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This paper proves that, under a monotonicity condition, the invariantprobability measure of a McKean--Vlasov process can be approximated by weightedempirical measures of some processes including itself. These processes aredescribed by distribution dependent or empirical measure dependent stochasticdifferential equations constructed from the equation for the McKean--Vlasovprocess. Convergence of empirical measures is characterized by upper boundestimates for their Wasserstein distance to the invariant measure. Thetheoretical results are demonstrated via a mean-field Ornstein--Uhlenbeckprocess.

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