A New Path Method for Exponential Ergodicity of Markov Processes on \({{ Zᵈ}}\), with Applications to Stochastic Reaction Networks

This paper provides a new path method that can be used to determine when an ergodic continuous-time Markov chain on [Formula: see text] converges exponentially fast to its stationary distribution in [Formula: see text]. Specifically, we provide general conditions that guarantee the positivity of the spectral gap. Importantly, our results do not require the assumption of time-reversibility of the Markov model. We then apply our new method to the well-studied class of stochastically modeled reaction networks. Notably, we show that each complex-balanced model that is also “open” has a positive spectral gap and is therefore exponentially ergodic. We further illustrate how our results can be applied for models that are not necessarily complex-balanced. Moreover, we provide an example of a detailed-balanced (in the sense of reaction network theory), and hence complex-balanced, stochastic reaction network that is not exponentially ergodic. We believe this to be the first such example in the literature.