A gradient flow approach of uniform in time propagation of chaos

In this talk I will describe a new method to obtain uniform in time rate of convergence from a system of interacting diffusion toward the McKean-Vlasov equation. This method is purely analytical, and is based on Brenier's polar factorization theorem. More precisely we estimate the dissipation of the Wasserstein 2 metric between the law of the N interacting particles and the N times tensorized product of the solution to the McKean-Vlasov equation. We can obtain in this way, a result of Durmus, Eberle, Guillin, and Zimmer concerning particles in a double well confinement proved by a probabilistic approach.