On absolute continuity of invariant measures associated with a piecewise-deterministic Markov process with random switching between flows

Abstract

We are concerned with the absolute continuity of stationary distributions corre-sponding to some piecewise deterministic Markov process, being typically encoun-tered in biological models. The process under investigation involves a deterministic motion punctuated by random jumps, occurring at the jump times of a Poisson process. The post-jump locations are obtained via random transformations of the pre-jump states. Between the jumps, the motion is governed by continuous semiflows, which are switched directly after the jumps. The main goal of this paper is to provide a set of verifiable conditions implying that any invariant distribution of the process under consideration that corresponds to an ergodic invariant measure of the Markov chain given by its post-jump locations has a density with respect to the Lebesgue measure.