Asymptotics of the Invariant Measure in Mean Field Models with Jumps

TL;DR

This paper investigates the asymptotic behavior of the invariant measure in large systems of coupled Markov chains with jumps, relevant to wireless networks and epidemic spread, using a control-theoretic approach to large deviations.

Contribution

It provides a novel analysis of the invariant measure's asymptotics in mean field models with jumps, linking stochastic behavior to deterministic McKean-Vlasov dynamics.

Findings

Invariant measure concentrates on the equilibrium when unique and globally stable.

Limit points are supported on the omega-limit sets of the McKean-Vlasov equation.

Large deviations are characterized using a control-theoretic framework.

Abstract

We consider the asymptotics of the invariant measure for the process of the empirical spatial distribution of $N$ coupled Markov chains in the limit of a large number of chains. Each chain reflects the stochastic evolution of one particle. The chains are coupled through the dependence of the transition rates on this spatial distribution of particles in the various states. Our model is a caricature for medium access interactions in wireless local area networks. It is also applicable to the study of spread of epidemics in a network. The limiting process satisfies a deterministic ordinary differential equation called the McKean-Vlasov equation. When this differential equation has a unique globally asymptotically stable equilibrium, the spatial distribution asymptotically concentrates on this equilibrium. More generally, its limit points are supported on a subset of the $\omega$-limit sets of the McKean-Vlasov equation. Using a control-theoretic approach, we examine the question of large deviations of the invariant measure from this limit.