Poisson stable solutions for stochastic functional evolution equations with infinite delay

Abstract

This work is devoted to Poisson stable (including stationary, periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent) solutions for stochastic functional evolution equations (SFEEs) with infinite delay. First, we prove the existence of bounded mild solutions for SFEEs with infinite delay. Then, according to the relationship between the solution and (drift and diffusion) coefficients, we obtain such Poisson stable solutions. Because the solutions of the delay equations are not Markov, we employ the solution maps in some phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability.