A New Path Method for Exponential Ergodicity of Markov Processes on \({{\mathbb Z^{d}}}\), 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 Zd converges exponentially fast to its stationary distribution in L2. 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.