A partition method for bounding continuous-time Markov chain models of general reaction network

TL;DR

This paper introduces a novel partition-based bounding method for analyzing multi-dimensional continuous-time Markov chains in reaction networks, enabling easier analysis of properties like stationarity and truncation errors.

Contribution

It presents a general, practical approach to bound complex Markov chains with simpler birth-death processes using coupling and transport theory, with explicit formulas for optimal bounds.

Findings

Bounding processes facilitate analysis of stationary distributions.

Explicit formulas improve practical application of the method.

Illustration on a chemical network demonstrates effectiveness.

Abstract

In this work, we present a general method to establish properties of multi-dimensional continuous-time Markov chains representing stochastic reaction networks. This method consists of grouping states together (via a partition of the state space), then constructing two one-dimensional birth and death processes that lower and upper bound the initial process under simple assumptions on the infinitesimal generators of the processes. The construction of these bounding processes is based on coupling arguments and transport theory. The bounding processes are easy to analyse analytically and numerically and allow us to derive properties on the initial continuous-time Markov chain. We focus on two important properties: the behavior of the process at infinity through the existence of a stationary distribution and the error in truncating the state space to numerically solve the master equation describing the time evolution of the probability distribution of the process. We derive explicit formulas for constructing the optimal bounding processes for a given partition, making the method easy to use in practice. We finally discuss the importance of the choice of the partition to obtain relevant results and illustrate the method on an example chemical reaction network.