October 2021 Hypocoercivity of piecewise deterministic Markov process-Monte Carlo
Christophe Andrieu, Alain Durmus, Nikolas Nüsken, Julien Roussel
Author Affiliations +
Ann. Appl. Probab. 31(5): 2478-2517 (October 2021). DOI: 10.1214/20-AAP1653


In this work, we establish L2-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.


CA acknowledges support from EPSRC “Intractable Likelihood: New Challenges from Modern Applications (ILike)” (EP/K014463/1) and “Computational Statistical Inference for Engineering and Security (CoSInES)”, (EP/R034710/1). AD acknowledges support of the Lagrange Mathematical and Computing Research Center. JR would like to thank Pierre Monmarché for showing him how ZZ and BPS fall under a general framework. All the authors acknowledge the support of the Institute for Statistical Science in Bristol.


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Christophe Andrieu. Alain Durmus. Nikolas Nüsken. Julien Roussel. "Hypocoercivity of piecewise deterministic Markov process-Monte Carlo." Ann. Appl. Probab. 31 (5) 2478 - 2517, October 2021. https://doi.org/10.1214/20-AAP1653


Received: 1 April 2019; Revised: 1 October 2020; Published: October 2021
First available in Project Euclid: 29 October 2021

Digital Object Identifier: 10.1214/20-AAP1653

Primary: 60J22 , 60J25 , 65C40

Keywords: geometric convergence , Hypoellipticity , PDMCMC

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.31 • No. 5 • October 2021
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