Random invariant manifolds of stochastic evolution equations driven by Gaussian and non-Gaussian noises

Abstract

The goal of this work is to compare the invariant manifold of the stochastic evolution equation driven by an α-stable process with the invariant manifold of the stochastic evolution equation forced by Brownian motion. First, we show that the solution of the Marcus stochastic evolution equation driven by a type of α-stable process converges to the solution of the related Stratonovich stochastic evolution equation forced by Brownian motion. Then, we study the invariant stable manifold of the stochastic evolution equation driven by an α-stable process. Finally, we prove that the invariant stable manifold of the Marcus stochastic evolution equation driven by an α-stable process converges in probability to the invariant stable manifold of the Stratonovich stochastic evolution equation forced by Brownian motion. The connection between the random dynamical system driven by non-Gaussian noise and the random dynamical system driven by Gaussian noise is established.