Substochastic semigroups and densities of piecewise deterministic Markov processes

TL;DR

This paper establishes conditions under which certain semigroups related to piecewise deterministic Markov processes are stochastic or stable, with applications to fragmentation equations and a probabilistic interpretation.

Contribution

It provides necessary and sufficient conditions for the stochasticity and stability of substochastic semigroups linked to PDMPs, expanding understanding of their behavior.

Findings

Conditions for semigroup stochasticity and stability are derived.

Semigroups are connected to PDMPs with a probabilistic interpretation.

Applications to fragmentation equations demonstrate practical relevance.

Abstract

Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.