How to construct decay rates for kinetic Fokker--Planck equations?

Abstract: We study time averages for the norm of solutions to kinetic Fokker--Planck equations associated with general Hamiltonians. We provide fully explicit and constructive decay estimates for systems subject to a confining potential, allowing fat-tail, sub-exponential and (super-)exponential local equilibria, which also include the classic Maxwellian case. The key step in our estimates is a modified Poincar\'e inequality, obtained via a Lions--Poincar\'e inequality and an averaging lemma.