Elementary coupling approach for non-linear perturbation of Markov processes with mean-field jump mechanims and related problems

TL;DR

This paper develops an elementary coupling method to analyze non-linear perturbations of Markov processes with mean-field jump mechanisms, establishing exponential convergence to equilibrium for both the equations and particle systems.

Contribution

It introduces a novel coupling approach for mean-field jump processes, proving exponential convergence and stability results in a unified abstract framework.

Findings

Unique equilibrium established for the mean-field equations.

Exponential convergence rate proven for the process and particle system.

Convergence rate is independent of the number of particles.

Abstract

Mean-field integro-differential equations are studied in an abstract framework, through couplings of the corresponding stochastic processes. In the perturbative regime, the equation is proven to admit a unique equilibrium, toward which the process converges exponentially fast. Similarly, in this case, the associated particle system is proven to converge toward its equilibrium at a rate independent from the number of particles.