Stepanov-like weighted asymptotic behavior of solutions to some stochastic differential equations in Hilbert spaces

Abstract

In this paper, we first introduce the notation and properties of -weighted pseudo almost automorphy for stochastic processes. And then, we apply the results obtained to consider the existence and uniqueness of -weighted pseudo almost automorphic solutions to some stochastic differential equations in a real separable Hilbert space under global Lipschitz conditions. Moreover, we also investigate asymptotic behavior of solutions to a stochastic differential equation under -weighted pseudo almost automorphic coefficients without global Lipschitz conditions. Our main results extend some known ones in the sense of square-mean weighted pseudo almost automorphy or -pseudo almost automorphy for stochastic processes.