Empirical approximation to invariant measures for McKean-Vlasov processes: Mean-field interaction vs self-interaction

This paper proves that, under a monotonicity condition, the invariant probability measure of a McKean-Vlasov process can be approximated by weighted empirical measures of some processes including itself. These processes are described by distribution dependent or empirical measure dependent stochastic differential equations constructed from the equation for the McKean-Vlasov process. Convergence of empirical measures is characterized by upper bound estimates for their Wasserstein distances to the invariant measure. Numerical simulations of the mean-field Ornstein-Uhlenbeck process are implemented to demonstrate the theoretical results.