Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations

TL;DR

This paper develops a non-autonomous honesty theory for substochastic evolution families in abstract state spaces, extending previous semigroup results and applying them to linear kinetic equations like Boltzmann and fragmentation equations.

Contribution

It introduces a non-autonomous honesty framework in abstract state spaces and connects it with existing semigroup theories, with applications to kinetic equations.

Findings

Established honesty theory for non-autonomous substochastic evolution families.

Linked honesty theory with perturbed substochastic semigroups.

Applied the theory to linear Boltzmann and fragmentation equations.

Abstract

We provide a honesty theory of substochastic evolution families in real abstract state space, extending to an non-autonomous setting the result obtained for $C_0$-semigroups in our recent contribution \textit{[On perturbed substochastic semigroups in abstract state spaces, \textit{Z. Anal. Anwend.} \textbf{30}, 457--495, 2011]}. The link with the honesty theory of perturbed substochastic semigroups is established. Several applications to non-autonomous linear kinetic equations (linear Boltzmann equation and fragmentation equation) are provided.