On collisionless transport semigroups with boundary operators of norm one (Q932065)

The study of generation results for collisionless streaming operators \(T_H = -(v,\text{grad})\) in subsets of \(\mathbb {R}^n\), with boundary conditions modeled by a positive contraction \(H\) connecting the incoming and outgoing fluxes of particles, goes back to J. Voigt's habilitation thesis (1980). In the context of \({L^1}\)-spaces and norm 1 operators \(H\), covering the case of conservative boundary conditions, some extension \(A \supset T_H\) generates a positive contraction semigroup which need not be stochastic, i.e., mass preserving on the positive cone. Recently, \textit{L. Arlotti} and \textit{B. Lods} [J. Evol. Equ. 5, No.~4, 485--508 (2005; Zbl 1117.47029)] obtained a description of Dom\((A)\). Dealing mainly with conservative \(H\) \((\|H\varphi\|=\|\varphi\|\), \(\varphi\geq0)\), they gave new characterizations of the stochasticity and rederived Voigt's sufficient condition. The present paper treats arbitrary positive \(H\) with \(\|H\|=1\) and, for initial \(f\geq0\), provides an analysis of the total mass carried by individual trajectories \(e^{tA}f\), \( t\geq0\). Among the related established results are sufficient conditions for stochasticity and for \(\overline{T_H}=A\).