On absolute continuity of invariant measures associated with a piecewise-deterministic Markov process with random switching between flows. (December 2021)

• Record Type: Journal Article
• Authors: Czapla, Dawid; Horbacz, Katarzyna; Wojewódka-Ściążko, Hanna
• Is Part Of: Nonlinear analysis. Volume 213(2021)
• Publication Date: 2021-12
• Subjects: 60J25 -- 60G30 -- 60J35 -- 60J05; Piecewise-deterministic Markov process -- Invariant measure -- Ergodic measure -- Absolute continuity -- Singularity -- Switching semiflows; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112522
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We are concerned with the absolute continuity of stationary distributions corresponding to some piecewise deterministic Markov process, being typically encountered in biological models. The process under investigation involves a deterministic motion punctuated by random jumps, occurring at the jump times of a Poisson process. The post-jump locations are obtained via random transformations of the pre-jump states. Between the jumps, the motion is governed by continuous semiflows, which are switched directly after the jumps. The main goal of this paper is to provide a set of verifiable conditions implying that any invariant distribution of the process under consideration that corresponds to an ergodic invariant measure of the Markov chain given by its post-jump locations has a density with respect to the Lebesgue measure.