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

Czapla, D.; Hille, S.C.; Horbacz, K.; Wojewodka-Sciazko, H. (2019)

Article / Letter to editor

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 λ. 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 ν∗λνλ∗. The aim of this paper is to prove that the map λ↦ν∗λλ↦νλ∗ 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.

• invariant measure
• piecewise-deterministic Markov process
• random dynamical system
• jump rate
• continuous dependence

• All authors: Czapla, D.; Hille, S.C.; Horbacz, K.; Wojewodka-Sciazko, H.
• Date: 2019-11-12
• Journal: Mathematical Biosciences and Engineering
• Volume: 17
• Issue: 2
• Pages: 1059 - 1073
• DOI: doi:10.3934/mbe.2020056
• Link: https://www.aimspress.com/article/10.3934/mbe.2020056