A.Bernou & N.Fournier: A Coupling Approach for the Convergence to Equilibrium for a Collisionless Gas

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 Aoki-Golse, 2011 and Kuo-Liu-Tsai, 2013. 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, since we allow for rather general diffusive reflections and for a specular reflection component.

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