A Wong-Zakai approximation for random invariant manifolds

Abstract

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. But these random objects are difficult to be visualized geometrically or computed numerically. The current work provides a perturbation approach to approximate these random invariant manifolds. We first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations. Then, we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise (i.e., integrated Ornstein-Uhlenbeck processes). The convergence in a pathwise Wong-Zakai sense is shown.