Elsevier

Statistics & Probability Letters

Volume 78, Issue 17, 1 December 2008, Pages 2844-2849
Statistics & Probability Letters

Existence of almost periodic solutions to some functional integro-differential stochastic evolution equations

https://doi.org/10.1016/j.spl.2008.04.008Get rights and content

Abstract

We investigate a class of abstract functional integro-differential stochastic evolution equations in a real separable Hilbert space. Under some suitable assumptions, we establish the existence and uniqueness of a quadratic mean almost periodic mild solution.

Introduction

Integro-differential equations arise naturally in mechanics, electromagnetic theory, heat flow, nuclear reactor dynamics, and population dynamics. A general abstract model for semi-linear functional stochastic integro-differential equations has been studied by Keck and McKibben, 2003, Keck and McKibben, 2005. In this paper, we consider some abstract semi-linear integro-differential equations in a real separable Hilbert space, and we address the problem of existence of quadratic mean almost periodic solutions.

The concept of an almost periodic function was introduced by Bohr for numerical valued functions and extended by Bochner for functions with values in Polish spaces. The basic aspects of the theory of almost periodic functions can be found for instance in Corduneanu (1989).

Recently in Bezandry and Diagana (2007), the case of semi-linear stochastic differential equations in a real separable Hilbert space was considered. Under some suitable conditions it was proved that such equations have unique quadratic mean almost periodic solutions. In this work we use the same kind of assumptions as in Bezandry and Diagana (2007) but in the stochastic integro-differential stochastic equation context in order to obtain the quadratic mean almost periodicity of the solutions.

The paper is organized as follows: in Section 2, we make precise the necessary notation and spaces, and review some basic definitions and results for the notion of quadratic mean almost periodicity. In Section 3, we give some sufficient conditions for the existence and uniqueness of a quadratic mean almost periodic solution to some abstract functional integro-differential stochastic evolution equation (2.1) in a real separable Hilbert space.

Section snippets

Preliminaries

For details of this section, we refer the reader to Bezandry and Diagana (2007), Corduneanu (1989), and the references therein. Throughout this paper, H and K will denote real separable Hilbert spaces with respective norms and K. Let (Ω,F,P) be a complete probability space. We denote by L2(K,H) the space of all Hilbert–Schmidt operators acting between K and H equipped with the Hilbert–Schmidt norm 2.

For a symmetric nonnegative operator QL2(K,H) with finite trace we assume that {W(t),t

Application to semilinear stochastic integro-differential equations

In this section we make extensive use of the results recalled in the previous section to study the existence and uniqueness of a quadratic mean almost periodic solution to (2.1).

Throughout the rest of this section, we impose the following conditions:

  • (H0)

    The operator A:D(A)L2(P;H)L2(P;H) is the infinitesimal generator of a uniformly exponentially stable semigroup (T(t))t0 defined on L2(P;H) such that there exist constant M>0,δ>0 with T(t)Meδt,t0.

  • (H1)

    The function Fi:R×L2(P;H)L2(P;H), (t,X)Fi(t,

Example

To illustrate Theorem 3.2 we consider the existence and uniqueness of quadratic-mean almost periodic solutions to the stochastic partial differential equation: Xt=2Xξ2+tC(tu)G(u,X(u,ξ))dW(u)+tB(tu)F2(u,X(u,ξ))du+F1(t,X(t,ξ)),tR,ξD. where D is a bounded domain in Rn with smooth boundary D. Set H=L2(D).

Define an operator A on L2(R;H) by Ax(t,)=2ξ2x(t,),xH2(D)H01(D). It is known (see Pazy (1983)) that A generates a strongly continuous semigroup {T(t)} on L2(D) satisfying (H0).

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