The Wong-Zakai approximations of invariant manifolds for stochastic evolution equations with linear noise

In this paper, we study the Wong-Zakai approximations given by a class of stationary processes and their associated dynamics of a class of stochastic evolution equations with linear noise. We first prove that the solutions of Wong-Zakai approximations converge almost surely to the solutions of the stochastic evolution equation, with a polynomial rate. Then we prove the existence and smoothness of invariant manifolds for the stochastic evolution equation and its Wong-Zakai approximation equations. We also show that the invariant manifolds of the Wong-Zakai approximations converge almost surely to the invariant manifolds of the stochastic evolution equation, with a polynomial rate.