) are jointly continuous satisfying some additional conditions, andw(t) is a Wiener process.

Bezandry and Diagana [5] have studied the existence and uniqueness ofa quadratic mean almost periodic solution to a non-autonomous semi-linearstochastic differential equations such as

dx(t) = A(t)x(t)dt + F (t, x(t)) dt + G(t, x(t))dw(t), t R,where A(t) for t R is a family of densely dened closed linear operators sat-isfying the so-called Acquistapace-Terreni condition in [3], F : RL2(;H) L2(;H) and G : R L2(;H) L2(;L02) are jointly continuous satisfyingsome additional conditions, and w(t) is a Wiener process. And Bezandry in [6]has considered the existence of quadratic mean almost periodic solutions to asemi-linear functional stochastic integro-differential equations in the form

x(t) = Ax(t) +

t

C(t u)G(u, x(u))dw(u)

+

t

B(t u)F2(u, x(u))du + F1(t, x(t)),

where t R, A : D(A) L2(;H) L2(;H) is a densely dened closed(possibly unbounded) linear operator; B and C are convolution-type kernelsin L1(0,) and L2(0,), respectively, satisfying Assumptions 3.2 in [21];F1, F2 : R L2(;H) L2(;H) and G : R L2(;H) L2(;L02) arejointly continuous functions. For more results on this topic, we refer the readerto the papers [7,8,12,33] and the references therein.

Motivated by the above mentioned works [46], the main purpose of thispaper is to deal with the existence and uniqueness of quadratic mean almostperiodic solutions to a class of neutral stochastic functional differential equa-tions in the abstract form

d[x(t) g(t, x(t))] = Ax(t)dt + G(t, x(t))dw(t), t R, (1.1)where A : D(A) L2(;H) L2(;H) is the innitesimal generator ofan analytic semigroup of linear operators {T (t)}t0 on L2(;H), g : R L2(;H) L2(;H) and G : R L2(;H) L2(;L02) are jointly con-tinuous functions, w(t) is a Brownian motion. The main technique is based

Vol. 63 (2013) Almost Periodic Solutions to a Stochastic Differential Equation 437

upon an appropriate composition theorem combined with the Banach con-traction mapping principle and an analytic semigroup of linear operators. Theobtained result can be seen as a contribution to this emerging eld.

The rest of this paper is organized as follows: In Sect. 2 we recall somebasic definitions, lemmas and preliminary facts which will be need in thesequel. Our main result and its proofs are arranged in Sect. 3. In the lastsection, an example is given to illustrate our main result.

2. Preliminaries

This section is mainly concerned with some notations, definitions, lemmasand preliminary facts which are used in what follows. For more details on thissection, we refer the reader to [4,5,10,11]

Throughout the paper, (H, ) and (K, K) denote two real Hilbertspaces. Let (,F , P ) be a complete probability space. We let L2(K,H) denotethe space of all HilbertSchmidt operators : K H, equipped with theHilbertSchmidt norm 2.

For a symmetric nonnegative operator Q L2(K,H) with nite trace wesuppose that {w(t) : t R} is a Q-Wiener process dened on (,F , P ) andwith values in K. So, actually, w can be obtained as follows: let wi(t), t R,i = 1, 2, be independent K-valued Q-Wiener processes, then

w(t) ={

w1(t) if t 0,w2(t) if t 0

is Q-Wiener process with R as time parameter. We then let Ft = {w(s) :s t} is the -algebra generated by w.

The collection of all strongly measurable, square integrable, H-valuedrandom variable, denoted by L2(;H), is a Banach space equipped with normxL2(;H)=(EX2) 12 , where the expectation E is dened E[x]=

x(w)dP (w).

Let K0 = Q12 K and L02 = L2(K0,H) with respect to the norm

2L02 = Q12 22 = Tr(Q).

Let 0 (A) where (A) is the resolvent of A. Then for 0 < 1, itis possible to dene the fractional power (A) , as a closed linear operatoron its domain D((A)). Furthermore, the subspace D((A)) is dense inL2(;H) and the expression

x = (A)xL2(;H), x D((A)),denes a norm on D((A)). Hereafter we denote by L2(;H) the Banachspace D((A)) with norm x.

The following properties hold by [26].


Lemma 2.1. Let 0 < 1. Then the following properties hold:(i) L2(;H) is a Banach space and L2(;H) L2(;H) is continuous.(ii) The function s (A)T (s) is continuous in the uniform operator

topology on (0,) and there exists M > 0 such that (A)T (t) Me

tt for each t > 0.(iii) For each x D((A)) and t 0, (A)T (t)x = T (t)(A)x.(iv) (A) is a bounded linear operator in L2(;H) with D((A)) =

Im((A)).In the following results and definitions, we let (X, X), (Y, Y) and

(Z, Z) be Banach spaces and let L2(;X) , L2(;Y) and L2(;Z) be theircorresponding L2-spaces, respectively.

Definition 2.1. [4] A stochastic process x : R L2(;X) is said to be contin-uous whenever

limts Ex(t) x(s)

2X

= 0.

Definition 2.2. [4] A continuous stochastic process x : R L2(;X) is said tobe quadratic mean almost periodic if for each > 0 there exists l() > 0 suchthat any interval of length l() contains at least a number for which

suptR

Ex(t + ) x(t)2X

< .

The collection of all stochastic processes x : R L2(;X) which arequadratic mean almost periodic is then denoted by AP

(R;L2(;X)

).

Lemma 2.2. [4] If x belongs to AP(R;L2(;X)

), then the following hold true:

(i) the mapping t Ex(t)2X

is uniformly continuous,(ii) there exists a constant N > 0, such that Ex(t)2

X N , for each t R,

(iii) x is stochastically bounded.

Let C(R, L2(;H)) denote the space of all continuous stochastic processesx : R L2(;H). The notation CUB(R;L2(;X)) stands for the collectionof all stochastic processes x : R L2(;X), which are continuous and uni-formly bounded. It is known from [4] that CUB(R;L2(;X)) is a Banach spaceendowed with the norm:

x = suptR

(Ex(t)2X)

12 .

Lemma 2.3. [4] AP (R;L2(;X)) CUB(R;L2(;X)) is a closed subspace.Lemma 2.4. [4] (AP (R;L2(;X)), AP (R;L2(;X))) is a Banach space endowedwith the norm:

xAP (R;L2(;X)) = suptR

(Ex(t)2X)

12 .


Definition 2.3. [4] A function F : R L2(;Y) L2(;Z), (t, y) F (t, y),which is jointly continuous, is said to be quadratic mean almost periodic int R uniformly in y B where B L2(;Y) is compact if for any > 0,there exists l(,B) > 0 such that any interval of length l(,B) contains at leasta number for which

suptR

EF (t + , y) F (t, y)2Z

<

for each stochastic process y : R B.Lemma 2.5. [4] Let F : RL2(;Y) L2(;Z), (t, y) F (t, y) be a quadraticmean almost periodic process in t R uniformly in y B, where B L2(;Y)is compact. Suppose that F is Lipschitz in the following sense:

EF (t, x) F (t, y)2Z

MEx y2Y

for all x, y L2(;Y) and for each t R, where M > 0. Then for any qua-dratic mean almost periodic process : R L2(;Y), the stochastic processt F (t,(t)) is quadratic mean almost periodic.Definition 2.4. A Ft-progressively process {x(t)}tR is called a mild solutionof the problem (1.1) on R if the function s AT (t s)g(s, x(s)) is integrableon (, t) for each t R, and x(t) satises

x(t) = T (t a)[x(a) g(a, x(a))] + g(t, x(t)) +t

a

AT (t s)g(s, x(s))ds

+

t

a

T (t s)G(s, x(s))dw(s)

for all t a and for each a R.Let us list the following assumptions:

(H1) The operator A : D(A) L2(;H) L2(;H) is the innitesimalgenerator of an analytic semigroup of linear operators {T (t)}t0 onL2(;H) and M, are positive numbers such that T (t) Met fort 0.

(H2) There exists a positive number (0, 1) such that g : RL2(;H) L2(;H) is quadratic mean almost periodic in t R uniformly inx B1 where B1 L2(;H) being a compact subspace. Moreover, g isLipschitz in the sense that: there exists Lg > 0 such that

E(A)g(t, x) (A)g(t, y)2 LgEx y2,for all t R and for each stochastic processes x, y L2(;H).

(H3) The function G : R L2(;H) L2(;L02) is quadratic mean almostperiodic in t R uniformly in x B2 where B2 L2(;H) being a


compact subspace. Moreover, G is Lipschitz in the sense that: thereexists LG > 0 such that

EG(t, x) G(t, y)2L02 LGEx y2,

for all t R and for each stochastic processes x, y L2(;H).

3. Main Results

In this section, we present and prove our main theorem.

Theorem 3.1. Assume the conditions (H1)(H3) are satisfied, then the prob-lem (1.1) admits a unique quadratic mean almost periodic mild solution on Rprovide that

L0 =[3Lg(A)2 + 3M21Lg2[()]2 +

3TrQM2LG2

]< 1, (3.1)

where () is the gamma function.Proof. Let : AP (R;L2(;H)) C(R, L2(;H)) be the operator dened by

x(t) = g(t, x(t)) +

t


+

t

T (t s)G(s, x(s))dw(s), t R.

First we prove that x is well dened. From Lemma 2.5, we infer thats g(s, x(s)) is in AP (R;L2(;H)). Thus using Lemma 2.2 (ii) it followsthat there exists a constant Ng > 0 such that E(A)g(t, x(t))2 Ng, forall t R. Moreover, from the continuity of s AT (t s) and s T (t s) inthe uniform operator topology on (, t) for each t R and the estimate

E

t


2

= E

t

(A)1T (t s)(A)g(s, x(s))ds

2

M21E

t

e(ts)(t s)1(A)g(s, x(s))ds

2

M21

t

e(ts)(t s)1ds


t

e(ts)(t s)1E(A)g(s, x(s))2ds

NgM21

t

e(ts)(t s)1ds

2

= NgM212[()]2,

it follows that s AT (ts)g(s, x(s)) and s T (ts)G(s, x(s)) are integrableon (, t) for every t R, therefore x is well dened and continuous.

Next, we show that x(t) AP (R;L2(;H)). We dene

1x(t) =

t


and

2x(t) =

t

T (t s)G(s, x(s))dw(s).

Let us show that 1x(t) is quadratic mean almost periodic. Now since g(, x()) AP (R;L2(;H)), by Definition 2.2, it follows that for any > 0, thereexists l() > 0 such that every interval of length l() contains at least a num-ber with the property that

E(A)g(t + , x(t + )) (A)g(t, x(t))2 < M212[()]2

,

for each t R.Now, using CauchySchwarz inequality, we obtain that

E 1x(t + ) 1x(t)2

= E

t

AT (t s)[g(s + , x(s + )) g(s, x(s))]ds

2

= E

t

(A)1T (t s)[(A)g(s + , x(s + ))(A)g(s, x(s))]ds

2

M21E

t

e(ts)(t s)1


(A)g(s + , x(s + )) (A)g(s, x(s))ds2

M21E

t

e(ts)(t s)1ds

t

e(ts)(t s)1(A)g(s + , x(s + ))

(A)g(s, x(s))2ds

M21

t

e(ts)(t s)1ds

t

e(ts)(t s)1E(A)g(s + , x(s + ))

(A)g(s, x(s))2ds M21

t

e(ts)(t s)1ds

2

suptR

E(A)g(t + , x(t + )) (A)g(t, x(t))2

2[()]2

t

e(ts)(t s)1ds

2

= .

Hence, 1x() is quadratic mean almost periodic.Similarly, by using Lemma 2.5, one can easily see that s G(s, x(s))

is quadratic mean almost periodic. Therefore, it follows from Definition 2.2that for any > 0, there exists l() > 0 such that every interval of length l()contains at least a number with the property that

EG(t + , x(t + )) G(t, x(t))2L02

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