Large deviations of the empirical flow for continuous time Markov chains

TL;DR

This paper establishes a large deviation principle for the empirical measure and flow in continuous time Markov chains, providing insights into the probabilities of rare events involving state transitions.

Contribution

It introduces a joint large deviation principle for empirical measure and flow, with direct and indirect proofs, advancing understanding of Markov chain fluctuations.

Findings

Proves a joint large deviation principle for empirical measure and flow.

Provides both direct and contraction-based proofs.

Enhances understanding of rare event probabilities in Markov processes.

Abstract

We consider a continuous time Markov chain on a countable state space and prove a joint large deviation principle for the empirical measure and the empirical flow, which accounts for the total number of jumps between pairs of states. We give a direct proof using tilting and an indirect one by contraction from the empirical process.