Long-Time Behaviors of Mean-Field Interacting Particle Systems Related to McKean–Vlasov Equations

In this paper, we investigate concentration inequalities, exponential convergence in the Wasserstein metric W₁, and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean–Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply to possibly multi-well confinement potentials, and interaction potentials W with bounded second mixed derivatives ∇xy2W which are not too big, so that there is no phase transition. Several examples are provided to illustrate the results.