How to construct decay rates for kinetic Fokker--Planck equations?
Résumé
We study time averages for the norm of the solution to kinetic Fokker--Planck equations associated with general Hamiltonians. We provide explicit and constructive decay estimates, allowing fat-tail, sub-exponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The evolution is subject to a transport operator induced by an external space-confining force. The key step in our estimates is a modified Poincaré inequality, obtained via a Lions-Poincaré inequality and an averaging lemma.