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On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows

Dawid Czapla, Katarzyna Horbacz, Hanna Wojewódka-Ściążko

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.

On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows

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.

Paper Structure

This paper contains 7 sections, 12 theorems, 126 equations.

Table of Contents

  1. Preliminaries
  2. Description of the model
  3. Main results
  4. Singularity/absolute continuity dichotomy of ergodic $P$-invariant measures
  5. A criterion on absolute continuity of ergodic invariant measures associated with the model
  6. The existence and uniqueness of invariant measures
  7. Examples

Key Result

Lemma 1.1

Suppose that $P:\mathcal{M}_{fin}(E)\to \mathcal{M}_{fin}(E)$ is a regular Markov operator which preserves absolute continuity of measures, i.e. $P(\mathcal{M}_{ac}(E,m))\subset \mathcal{M}_{ac}(E,m)$. Then, every ergodic invariant probability measure of $P$ is either absolutely continuous or singul

Theorems & Definitions (34)

  • Remark 1.1
  • proof
  • Remark 1.2
  • Lemma 1.1
  • proof
  • Theorem 2.1: b:czapla_erg
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 24 more