Wong-Zakai approximations and periodic solutions in distribution of dissipative stochastic differential equations

Abstract

In this paper, we study the smooth Wong-Zakai approximations given by a stochastic process via Wiener shift and mollifier of Brownian motions. We show that solutions of random differential equations driven by such processes generate random dynamical systems and converge in mean square to solutions of Stratonovich stochastic differential equations with sub-linear drift and bounded diffusion. With the help of this result, we obtain the existence of periodic solutions in distribution and stationary measures for time-inhomogeneous and time-homogeneous stochastic systems with dissipativity respectively. As an application, we verify Levinson's conjecture to second order stochastic Newtonian systems via Lyapunov's method and truncation method.