Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations

Abstract

The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equationdx(t)=(Ax(t)+f(t,x(t)))dt+g(t,x(t))dW(t)(⁎) with exponentially stable linear operator A and Poisson stable in time coefficients f and g. We prove that if the functions f and g are appropriately “small”, then equation (⁎) admits at least one solution which has the same character of recurrence as the functions f and g. We also discuss the asymptotic stability of these Poisson stable solutions.