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This paper provides a new path method that can be used to determine when anergodic continuous-time Markov chain on $\mathbb Z^d$ converges exponentiallyfast to its stationary distribution in $L^2$. Specifically, we provide generalconditions that guarantee the positivity of the spectral gap. Importantly, ourresults do not require the assumption of time-reversibility of the Markovmodel. We then apply our new method to the well-studied class of stochasticallymodeled reaction networks. Notably, we show that each complex-balanced modelthat is also ``open'' has a positive spectral gap, and is thereforeexponentially ergodic. We further illustrate how our results can be applied formodels that are not necessarily complex-balanced. Moreover, we provide anexample of a detailed-balanced (in the sense of reaction network theory), andhence complex-balanced, stochastic reaction network that is not exponentiallyergodic. We believe this to be the first such example in the literature.

A new path method for exponential ergodicity of Markov processes on $\mathbb Z^d$, with applications to stochastic reaction networks