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 (Kinet. Relat. Models 4 (2011) 87–107) and (Comm. Math. Phys. 318 (2013) 375–409). 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.
A.B. gratefully acknowledges support by grants from Région Ile-de-France.
The authors would like to thank the anonymous referee for their careful reading and useful comments.
"A coupling approach for the convergence to equilibrium for a collisionless gas." Ann. Appl. Probab. 32 (2) 764 - 811, April 2022. https://doi.org/10.1214/21-AAP1696