Stationary distributions via decomposition of stochastic reaction networks

TL;DR

This paper introduces a method for analyzing the stationary distributions of stochastic reaction networks by decomposing their associated Markov processes, enabling easier calculation and understanding of their long-term behavior.

Contribution

It develops a recursive decomposition approach for Markov chains of CRNs, allowing the derivation of stationary distributions from parts of the network, with broad applicability.

Findings

Decomposition simplifies stationary distribution calculations.

Conditions for deriving distributions are easily checkable.

Method applies to various stochastic models beyond CRNs.

Abstract

We examine reaction networks (CRNs) through their associated continuous-time Markov processes. Studying the dynamics of such networks is in general hard, both analytically and by simulation. In particular, stationary distributions of stochastic reaction networks are only known in some cases. We analyze class properties of the underlying continuous-time Markov chain of CRNs under the operation of join and examine conditions such that the form of the stationary distributions of a CRN is derived from the parts of the decomposed CRNs. The conditions can be easily checked in examples and allow recursive application. The theory developed enables sequential decomposition of the Markov processes and calculations of stationary distributions. Since the class of processes expressible through such networks is big and only few assumptions are made, the principle also applies to other stochastic models. We give examples of interest from CRN theory to highlight the decomposition.