Hypocoercivity of linear kinetic equations via Harris's Theorem
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.