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

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.