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We show fundamental properties of the Markov semigroup of recently proposedMCMC algorithms based on Piecewise-deterministic Markov processes (PDMPs) suchas the Bouncy Particle Sampler, the Zig-Zag process or the RandomizedHamiltonian Monte Carlo method. Under assumptions typically satisfied in MCMCsettings, we prove that PDMPs are Feller and that their generator admits thespace of infinitely differentiable functions with compact support as a core. Aswe illustrate via martingale problems and a simplified proof of the invarianceof target distributions, these results provide a fundamental tool for therigorous analysis of these algorithms and corresponding stochastic processes.

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