Ergodic approximation for the invariant distribution: An abstract framework for law-dependent dynamics

Abstract

This paper studies the approximation of invariant distributions for a broad class of law-dependent dynamics, including McKean-Vlasov stochastic differential equations and Boltzmann-type equations. We consider discrete-time approximation schemes with decreasing time steps and analyse the convergence of their associated ergodic (or occupation) measure towards the invariant distribution of the underlying continuous-time process. Under a general coupling assumption, we prove convergence in the expected $p$-Wasserstein distance ($p \ge 1$) and derive explicit convergence rates. Our approach combines estimates on ergodic averages, regularization techniques for discrete measures, and a generalized discrete Gronwall lemma to control the error between the self-interacting scheme and the target invariant measure. We show that our framework applies to a wide range of models: McKean-Vlasov SDEs, a Boltzmann type equation, and a neuronal model.