# Hypocoercivity of linear kinetic equations via Harris's Theorem

## TL;DR

This paper establishes explicit convergence rates to equilibrium for linear kinetic equations on both compact and unbounded domains using Harris's Theorem, with exponential rates on the torus and algebraic rates for subquadratic potentials.

## Contribution

It applies Harris's Theorem to derive explicit convergence rates for linear kinetic equations in various settings, including on the torus and with different potential growth conditions.

## Key findings

- Exponential convergence on the torus and with quadratic potentials.
- Algebraic convergence for subquadratic potentials.
- Explicit convergence rates in total variation and weighted norms.

## Abstract

We study convergence to equilibrium of the linear relaxation Boltzmann (also known as linear BGK) and the linear Boltzmann equations either on the torus $(x,v) \in \mathbb{T}^d \times \mathbb{R}^d$ or on the whole space $(x,v) \in \mathbb{R}^d \times \mathbb{R}^d$ with a confining potential. We present explicit convergence results in total variation or weighted total variation norms (alternatively $L^1$ or weighted $L^1$ norms). The convergence rates are exponential when the equations are posed on the torus, or with a confining potential growing at least quadratically at infinity. Moreover, we give algebraic convergence rates when subquadratic potentials considered. We use a method from the theory of Markov processes known as Harris's Theorem.