ABSTRACT

Wasserstein gradient flows of free energies are solutions of non-linear parabolic PDE used to solve optimization problems in spaces of probability measures. In practice, in high dimension, they are approximated by mean-field interacting particle systems. We are interested in obtaining non-asymptotic convergence estimates for long trajectories in the situation where the free energy has several critical points, by contrast to previous works. In that case the convergence of the non-linear flow cannot be global, and the particle system is metastable, in the sense that it performs rare transitions between the local minimizers. In particular, propagation of chaos (i.e. the convergence of the particle system to its mean-field limit) cannot be uniform in time, and the mixing time of the particle system towards its ergodic Gibbs measure is extremely slow when the number of particles is large. This talk presents some recent works devoted to the obtention of relevant bounds describing the fast local convergence of the non-linear system and the particle system. The method relies on entropy arguments and functional inequality (specifically, non-linear variations of log-Sobolev inequalities, which are the Polyak-Lojasiewicz inequalities in this context). The same analysis works in the kinetic case (i.e. the Vlasov-Fokker-Planck equation).