Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

TL;DR

This paper proves that the invariant measure of a piecewise-deterministic Markov process depends continuously on the jump rate parameter, extending understanding of stochastic models in biological systems.

Contribution

It establishes the weak convergence continuity of the invariant measure with respect to the jump rate parameter in a general class of PDMPs.

Findings

Invariant measure depends continuously on jump rate λ

Continuity holds in the topology of weak convergence

Applicable to models of cell division and gene expression

Abstract

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity $\lambda$. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.