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.
Highlights
Markov chain Monte Carlo (MCMC) is a widely-applied class of algorithms which use Markov chains to sample from a probability distribution
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
Summary
Markov chain Monte Carlo (MCMC) is a widely-applied class of algorithms which use Markov chains to sample from a probability distribution. Similiar to classical MCMC algorithms, we can use PDMPs to sample from a target distribution by designing the deterministic dynamics and the jump mechanism in a way such that the target distribution is an invariant distribution of the PDMP Among those recently proposed piecewise-deterministic MCMC schemes are the Bouncy Particle Sampler (BPS) [8], the Zig-Zag process [5], the Randomized Hamiltonian Monte Carlo method (RHMC) [7], and their variations [25, 24]. The goal of this work is to help out here: we give sufficient conditions for the PDMP to be a Feller process and for the generator L to admit the space Cc∞(Rd) of compactly supported, infinitely differentiable functions as a core
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