Large deviations of Markov chains with multiple time-scales

TL;DR

This paper develops a general framework for establishing large deviation principles in path space for multi-scale Markov processes, with applications to systems biology and chemical reaction networks.

Contribution

It introduces a new approach to large deviations for multi-scale Markov processes, extending previous laws of large numbers and central limit theorems.

Findings

Large deviation principles are established for multi-scale Markov processes.

Explicit calculations are provided for chemical reaction systems with multiple time-scales.

The framework applies to models in systems biology and related fields.

Abstract

For Markov processes evolving on multiple time-scales a combination of large component scalings and averaging of rapid fluctuations can lead to useful limits for model approximation. A general approach to proving a law of large numbers to a deterministic limit and a central limit theorem around it have already been proven in [16] and [17]. We present here a general approach to proving a large deviation principle in path space for such multi-scale Markov processes. Motivated by models arising in systems biology, we apply these large deviation results to general chemical reaction systems which exhibit multiple time-scales, and provide explicit calculations for several relevant examples.