2nd International Conference on Reaction Kinetics, Mechanisms and Catalysis (RKMC2021)

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 (a notable example of where these properties are heavily used is the study of birth and death chains).

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.

Budapest, Hungary

Budapest, Hungary