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 Ω⊂Rd (d⩾2) with C1 boundary ∂Ω, orthogonally invariant velocity measure m(dv) with support V⊂Rd and stochastic partly diffuse boundary operators H relating the outgoing and incoming fluxes. Under very general conditions, such equations are governed by stochastic C0-semigroups (UH(t))t⩾0 on L1(Ω×V,dx⊗m(dv)). We give a general criterion of irreducibility of (UH(t))t⩾0 and we show that, under very natural assumptions, if an invariant density exists then (UH(t))t⩾0 converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then (UH(t))t⩾0 is sweeping in the sense that, for any density φ, the total mass of UH(t)φ concentrates near suitable sets of zero measure as t→+∞. 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 (UH(t))t⩾0.