Invariant measures of fractional stochastic delay reaction–diffusion equations on unbounded domains

In this paper, existence of invariant measure is mainly investigated for a fractional stochastic delay reaction–diffusion equation defined on unbounded domains. We first establish the mean-square uniform smallness of the tails of the solutions in order to overcome the non-compactness of standard Sobolev embeddings on unbounded domains. We then show the weak compactness of a family of probability distributions of the solutions by combining the Ascoli–Arzelà theorem, the uniform tail-estimates as well as the technique of dyadic division. (paper)