Abstract
A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.
Highlights
Using Limit of Relative Entropies.The study of heterogeneous mean-field systems is a growing area of research
We introduced in this paper a family of Gibbs systems constructed on block graphs together with their asymptotics
As the total number of particles in the system goes to infinity, the law of large numbers was proven to hold, giving rise to a McKean–Vlasov system of equations (27)
Summary
Classical questions in the study of mean-field systems include their asymptotic behavior when the total number of particles N in the system and/or the time t tends to infinity and under which conditions one can justify the interchangeability of the limits. We will show for the specific family of Gibbs systems introduced in Section 2 that, under mild assumptions, the associated empirical vector converges weakly and uniformly over compact time intervals, as N → ∞, towards the solution of a McKean–Vlasov system of equations (see Theorem 1). This kind of result is known in the literature as the law of large numbers. Proposition 5 shows that this limiting function satisfies a descent property, which, combined with mild assumptions, shows that it is a local Lyapunov function for the McKean–Vlasov system of equations
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