Wong–Zakai approximations and center manifolds of stochastic differential equations

Abstract

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated dynamics of the stochastic differential equation driven by a l-dimensional Brownian motion. We prove that the solutions of Wong–Zakai approximations converge in the mean square to the solutions of the Stratonovich stochastic differential equation. We also show that for a simple multiplicative noise, the center-manifold of the Wong–Zakai approximations converges to the center-manifold of the Stratonovich stochastic differential equation.