Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation.