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

TL;DR

This paper introduces a novel path method to establish exponential ergodicity in Markov processes on , particularly applied to stochastic reaction networks, demonstrating conditions for spectral gap positivity without requiring time-reversibility.

Contribution

The paper develops a new path method for exponential ergodicity of Markov chains on , applicable to reaction networks, and provides the first example of a detailed-balanced network that is not exponentially ergodic.

Findings

Complex-balanced open models have positive spectral gap.

The new method does not require time-reversibility.

An example of a detailed-balanced network that is not exponentially ergodic.

Abstract

This paper provides a new path method that can be used to determine when an ergodic continuous-time Markov chain on $\mathbb Z^d$ converges exponentially fast to its stationary distribution in $L^2$. 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.