Substochastic semigroups and densities of piecewise deterministic Markov processes

Necessary and sufficient conditions are given for a substochastic semigroup on L1 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.