This paper deals with collisionless transport equations in bounded open domains $\mathrm{\Omega }\subset {\mathbb{R}}^d$ $(d⩾2)$ with ${\mathcal{C}}^1$ boundary ∂Ω, orthogonally invariant velocity measure $\mathit{m}(\mathrm{d}v)$ with support $V\subset {\mathbb{R}}^d$ 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 ${(U_{\mathsf{H}}(t))}_{t⩾0}$ on $L^1(\mathrm{\Omega }\times V,\mathrm{d}x\otimes \mathit{m}(\mathrm{d}v))$. We give a general criterion of irreducibility of ${(U_{\mathsf{H}}(t))}_{t⩾0}$ and we show that, under very natural assumptions, if an invariant density exists then ${(U_{\mathsf{H}}(t))}_{t⩾0}$ converges strongly (not simply in Cesarò means) to its ergodic projection. We show also that if no invariant density exists then ${(U_{\mathsf{H}}(t))}_{t⩾0}$ is sweeping in the sense that, for any density φ, the total mass of $U_{\mathsf{H}}(t)\phi$ concentrates near suitable sets of zero measure as $t\to +\infty$. 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 ${(U_{\mathsf{H}}(t))}_{t⩾0}$.

本文讨论有界开放域中的无碰撞输运方程 $\mathrm{\Omega }\subset {\mathbb{[R}}^d$ $（d⩾2）$ 与 ${\mathcal{C}}^{\mathrm{1个}}$ 边界∂Ω，正交不变速度测度 $\mathit{米}（\mathrm{d}v）$ 在支持下 $V\subset {\mathbb{[R}}^d$ 和随机的部分扩散边界算子 $\mathsf{H}$关联输出和输入通量。在非常普遍的条件下，这些方程受随机控制$C_0$-半组 ${（{ü}_{\mathsf{H}}（Ť））}_{Ť⩾0}$ 上 ${\mathrm{大号}}^{\mathrm{1个}}（\mathrm{\Omega }\times V，\mathrm{d}X\otimes \mathit{米}（\mathrm{d}v））$。我们给出了不可约的一般判据${（{ü}_{\mathsf{H}}（Ť））}_{Ť⩾0}$ 我们证明，在非常自然的假设下，如果存在不变密度，则 ${（{ü}_{\mathsf{H}}（Ť））}_{Ť⩾0}$强烈地趋同于其遍历投影（不只是以塞萨罗语为代表）。我们还表明，如果不存在不变密度，则${（{ü}_{\mathsf{H}}（Ť））}_{Ť⩾0}$在某种意义上是扫掠的，对于任何密度φ，总质量${ü}_{\mathsf{H}}（Ť）\phi$ 集中在零度量的合适集合附近 $Ť\to +\infty$。我们还显示了不变密度的存在所关心的一般弱紧致性定理。这个定理是基于与流动相关的动力学流动的光滑性和横向性的几个结果。${（{ü}_{\mathsf{H}}（Ť））}_{Ť⩾0}$。