Large-time behavior of finite-state mean-field systems with multi-classes

TL;DR

This paper analyzes the long-term behavior of multi-class finite-state mean-field systems, focusing on convergence, large deviations, and metastability as the number of particles grows large.

Contribution

It provides new insights into the large-time asymptotics, large deviations, and metastable phenomena of multi-class mean-field systems, with results in a product space setting.

Findings

Convergence of empirical vector to McKean-Vlasov solution

Large deviations principles for invariant distributions

Estimates of convergence rates to invariant measures

Abstract

We study in this paper the large-time asymptotics of the empirical vector associated with a family of finite-state mean-field systems with multi-classes. The empirical vector is composed of local empirical measures characterizing the different classes within the system. As the number of particles in the system goes to infinity, the empirical vector process converges towards the solution to a McKean-Vlasov system. First, we investigate the large deviations principles of the invariant distribution from the limiting McKean-Vlasov system. Then, we examine the metastable phenomena arising at a large scale and large time. Finally, we estimate the rate of convergence of the empirical vector process to its invariant measure. Given the local homogeneity in the system, our results are established in a product space.