Hypocoercivity of piecewise deterministic Markov process-Monte Carlo

Abstract

In this work, we establish ${\mathrm{L}}^2$-exponential convergence for a broad class of piecewise deterministic Markov processes recently proposed in the context of Markov process Monte Carlo methods and covering in particular the randomized Hamiltonian Monte Carlo (Trans. Amer. Math. Soc. 367 (2015) 3807–3828; Ann. Appl. Probab. 27 (2017) 2159–2194), the zig-zag process (Ann. Statist. 47 (2019) 1288–1320) and the bouncy particle Sampler (Phys. Rev. E 85 (2012) 026703; J. Amer. Statist. Assoc. 113 (2018) 855–867). The kernel of the symmetric part of the generator of such processes is nontrivial, and we follow the ideas recently introduced in (C. R. Math. Acad. Sci. Paris 347 (2009) 511–516; Trans. Amer. Math. Soc. 367 (2015) 3807–3828) to develop a rigorous framework for hypocoercivity in a fairly general and unifying set-up, while deriving tractable estimates of the constants involved in terms of the parameters of the dynamics. As a by-product we characterize the scaling properties of these algorithms with respect to the dimension of classes of problems, therefore providing some theoretical evidence to support their practical relevance.