Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains

In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.