  This paper deals with collisionless transport equations in bounded open
domains $\Omega \subset \R^{d}$ $(d\geq 2)$ with $\mathcal{C}^{1}$ boundary
$\partial \Omega $, orthogonally invariant velocity measure $\bm{m}(\d v)$ with
support $V\subset \R^{d}$ and stochastic partly diffuse
  boundary operators $\mathsf{H}$ relating the outgoing and incoming fluxes.
Under very general conditions, such equations are governed by stochastic
$C_{0}$-semigroups $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ on $%
L^{1}(\Omega \times V,\d x \otimes \bm{m}(\d v)).$ We give a general criterion
of irreducibility of $% \left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ and we show
that, under very natural assumptions, if an invariant density exists then
$\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}$ converges strongly (not simply in
Cesar\`o means) to its ergodic projection. We show also that if no invariant
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 $ U_{\mathsf{H}}(t)\varphi $ concentrates near suitable sets of zero measure
as $ t\rightarrow +\infty .$ 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 $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}.$
