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A family of heterogeneous mean-field systems with jumps is analyzed. Thesesystems are constructed as a Gibbs measure on block graphs. When the totalnumber of particles goes to infinity, a law of large numbers is shown to holdin a multi-class context resulting in the weak convergence of the empiricalvector towards the solution of a McKean-Vlasov system of equations. We theninvestigate the local stability of the limiting McKean-Vlasov system throughthe construction of a local Lyapunov function. We first compute the limit ofadequately scaled relative entropy functions associated with the explicitstationary distribution of the N-particles system. Using a Laplace principlefor empirical vectors we show that the limit takes an explicit form. Then wedemonstrate that this limit satisfies a descent property which, combined withsome mild assumptions shows that it is indeed a local Lyapunov function.

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