= Product-Poisson stationary distribution
{title2=$\pi(n)=\prod_i e^{-c_i}c_i^{n_i}/n_i!$}
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 $\pi(n,m)=e^{-a-b}a^nb^m/(n!m!)$, 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>.
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