Infinite Dimensional Piecewise Deterministic Markov Processes

TL;DR

This paper develops an infinite-dimensional framework for Piecewise Deterministic Markov Processes, extending Monte Carlo methods like the Bouncy Particle, Zig-Zag, and Boomerang Samplers, and analyzes their convergence and approximation properties.

Contribution

It introduces a novel infinite-dimensional construction for PDMPs and demonstrates convergence and finite-dimensional approximation for the Boomerang Sampler.

Findings

Established exponential convergence to equilibrium for the infinite-dimensional Boomerang Sampler.

Provided a finite-dimensional approximation suitable for simulation.

Extended PDMP methods to infinite-dimensional settings.

Abstract

In this paper we aim to construct infinite dimensional versions of well established Piecewise Deterministic Monte Carlo methods, such as the Bouncy Particle Sampler, the Zig-Zag Sampler and the Boomerang Sampler. In order to do so we provide an abstract infinite-dimensional framework for Piecewise Deterministic Markov Processes (PDMPs) with unbounded event intensities. We further develop exponential convergence to equilibrium of the infinite dimensional Boomerang Sampler, using hypocoercivity techniques. Furthermore we establish how the infinite dimensional Boomerang Sampler admits a finite dimensional approximation, rendering it suitable for computer simulation.