Central limit theorems and diffusion approximations for multiscale Markov chain models

TL;DR

This paper develops a general framework for proving central limit theorems for multiscale Markov chain models, enabling the derivation of diffusion approximations in systems with multiple time-scales, especially in systems biology.

Contribution

It introduces a unified approach to establish CLTs for multiscale Markov processes, facilitating diffusion approximations in complex systems.

Findings

Central limit theorem for multiscale Markov chains

Diffusion (Langevin) approximations derived from CLTs

Applicable to systems with multiple time-scales in biology

Abstract

Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a combination of the two. Motivated by models with multiple time-scales arising in systems biology, we present a general approach to proving a central limit theorem capturing the fluctuations of the original model around the deterministic limit. The central limit theorem provides a method for deriving an appropriate diffusion (Langevin) approximation.