On Mean Field Convergence and Stationary Regime

TL;DR

This paper establishes that for a broad class of stochastic processes converging to deterministic limits, any invariant measure of the stochastic process converges to an invariant measure of the deterministic process, applicable in both discrete and continuous time.

Contribution

It proves that under weak convergence assumptions, invariant measures of stochastic processes converge to those of the deterministic limit, extending previous results.

Findings

Invariant measures of stochastic processes converge to deterministic invariants.

Results apply to both discrete and continuous time models.

Weak convergence assumptions are sufficient for convergence of invariant measures.

Abstract

Assume that a family of stochastic processes on some Polish space $E$ converges to a deterministic process; the convergence is in distribution (hence in probability) at every fixed point in time. This assumption holds for a large family of processes, among which many mean field interaction models and is weaker than previously assumed. We show that any limit point of an invariant probability of the stochastic process is an invariant probability of the deterministic process. The results are valid in discrete and in continuous time.