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

TL;DR

This paper establishes verifiable conditions for exponential ergodicity in a subclass of piecewise-deterministic Markov processes with random switching, focusing on flows and transition laws, with applications to biological models.

Contribution

It provides new criteria for exponential ergodicity based on properties of flows and transition laws, applicable to biological models with place-dependent jumps.

Findings

Derived verifiable conditions for exponential ergodicity.

Established a criterion for biological models with iterated function systems.

Demonstrated applicability to processes with random switching between flows.

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.