Long run convergence of discrete-time interacting particle systems of the McKean–Vlasov type

We consider a discrete-time system of $n$ coupled random vectors, a.k.a. interacting particles. The dynamics involve a vanishing step size, some random centered perturbations, and a mean vector field which induces the coupling between the particles. We study the doubly asymptotic regime where both the number of iterations and the number $n$ of particles tend to infinity, without any constraint on the relative rates of convergence of these two parameters. We establish that the empirical measure of the interpolated trajectories of the particles converges in probability, in an ergodic sense, to the set of recurrent McKean–Vlasov distributions. We also consider the pointwise convergence of the empirical measures of the particles. We consider the example of the granular media equation, where the particles are shown to converge to a critical point of the Helmholtz energy.