Asymptotics of the Invariant Measure in Mean Field Models with Jumps

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.

💡 Research Summary

The paper investigates the asymptotic behavior of the invariant (stationary) measure for a class of mean‑field interacting Markov jump processes when the number of particles N tends to infinity. Each particle evolves on a finite state space Z={0,…,r‑1} and its transition rates λ_{i,j}(μⁿ(t)) depend only on the current empirical distribution μⁿ(t) of all particles. Under three standing assumptions—(A1) the directed graph (Z,E) is irreducible, (A2) the rate functions are Lipschitz in the measure argument, and (A3) the rates are uniformly bounded away from zero and infinity—the empirical measure process μⁿ(t) converges, as N→∞, to the deterministic solution of a McKean‑Vlasov ordinary differential equation (ODE). This ODE describes the fluid limit of the system and is the cornerstone for the subsequent analysis.

The authors’ main contributions are twofold. First, they establish a uniform large‑deviation principle (LDP) for the empirical path measures on the Skorokhod space D(