We consider a collisionless kinetic equation describing the probability density of particles moving in a one-dimensional domain subject to partly diffusive reflection at the boundary. It was shown in 2017 by Mokhtar-Kharroubi and Rudnicki that for large times such systems either converge to an invariant density or, if no invariant density exists, exhibit a so-called “sweeping phenomenon” in which the mass concentrates near small velocities. This dichotomy is obtained by means of subtle arguments relying on the theory of positive operator semigroups. In this talk I shall review some of these results before discussing how, under suitable assumptions both on the boundary operators (which in particular ensure that an invariant density exists) and on the initial density, one may even obtain estimates on the rate at which the system converges to its equilibrium. This is joint work with Mustapha Mokhtar-Kharroubi (Besançon).
