Invariant Measures for Stochastic Reaction–Diffusion Problems on Unbounded Thin Domains Driven by Nonlinear Noise

Abstract

This article is concerned with the limiting behavior of invariant measures for stochastic reaction–diffusion equations driven by nonlinear noise on unbounded thin domains. We first show the existence of invariant measures when the diffusion terms are globally Lipschitz continuous. The uniform estimates on the tails of solutions are employed to present the tightness of a family of probability distributions of solutions in order to overcome the non-compactness of usual Sobolev embeddings on unbounded domains. Then, we prove any limit of invariant measures of the equations defined on \((n+1)\)-dimensional unbounded thin domains must be an invariant measure of the limiting system as the thin domains collapse onto the space \(\mathbb {R}^n\).