Cores for Piecewise-Deterministic Markov Processes used in Markov Chain Monte Carlo

TL;DR

This paper establishes fundamental mathematical properties of PDMP-based MCMC algorithms, such as Feller property and generator cores, providing a rigorous foundation for analyzing these stochastic sampling methods.

Contribution

It proves that PDMP-based MCMC algorithms are Feller processes with generators having smooth compactly supported functions as cores, aiding their theoretical analysis.

Findings

PDMPs are Feller processes under typical MCMC assumptions.

The generator of PDMPs admits smooth functions with compact support as a core.

Results facilitate rigorous analysis of PDMP-based MCMC algorithms.

Abstract

We show fundamental properties of the Markov semigroup of recently proposed MCMC algorithms based on Piecewise-deterministic Markov processes (PDMPs) such as the Bouncy Particle Sampler, the Zig-Zag process or the Randomized Hamiltonian Monte Carlo method. Under assumptions typically satisfied in MCMC settings, we prove that PDMPs are Feller and that their generator admits the space of infinitely differentiable functions with compact support as a core. As we illustrate via martingale problems and a simplified proof of the invariance of target distributions, these results provide a fundamental tool for the rigorous analysis of these algorithms and corresponding stochastic processes.