# Densities for piecewise deterministic Markov processes with boundary

## TL;DR

This paper investigates the existence of probability densities for piecewise deterministic Markov processes with boundaries, establishing relationships between invariant densities and developing a new perturbation theorem for substochastic semigroups.

## Contribution

It introduces a novel perturbation theorem for substochastic semigroups and applies functional-analytic methods to analyze densities of PDMPs with boundary conditions.

## Key findings

- Existence of densities for PDMP distributions established.
- Relationships between invariant densities at different observation times derived.
- A new perturbation theorem for substochastic semigroups developed.

## Abstract

We study the existence of densities for distributions of piecewise deterministic Markov processes. We also obtain relationships between invariant densities of the continuous time process and that of the process observed at jump times. In our approach we use functional-analytic methods and the theory of linear operator semigroups. By imposing general conditions on the characteristics of a given Markov process, we show the existence of a substochastic semigroup describing the evolution of densities for the process and we identify its generator. Our main tool is a new perturbation theorem for substochastic semigroups, where we perturb both the action of the generator and of its domain, allowing to treat general transport-type equations with non-local boundary conditions. A couple of particular examples illustrate our general results.