Subgeometric hypocoercivity for piecewise-deterministic Markov process Monte Carlo methods

TL;DR

This paper extends hypocoercivity techniques for piecewise-deterministic Markov process Monte Carlo methods to heavy-tailed distributions, enabling analysis of subgeometric convergence rates using weak Poincaré inequalities.

Contribution

It adapts hypocoercivity framework to heavy-tailed distributions and develops potential-independent bounds for solutions of the Poisson equation in Langevin diffusions.

Findings

Extended hypocoercivity to heavy-tailed distributions.

Derived potential-independent bounds for Poisson equation solutions.

Analyzed subgeometric convergence rates for PDMP Monte Carlo.

Abstract

We extend the hypocoercivity framework for piecewise-deterministic Markov process (PDMP) Monte Carlo established in [Andrieu et. al. (2018)] to heavy-tailed target distributions, which exhibit subgeometric rates of convergence to equilibrium. We make use of weak Poincar\'e inequalities, as developed in the work of [Grothaus and Wang (2019)], 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.