## The Annals of Probability

- Ann. Probab.
- Volume 43, Number 3 (2015), 1121-1156.

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

Paul Dupuis and Yufei Liu

#### 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.

#### Article information

**Source**

Ann. Probab., Volume 43, Number 3 (2015), 1121-1156.

**Dates**

First available in Project Euclid: 5 May 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1430830279

**Digital Object Identifier**

doi:10.1214/13-AOP883

**Mathematical Reviews number (MathSciNet)**

MR3342660

**Zentralblatt MATH identifier**

1325.60023

**Subjects**

Primary: 60E10: Characteristic functions; other transforms 60J75: Jump processes

**Keywords**

Large deviation rate function reversible Markov process pure jump process empirical measure weak convergence

#### Citation

Dupuis, Paul; Liu, Yufei. On the large deviation rate function for the empirical measures of reversible jump Markov processes. Ann. Probab. 43 (2015), no. 3, 1121--1156. doi:10.1214/13-AOP883. https://projecteuclid.org/euclid.aop/1430830279