Empirical approximation to invariant measures of mean-field Langevin dynamics

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.