Transport semigroup associated to positive boundary conditions of unit norm: a Dyson-Phillips approach

Abstract

We revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\T_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\T_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t){)}_{t\ge 0}$. We show here that $(V_H(t){)}_{t\ge 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t){)}_{t\ge 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J\subseteq [0,\infty )$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.