Square-mean weighted pseudo almost automorphic solutions for stochastic semilinear integral equations

Abstract

In this paper, we introduce the concept of $S^2$-weighted pseudo almost automorphy for stochastic processes. We study the existence and uniqueness of square-mean weighted pseudo almost automorphic solutions for the semilinear stochastic integral equation $x(t)={\int }_{-\infty }^{t}a(t-s)[Ax(s)+f(s,x(s))]ds+{\int }_{-\infty }^{t}a(t-s)\phi (s,x(s))dw(s),t\in \mathbb{R}$, where $a\in L^1({\mathbb{R}}_{+})$, $A$ is the generator of an integral resolvent family on a Hilbert space $H$, $w(t)$ is the two-sided $Q$-Wiener process, $f,\phi :\mathbb{R}\times L^2(P,H)\to L^2(P,H)$ are two $S^2$-weighted pseudo almost automorphic functions.