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

TL;DR

This paper establishes that, under certain conditions, the invariant measure of a McKean--Vlasov process can be approximated by weighted empirical measures derived from related stochastic processes, with convergence analyzed via Wasserstein distances.

Contribution

It provides a novel approximation method for invariant measures of McKean--Vlasov processes using empirical measures from distribution-dependent SDEs, under a monotonicity condition.

Findings

Convergence of empirical measures to the invariant measure is characterized by Wasserstein distance bounds.

Theoretical results are illustrated with a mean-field Ornstein--Uhlenbeck process.

Weighted empirical measures effectively approximate invariant measures under the specified conditions.

Abstract

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 distance to the invariant measure. The theoretical results are demonstrated via a mean-field Ornstein--Uhlenbeck process.