A note on the jump locations of Markov processes

TL;DR

This paper characterizes the distribution of the first jump location in continuous-time Markov processes using an auxiliary process, aiding in the analysis of advanced Monte Carlo methods.

Contribution

It introduces a novel characterization of first jump locations for Markov processes, relevant for improving Monte Carlo sampling techniques.

Findings

Law of first jump location expressed via invariant distribution

Applicable to piecewise-deterministic Markov chain Monte Carlo methods

Potential for enhanced analysis of quasi-stationary Monte Carlo methods

Abstract

For a continuous-time Markov process, we characterize the law of the first jump location when started from an arbitrary initial distribution, in terms of the invariant distribution of an auxiliary Markov process. This could be of interest in the burgeoning fields of piecewise-deterministic Markov chain Monte Carlo methods and quasi-stationary Monte Carlo methods.