Product-form stationary distributions for deficiency zero chemical reaction networks

TL;DR

This paper proves the existence of product-form stationary distributions for certain chemical reaction networks modeled stochastically, linking deterministic complex balance conditions to stochastic stationary distributions, and extends results beyond mass-action kinetics.

Contribution

It establishes a connection between deterministic complex balanced equilibria and stochastic stationary distributions for deficiency zero, weakly reversible networks, including some non-mass-action kinetics.

Findings

Product-form stationary distributions exist under specified conditions.

Feinberg's deficiency zero theorem implies distribution existence for weakly reversible networks.

Generalization to non-mass-action kinetics broadens applicability.

Abstract

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg's deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.