Asymptotic behavior of random lattice dynamical systems and their Wong-Zakai approximations

Abstract

In this paper, we study the Wong-Zakai approximations given by a smoothed approximation of the white noise and their associated long term pathwise behavior for the stochastic lattice dynamical systems. To be exactly, we first establish the existence of the random attractor for the random lattice dynamical system driven by the smoothed noise and then show the convergence of solutions and random attractors to these of stochastic lattice dynamical systems driven by a multiplicative noise and an additive white noise, respectively, when the perturbation parameters tend to zero.