Existence of invariant densities for semiflows with jumps

TL;DR

This paper investigates conditions for the existence and uniqueness of invariant densities in certain jump processes, using semigroup theory, and applies results to a gene expression model.

Contribution

It introduces a new criterion for the existence of a unique invariant density for piecewise deterministic Markov processes.

Findings

Established a criterion for invariant density existence

Proved uniqueness of invariant densities under certain conditions

Applied results to a gene expression model with bursting

Abstract

The problem of existence and uniqueness of absolutely continuous invariant measures for a class of piecewise deterministic Markov processes is investigated using the theory of substochastic semigroups obtained through the Kato--Voigt perturbation theorem on the $L^1$-space. We provide a new criterion for the existence of a strictly positive and unique invariant density for such processes. The long time qualitative behavior of the corresponding semigroups is also considered. To illustrate our general results we give a detailed study of a two dimensional model of gene expression with bursting.