Invariant measures, periodic measures and pullback measure attractors of McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains

TL;DR

This paper investigates the long-term behavior of McKean-Vlasov stochastic reaction-diffusion equations on unbounded domains, establishing the existence of attractors and measures, and analyzing their stability and convergence properties.

Contribution

It proves the existence and uniqueness of pullback measure attractors, invariant measures, and periodic measures for these equations, including their stability and convergence analysis.

Findings

Existence and uniqueness of pullback measure attractors

Existence and uniqueness of invariant and periodic measures

Upper semi-continuity and convergence of measures

Abstract

This paper deals with the long term dynamics of the non-autonomous McKean-Vlasov stochastic reaction-diffusion equations on R^n. We first prove the existence and uniqueness of pullback measure attractors of the non-autonomous dynamical system generated by the solution operators defined in the space of probability measures. We then prove the existence and uniqueness of invariant measures and periodic measures of the equation under further conditions. We finally establish the upper semi-continuity of pullback measure attractors as well as the convergence of invariant measures and periodic measures when the distribution dependent stochastic equations converge to a distribution independent system.