Asymptotics of the Invariant Measure in Mean Field Models with Jumps

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 ω-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.

Local Stability of McKean–Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

“…Thus, F j,ι q (q j,ι ) = q j,ι A j,ι q for all ι ∈ {c, p} and 1 ≤ j ≤ r, which in turn corresponds to the z-th component of the raw vector q j,ι A j,ι q . Finally, notice that: sup in (20) and (21) and the fact that Z is finite. Thus, condition (2.9) in [19] is verified.…”

“…In the case that there are multiple equilibria or a unique unstable equilibrium, metastable phenomena can arise and care must be taken in using the approximation. One can consult [7,20] for a detailed discussion. The long-time behavior of the large N-particles system is thus related to the stability of the McKean-Vlasov system (27).…”

A sufficient condition for the quasipotential to be the rate function of the invariant measure of countable-state mean-field interacting particle systems

“…The starting point of our study of the asymptotics of {℘ N , N ≥ 1} is the process-level LDP for {µ N ν N , ν N ∈ M N 1 (Z), N ≥ 1}, whenever ν N converges to ν in M 1 (Z). This LDP was established by Léonard [21] when the initial conditions are fixed, and by Borkar and Sundaresan [7] when the initial conditions converge 3 in M 1 (Z). The rate function of this LDP is governed by "costs" associated with trajectories on [0, T ] with initial condition ν, which we denote by S [0,T ] (ϕ|ν), ϕ ∈ D([0, T ], M 1 (Z)) (see (2.1) for its definition).…”

“…For a broad class of Markov processes such as small-noise diffusions, finite-state mean-field models, simple exclusion processes, etc., it is well-known that the Freidlin-Wentzell quasipotential is the rate function that governs the large deviation principle (LDP) for the family of invariant measures [18,32,7,17]. The quasipotential is the minimum cost (arising from the rate function for a process-level large deviation principle) associated with trajectories of arbitrary but finite duration, with fixed initial and terminal conditions.…”

“…From the theory of large deviations of the invariant measure of Markov processes [18,32,11,7], V is a natural candidate for the rate function of the family {℘ N , N ≥ 1}.…”