This paper deals with collisionless transport equations in bounded open domains with boundary ∂Ω, orthogonally invariant velocity measure with support and stochastic partly diffuse boundary operators relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic -semigroups on . We give a general criterion of irreducibility of and we show that, under very natural assumptions, if an invariant density exists then converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then is sweeping in the sense that, for any density φ, the total mass of concentrates near suitable sets of zero measure as . 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 .
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B. Lods, M. Mokhtar-Kharroubi, R. Rudnicki, Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 4, pp. 877–923DOI 10.1016/J.ANIHPC.2020.02.004