Boundary-Induced Slow Mixing for Markov Chains and Its Application to Stochastic Reaction Networks

Abstract. Markov chains on the nonnegative quadrant of dimension [Formula: see text] are often used to model the stochastic dynamics of the number of [Formula: see text] entities, such as [Formula: see text] chemical species in stochastic reaction networks. The infinite state space poses technical challenges, and the boundary of the quadrant can have a dramatic effect on the long-term behavior of these Markov chains. For instance, the boundary can slow down the convergence speed of an ergodic Markov chain toward its stationary distribution due to the extinction or the lack of an entity. In this paper, we quantify this slowdown by establishing general criteria for a continuous-time Markov chain on the nonnegative integer grids to exhibit a power-law lower bound for its mixing time. The lower bound is of order [Formula: see text] for all initial states [Formula: see text] on a boundary face of the quadrant, where [Formula: see text] is characterized by the local behavior of the Markov chain near the boundary of the quadrant. We then verify the general criteria for a cyclic reaction network with two species and an arbitrary number of complexes under mass-action kinetics. A better understanding of how these lower bounds arise leads to insights into how the structure of chemical reaction networks contributes to slow mixing.