 Product-Poisson stationary distribution

An invariant distribution of a continuous-time Markov chain whose coordinates are independent Poisson distributions. This property needs verification from the reaction propensity functions; it does not follow merely from having an equilibrium for the deterministic reaction-rate equations.

For example, consider the cyclic chemical reaction network with stoichiometric vectors $(1,0)$, $(0,1)$ and $(-1,-1)$ and respective reaction propensity functions $\lambda$, $\beta n$, $\gamma nm$, where all three constants are positive. Put $a=\lambda /\beta$ and $b=\beta /\gamma$. For $\pi (n,m)=e^{-a-b}{a}^n{b}^m/(n!m!)$, the three incoming terms in the chemical master equation are respectively $\beta n\pi (n,m)$, $\gamma nm\pi (n,m)$ and $\lambda \pi (n,m)$. They exactly cancel the outgoing terms. Boundary terms vanish by the zero negative-index convention, and normalization follows from the two Poisson distributions. Establishing a limiting probability distribution additionally requires nonexplosion, an irreducible Markov chain and positive recurrence.