Local convergence and metastability for mean-field particles in a multi-well potential

We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.