## The Annals of Probability

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

#### Article information

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

Dates
First available in Project Euclid: 5 May 2015

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

Digital Object Identifier
doi:10.1214/13-AOP883

Mathematical Reviews number (MathSciNet)
MR3342660

Zentralblatt MATH identifier
1325.60023

#### 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

#### References

• [1] Chen, M. F. and Lu, Y. G. (1990). On evaluating the rate function of large deviations for jump processes. Acta Math. Sinica (N.S.) 6 206–219.
• [2] Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I. Comm. Pure Appl. Math. 28 1–47.
• [3] Donsker, M. D. and Varadhan, S. R. S. (1976). Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 389–461.
• [4] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
• [5] Ellis, R. S. and Wyner, A. D. (1989). Uniform large deviation property of the empirical process of a Markov chain. Ann. Probab. 17 1147–1151.
• [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
• [7] Folland, G. B. (1999). Real Analysis: Modern Techniques and their Applications, 2nd ed. Wiley, New York.
• [8] Liu, Y. (2013). Large deviations rate functions for the empirical measure: Explicit formulas and an application to Monte Carlo. Ph.D. thesis, Brown Univ., Providence, RI.
• [9] Pinsky, R. (1985). On evaluating the Donsker–Varadhan $I$-function. Ann. Probab. 13 342–362.
• [10] Rudin, W. (1991). Functional Analysis, 2nd ed. McGraw-Hill, New York.
• [11] Stroock, D. W. (1984). An Introduction to the Theory of Large Deviations. Springer, New York.