Wong–Zakai approximations and long term behavior of stochastic FitzHugh–Nagumo system

Abstract

This paper is concerned with the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of stochastic FitzHugh–Nagumo system driven by white noise. We prove the existence and uniqueness of pullback random attractors for the approximate system under much weaker conditions than the original system. When the system is driven by additive white noise, we also prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.