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

TitleContinuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process
Publication TypeJournal Article
Year of Publication2020
AuthorsCzapla D, Hille SC, Horbacz K, Wojewódka-Ściążko H
JournalMathematical Biosciences and Engineering
Volume17
Issue2
Start Page1059-1073
Date Published11/2019
Keywordscontinuous dependence, invariant measure, jump rate, piecewise-deterministic Markov process, random dynamical system
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 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_λ. The aim of this paper is to prove that the map λ↦ν_λ 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.

DOI10.3934/mbe.2020056