SIAM Conference on Applications of Dynamical Systems (DS21)

Extensions of Detailed Balance to Not Necessarily Reversible Markov Chains

New theoretical results on stochastic reaction networks

We say that a measure $\pi$ for a continuous-time Markov chain with transition rates $q(x,y)$ is detailed balanced if for all states $x,y$ we have $\pi(x)q(x,y)=\pi(y)q(y,x)$. If this property holds for a stationary distribution $\pi$, then the stationary Markov chain is time reversible and $\pi$ can be explicitly calculated. In the setting of stochastic reaction networks, the existence of a detailed balanced distribution and its connections with the detailed balanced steady states of the associated deterministic model have been extensively studied. In this work, we extend the notion of detailed balanced distribution to cover not necessarily reversible Markov chain models of reaction networks (that is, we do not require that $q(x,y)>0$ whenever $q(y,x)>0$). While a generalization of the concept of time reversibility does not seem attainable, we fruitfully use this extension to explicitly calculate the stationary distribution of certain families of stochastic reaction networks. As a consequence, we are able to explicitly calculate the stationary distributions of generalizations of birth and death chains where the forward and backward jump sizes may not be equal to one another.