On explicit $L^2$-convergence rate estimate for piecewise deterministic Markov processes in MCMC algorithms

We establish $L^2$-exponential convergence rate for three popular piecewise deterministic Markov processes for sampling: the randomized Hamiltonian Monte Carlo method, the zigzag process, and the bouncy particle sampler. Our analysis is based on a variational framework for hypocoercivity, which combines a Poincaré-type inequality in time-augmented state space and a standard $L^2$ energy estimate. Our analysis provides explicit convergence rate estimates, which are more quantitative than existing results.

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