Abstract stochastic integrodifferential delay equations

David N Keck,Mark A Mckibben

doi:10.1155/jamsa.2005.275

Abstract

We investigate a class of abstract stochastic integrodifferential delay equations dependent upon a family of probability measures in a separable Hilbert space. We establish the existence and uniqueness of a mild solution, along with various continuous dependence estimates and Markov (weak and strong) properties of this solution. This is followed by a convergence result concerning the strong solutions of the Yosida approximations of our equation, from which we deduce the weak convergence of the measures induced by these strong solutions to the measure induced by the mild solution of the primary problem under investigation. Next, we establish the pth moment and almost sure exponential stability of the mild solution. Finally, an analysis of two examples, namely a generalized stochastic heat equation and a Sobolev‐type evolution equation, is provided to illustrate the applicability of the general theory.

Highlights

We will initiate an investigation of a class of abstract delay integrodifferential stochastic evolution equations of the general form t x (t) + Ax(t) = f1 t, xt, μ(t) + K1(t, s) f2 s, xs, μ(s) ds
W is a given K-valued Wiener process associated with a positive, nuclear covariance operator Q; A is a linear operator which generates a strongly continuous semigroup {S(t) : t ≥ 0} on H; K1(t, s) and K2(t, s) are bounded, linear operators on H; fi : [0, T] × Cr × ℘λ2 (H) → H (i = 1, 2) and f3 : [0, T] × Cr × ℘λ2 (H) → BL(K; H) (where K is a real separable Hilbert space, BL(K; H) denotes the space of all bounded, linear operators from K into H, and ℘λ2(H) denotes a particular subset of probability measures on H) are given mappings; μ(t) is
A prototypical example in the finite-dimensional setting would be an interacting N-particle system in which (1.1) describes the dynamics of the particles x1, . . . , xN moving in the space H in which the probability measure μ is taken to be the empirical measure μN (t) = (1/N)

Summary

Introduction

We will initiate an investigation of a class of abstract delay integrodifferential stochastic evolution equations of the general form t x (t) + Ax(t) = f1 t, xt, μ(t) + K1(t, s) f2 s, xs, μ(s) ds 0 t. Stochastic partial functional differential equations with finite delay arise naturally in the mathematical modeling of phenomena in the natural sciences (see [22, 30, 34]) and have begun to receive a significant amount of attention. The motivation of the present work lies primarily in formulating an extension of the work in [1, 7, 13, 18] to a more general class of abstract integrodifferential stochastic evolution equations with finite delay.

Preliminaries

Cr d s ts

Asymptotic stability

Examples