Effective splitting of invariant measures for a stochastic reaction diffusion equation with multiplicative noise

Abstract

This work is concerned with the effective dynamics for the stochastic reaction diffusion equations with cubic nonlinearity driven by a multiplicative noise. By splitting the solution into the finite dimension kernel space and its complement space with some appropriate multi-scale, it derives the dominant solution and the effective invariant measure in the sense of the Wasserstein distance, which capture the complex dynamical behaviors of the original system as a singular parameter is enough small. Furthermore, the effective invariant measure is decomposed to a product of two invariant measures.