An algebraic approach to product-form stationary distributions for some reaction networks

TL;DR

This paper develops an algebraic framework to characterize when certain chemical reaction networks have product-form stationary distributions, providing necessary and sufficient conditions and identifying parameter regions.

Contribution

It introduces an algebraic approach to fully characterize classes of reaction networks with product-form stationary distributions, extending beyond known special cases.

Findings

Provides algebraic conditions for product-form distributions in CRNs

Identifies parameter regions where product-form holds

Applies theory to models in statistical mechanics

Abstract

Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative ergodic CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.