On the large deviation rate function for the empirical measures of reversible jump Markov processes

TL;DR

This paper derives an explicit large deviation rate function for the empirical measures of reversible pure jump Markov processes, extending previous results to include these processes which were not covered before.

Contribution

It provides a new explicit formula for the large deviation rate function specifically for reversible pure jump Markov processes, filling a gap in existing literature.

Findings

Explicit rate function derived for reversible pure jump Markov processes.

Extends large deviations results to include continuous time pure jump processes.

Enhances understanding of empirical measure deviations in Markov processes.

Abstract

The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a reversible finite state Markov chain. In this paper, we study the large deviations principle for the empirical measure of pure jump Markov processes and provide an explicit formula of the rate function under reversibility.