Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations

Abstract

In this paper, the concept of p-mean almost automorphy for stochastic processes is firstly introduced. Some properties of the p-mean almost automorphic stochastic processes are further studied. Based on these properties, a class of stochastic functional differential equations given by is investigated. Under suitable assumptions, the existence, uniqueness and exponential stability of quadratic-mean almost automorphic mild solutions for the equations are discussed by means of semigroups of operators and by Banach contraction principle. Moreover, the quadratic-mean almost automorphic mild solutions for semilinear stochastic partial functional differential equations, which is illustrated by example, are in good agreement with the theoretical analysis.