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

International audienceThis paper deals with collisionless transport equationsin bounded open domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary $\partial \Omega$, orthogonallyinvariant velocity measure \bm{m}(\d v) with support $V\subset \R^{d}$ and stochastic partly diffuse boundary operators $\mathsf{H}$ relating the outgoing andincoming fluxes. Under very general conditions, such equations are governedby stochastic $C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $$ We give a general criterion of irreducibility of $$ and we show that, under very natural assumptions, if an invariant densityexists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (notsimply in Cesar\`o means) to its ergodic projection. We show also that if noinvariant density exists then $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ is\emph{sweeping} in the sense that, for any density $\varphi$, the total mass of $$ concentrates near suitable sets of zero measure as $$ We show also a general weak compactness theoremwhich provides a basis for a general theory on existence of invariantdensities. This theorem is based on a series of results on smoothness andtransversality of the dynamical flow associated to $\left( U_{\mathsf{H}}(t)\right) _{t\geq0}.