Approximation for random stable manifolds under multiplicative correlated noises

The usual Wong-Zakai approximation is about simulating individual solutionsof stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariantmanifolds which are geometric objects useful for understanding how complex dynamics evolve under stochasticinfluences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on theinvariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.