Cores for Piecewise-Deterministic Markov Processes used in Markov Chain Monte Carlo
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This article provides a tool for the analysis of Piecewise-deterministic Markov processes (PDMPs) which have recently gained attention with the proposal of new efficient Markov Chain Monte Carlo schemes such as the Bouncy Particle Sampler (BPS), the Zig-Zag process or the Randomized Hamiltonian Monte Carlo method (RHMC). A thorough theoretical analysis often relies on the generator of the semigroup of these processes. In this work, the main theorem establishes that the infinitely differentiable functions with compact support form a core of the generator of a PDMP under assumptions which are typically fulfilled in piecewise-deterministic MCMC schemes such as the RHMC, the BPS or the Zig-Zag. Due to this result, one is often able to focus on test functions in the analysis of these MCMC algorithms which is illustrated on the example of martingale problems at the end of this work.
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