A coupling approach for the convergence to equilibrium for a collisionless gas

TL;DR

This paper introduces a probabilistic coupling method to quantify the polynomial rate of convergence to equilibrium for collisionless gases in higher dimensions, without symmetry or monokinetic assumptions, and with general boundary reflections.

Contribution

It provides the first dimension-agnostic quantitative convergence result in collisionless kinetic theory using coupling techniques.

Findings

Refined polynomial convergence rates in total variation distance.

Applicable to general boundary reflection conditions.

No symmetry or monokinetic assumptions required.

Abstract

We use a probabilistic approach to study the rate of convergence to equilibrium for a collisionless (Knudsen) gas in dimension equal to or larger than 2. The use of a coupling between two stochastic processes allows us to extend and refine, in total variation distance, the polynomial rate of convergence given in [AG11] and [KLT13]. This is, to our knowledge, the first quantitative result in collisionless kinetic theory in dimension equal to or larger than 2 that does not require any symmetry of the domain, nor a monokinetic regime. Our study is also more general in terms of reflection at the boundary: we allow for rather general diffusive reflections and for a specular reflection component.