Eyring-Kramers type formulas for some piecewise deterministic Markov processes

TL;DR

This paper derives precise asymptotic formulas for the smallest eigenvalues of generators of certain piecewise deterministic Markov processes, such as ZigZag and Bouncy Particle Sampler, in the small temperature limit, including cases with zero refreshment rate.

Contribution

It provides sharp Eyring-Kramers type asymptotic formulas for eigenvalues of PDMP generators on the 1D torus, extending to zero refreshment rate scenarios.

Findings

Derived asymptotic equivalents for eigenvalues in small temperature regime.

Extended formulas to cases with vanishing refreshment rate.

Applicable to processes like ZigZag and Bouncy Particle Sampler.

Abstract

In this work, we give sharp asymptotic equivalents in the small temperature regime of the smallest eigenvalues of the generator of some piecewise deterministic Markov processes (including the ZigZag process and the Bouncy Particle Sampler process) with refreshment rate $\alpha$ on the one-dimensional torus T. These asymptotic equivalents are usually called Eyring-Kramers type formulas in the literature. The case when the refreshment rate $\alpha$ vanishes on T is also considered.