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
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
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