Pullback measure attractors and limiting behaviors of McKean-Vlasov stochastic delay lattice systems

TL;DR

This paper investigates the long-term distributional behavior of non-autonomous McKean-Vlasov stochastic delay lattice systems, establishing well-posedness, attractor existence, ergodicity, and stability properties of the system's invariant measures.

Contribution

It introduces the first analysis of pullback measure attractors and their stability for non-autonomous distribution-dependent stochastic delay lattice systems.

Findings

Existence and uniqueness of pullback measure attractors.

Ergodicity and exponential mixing of invariant measures.

Upper semi-continuity of attractors under system convergence.

Abstract

We study the long-term behavior of the distribution of the solution process to the non-autonomous McKean-Vlasov stochastic delay lattice system defined on the integer set $\mathbb{Z}$. Specifically, we first establish the well-posedness of solutions for this non-autonomous, distribution-dependent stochastic delay lattice system. Then, we prove the existence and uniqueness of pullback measure attractors for the non-autonomous dynamical system generated by the solution operators, defined in the space of probability measures. Furthermore, as an application of the pullback measure attractor, we prove the ergodicity and exponentially mixing of invariant measures for the system under appropriate conditions. Finally, we establish the upper semi-continuity of these attractors as the distribution-dependent stochastic delay lattice system converges to a distribution-independent system.