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

Dawid Czapla, Katarzyna Horbacz, Hanna Wojewódka-Ściążko

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.

Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov 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.

Paper Structure

This paper contains 9 sections, 9 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model under study
  4. Correspondence between invariant distributions of $\Psi$ and $\Phi$
  5. A coupling argument (involving $\Phi$) for establishing the exponential ergodicity in $d_{FM,c}$ of both $\Phi$ and $\Psi$
  6. Sufficient conditions for exponential ergodicity in $d_{FM,c}$
  7. The main result
  8. A model with jumps generated by random iterations
  9. Proof of Proposition \ref{['lem:coup']}

Key Result

Theorem 4.1

Let $P$ and $\{P_t\}_{t\in\mathbb{R}_+}$ denote the transition operator of the chain $\Phi$ and the transition semigroup of the PDMP $\Psi$, respectively. Further, let $J$ be the kernel appearing in def:P_bar, and suppose that Then

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 2.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.1
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 15 more