Empirical approximation to invariant measures of mean-field Langevin dynamics

TL;DR

This paper studies how empirical measures of mean-field Langevin dynamics approximate invariant measures, proving convergence under certain conditions and supporting results with numerical experiments.

Contribution

It provides the first rigorous proof of convergence of empirical measures to invariant measures for McKean--Vlasov Langevin dynamics under dissipativity and Lipschitz assumptions.

Findings

Empirical measures converge to invariant measures in Wasserstein distance.

Convergence holds for both mean-field and self-interacting Langevin dynamics.

Numerical experiments validate theoretical convergence results.

Abstract

This paper is concerned with the approximation to invariant measures for Langevin dynamics of McKean--Vlasov type. Under dissipativity and Lipschitz conditions, we prove that the empirical measures of both the mean-field and self-interacting Langevin dynamics converge to the invariant measure in the Wasserstein distance. Numerical experiments are conducted to illustrate theoretical results.