Stochastic Analysis Seminar – Pierre Monmarché (Univeristé Gustave Eiffel)

For the granular media equation, or other Wasserstein gradient flows associated to some free energy, the exponential convergence towards a unique global minimizer is known to follow from a suitable non-linear log-Sobolev inequality. However this inequality cannot hold when the free energy admits non-global local minimizers, as in the granular media case in a double-well potential with attractive interaction below the critical temperature. We will discuss how local inequalities can still be established in this context to obtain local convergence rates for initial conditions in a Wasserstein ball centered at local minimizers. The same analysis works in the kinetic case (i.e. the Vlasov-Fokker-Planck equation). In practice, the flow is approximated by interacting particles, which are metastable (they eventually leave the local minimizer) and thus uniform-in-time convergence results cannot hold. We will see that it is still possible to prove such results over very long times, of practical relevance.