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2021 Cores for piecewise-deterministic Markov processes used in Markov chain Monte Carlo
Peter Holderrieth
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Electron. Commun. Probab. 26: 1-12 (2021). DOI: 10.1214/21-ECP430

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.

Acknowledgments

I want to express my gratitude to Andreas Eberle for his helpful advice and extensive guidance across the whole course of this project. I also want to thank Nawaf Bou-Rabee for his advice on improving the manuscript. Finally, I would like to thank Benedikt Geiger for his detailed suggestions for improvements of this work.

Citation

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Peter Holderrieth. "Cores for piecewise-deterministic Markov processes used in Markov chain Monte Carlo." Electron. Commun. Probab. 26 1 - 12, 2021. https://doi.org/10.1214/21-ECP430

Information

Received: 17 July 2021; Accepted: 24 September 2021; Published: 2021
First available in Project Euclid: 11 November 2021

Digital Object Identifier: 10.1214/21-ECP430

Subjects:
Primary: 60G53 , 60J25 , 60J35 , 60J76

Keywords: Bouncy particle sampler , Cores , Feller process , Markov chain Monte Carlo , Markov semigroup , piecewise-deterministic Markov process , Randomized Hamiltonian Monte Carlo

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