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

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.

Local stability of McKean-Vlasov equations arising from heterogeneous Gibbs systems using limit of relative entropies

“…In the next result, we state that a law of large numbers holds for the empirical vector µ N (t) as N → ∞. It is worthwhile to mention that contrary to the models studied in Dawson et al (2020Dawson et al ( , 2021, the rate functions here are convergent.…”

“…This family is a particular instance of the models introduced in Dawson et al (2020Dawson et al ( , 2021 with the particularity laying in the specification of a heterogeneous Gibbs deterministic models was observed in the literature; see, e.g., DeMarco & Qian (1989); Marco & Bergen (1987). Therefore, for the general models introduced in Dawson et al (2020Dawson et al ( , 2021, one might view the finite N -particles system as a small noise perturbation of the limiting McKean-Vlasov system. Notice that the specific quasipotential for these models was introduced in Dawson et al (2021).…”

“…In the case where 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 Borkar & Sundaresan (2012); Dawson et al (2021) 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 (2.1).…”

“…It is then of interest to estimate quantities such as the mean time spent by the process near a stable point, the probability of reaching a given stable point before reaching another one, or also the probability of transiting between a collection of ω-limit sets in a particular order, and so on. Notice that the metastability analysis was conducted in Dawson et al (2021) for the general finite-state mean-field systems on block graphs under the classical conditions of Friedlin and Wentzell Freidlin & Wentzell (2012).…”

“…Therefore, for the general models introduced in Dawson et al (2020Dawson et al ( , 2021, one might view the finite N -particles system as a small noise perturbation of the limiting McKean-Vlasov system. Notice that the specific quasipotential for these models was introduced in Dawson et al (2021). The idea is then to investigate under which condition the quasipotential can serve as a Lyapunov function for the McKean-Vlasov system.…”