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We study ergodic properties of nonlinear Markov chains and stochastic McKean--Vlasov equations. For nonlinear Markov chains we obtain sufficient conditions for existence and uniqueness of an invariant measure and uniform ergodicity. We also prove optimality of these conditions. For stochastic McKean--Vlasov equations we establish exponential convergence of their solutions to stationarity in the total variation metric under Veretennikov--Khasminskii-type conditions.
Oleg Butkovsky (Wed,) studied this question.