Periodic, Quasi-Periodic, Almost Periodic, Almost Automorphic, Birkhoff Recurrent and Poisson Stable Solutions for Stochastic Differential Equations

Original · EN

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 equation dx(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.