We extend the hypocoercivity framework for piecewise-deterministic Markov process (PDMP) Monte Carlo established in  to heavy-tailed target distributions, which exhibit subgeometric rates of convergence to equilibrium. We make use of weak Poincaré inequalities, as developed in the work of , the ideas of which we adapt to the PDMPs of interest. On the way we report largely potential-independent approaches to bounding explicitly solutions of the Poisson equation of the Langevin diffusion and its first and second derivatives, required here to control various terms arising in the application of the hypocoercivity result.
Research of CA supported by EPSRC grants Bayes4Health, ‘New Approaches to Bayesian Data Science: Tackling Challenges from the Health Sciences’ (EP/R018561/1) and ‘CoSInES (COmputational Statistical INference for Engineering and Security)’ (EP/R034710/1). Research of PD supported by the research programme ‘Zigzagging through computational barriers’ with project number 016.Vidi.189.043, financed by the Dutch Research Council (NWO). Research of AQW supported by EPSRC grant CoSInES (EP/R034710/1).
We are very grateful to the anonymous reviewer, whose helpful remarks have improved the paper. We would like to thank the Heilbronn Institute for Mathematical Research Research for funding the Hypocoercivity Workshop held at the University of Bristol, March 2020, which initiated this research. We would like to thank Joris Bierkens, Anthony Lee and Sam Power for interesting discussions related to this work.
"Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods." Electron. J. Probab. 26 1 - 26, 2021. https://doi.org/10.1214/21-EJP643