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

## TL;DR

This paper develops explicit decay estimates for solutions to kinetic Fokker-Planck equations with various potentials, using a modified Poincaré inequality to handle different equilibrium types.

## Contribution

It introduces a new approach to construct decay rates for kinetic Fokker-Planck equations with general Hamiltonians, including non-Maxwellian equilibria.

## Key findings

- Derived explicit decay estimates for solutions.
- Applicable to fat-tail, sub-exponential, and super-exponential equilibria.
- Unified approach using a modified Poincaré inequality.

## 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.