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

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 Poincaré inequality, obtained via a Lions–Poincaré inequality and an averaging lemma.