Exponential ergodicity in the bounded-Lipschitz distance for some piecewise-deterministic Markov processes with random switching between flows

TitleExponential ergodicity in the bounded-Lipschitz distance for some piecewise-deterministic Markov processes with random switching between flows
Publication TypeJournal Article
Year of Publication2022
AuthorsCzapla D, Horbacz K, Wojewódka-Ściążko H
JournalNonlinear Analysis
Volume213
Start Page112678
Date Published02/2022
Keywordscoupling, exponential ergodicity, Fortet–Mourier distance, gene expression, piecewise-deterministic Markov process, switching semiflows
Abstract

 In this paper, we study a subclass of piecewise-deterministic Markov processes with a Polish state space, involving deterministic motion punctuated by random jumps that occur at exponentially distributed time intervals. Over each of these intervals, the process follows a flow, selected randomly among a finite set of all possible ones. Our main goal is to provide a set of verifiable conditions guaranteeing the exponential ergodicity for such processes (in terms of the bounded Lipschitz distance), which would refer only to properties of the flows and the transition law of the Markov chain given by the post-jump locations. Moreover, we establish a simple criterion on the exponential ergodicity for a particular instance of these processes, applicable to certain biological models, where the jumps result from the action of an iterated function system with place-dependent probabilities.

DOI10.1016/j.na.2021.112678