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

## TL;DR

This paper analyzes the long-term behavior of collisionless kinetic equations with partly diffuse boundary conditions, establishing criteria for invariant densities, convergence, and mass concentration phenomena in bounded domains.

## Contribution

It provides a general criterion for irreducibility, conditions for convergence to ergodic projections, and a sweeping result for equations with partly diffuse boundary operators.

## Key findings

- Convergence to invariant densities under natural assumptions
- Mass concentration near zero-measure sets if no invariant density exists
- A weak compactness theorem related to invariant density existence

## Abstract

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 $\left( U_{\mathsf{H}}(t)\right) _{t\geq 0}.$