Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators

This paper deals with collisionless transport equations in bounded open domains Omega subset of R-d (d >= 2) with C-1 boundary partial derivative Omega, orthogonally invariant velocity measure m(dv) with support V subset of R-d and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C-0-semigroups (U-H(t))(t >= 0) on L-1 (Omega x V, dx circle times m(dv)). We give a general criterion of irreducibility of (U-H(t))(t >= 0) and we show that, under very natural assumptions, if an invariant density exists then (U-H(t))(t >= 0) converges strongly (not simply in Cesaro means) to its ergodic projection. We show also that if no invariant density exists then (U-H(t))(t >= 0) is sweeping in the sense that, for any density phi, the total mass of U-H(t)phi concentrates near suitable sets of zero measure as t -> +oo. We show also a general weak compactness theorem of interest for the existence of invariant densities. This theorem is based on several results on smoothness and transversality of the dynamical flow associated to (U-H(t))(t >= 0). (C) 2020 Elsevier Masson SAS. All rights reserved.