How to construct explicit decay rates for kinetic Fokker—Planck equations?

In this talk, I will discuss a few results (mostly published in the last three years) in the study of kinetic Fokker—Planck equations, addressing the long-time behaviour of solutions. After recalling the classic hypocoercivity theory, I will stick to the recent variational framework of Albritton, Armstrong, Mourrat, and Novack. Then, I will present a constructive version of it, derived from works of Cao, Lu, L. Wang, Stoltz, A. Wang, Eberle, Lörler, and myself. 

The main goal of my talk is to show how this framework yields explicit convergence rates to equilibrium for kinetic Fokker—Planck equations, according to different concentration properties of the equilibrium measures.

Also by making a link with the novel theory of « lifts » for diffusion processes, I will discuss dependence of the convergence rates on key-parameters of the equation, such as the dimension or the relaxation time of the associated overdamped limit.