Explicit formula for the density of local times of Markov Jump Processes

TL;DR

This paper derives a simple explicit formula involving Bessel functions for the joint density of local times, last exit trees, and cycling numbers in finite-state continuous-time Markov chains.

Contribution

It provides a novel explicit formula for the joint density of key Markov chain quantities, linking local times with Bessel functions.

Findings

Explicit formula involving Bessel functions for joint density.

Simplifies analysis of local times in Markov processes.

Applicable to finite-state continuous-time Markov chains.

Abstract

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov Chains on finite graphs, which involves the modified Bessel function of the first type.