Piecewise Deterministic Markov Processes and their invariant measures

TL;DR

This paper provides a comprehensive analysis of Piecewise Deterministic Markov Processes (PDMPs), establishing their constructions, coupling methods, and properties of their invariant measures, with applications to PDMP-based MCMC algorithms.

Contribution

It introduces equivalent constructions of PDMPs, develops coupling techniques for bounding distributions, and presents new results on invariant measures and their applications to MCMC.

Findings

Equivalent constructions of PDMPs are proven.

Coupling bounds on total variation are established.

Bounds on invariant measures are derived and applied to MCMC bias analysis.

Abstract

Piecewise Deterministic Markov Processes (PDMPs) are studied in a general framework. First, different constructions are proven to be equivalent. Second, we introduce a coupling between two PDMPs following the same differential flow which implies quantitative bounds on the total variation between the marginal distributions of the two processes. Finally two results are established regarding the invariant measures of PDMPs. A practical condition to show that a probability measure is invariant for the associated PDMP semi-group is presented. In a second time, a bound on the invariant probability measures in $V$-norm of two PDMPs following the same differential flow is established. This last result is then applied to study the asymptotic bias of some non-exact PDMP MCMC methods.