Boundary-induced slow mixing for Markov chains and its application to stochastic reaction networks

TL;DR

This paper investigates how boundaries in non-negative quadrant Markov chains, especially in stochastic reaction networks, cause slow mixing times, providing criteria to quantify this slow-down and insights into network structure effects.

Contribution

The paper introduces general criteria for power-law lower bounds on mixing times of Markov chains near boundaries, linking boundary behavior to slow convergence.

Findings

Boundary effects can cause polynomial slow-down in mixing times.

Criteria for power-law lower bounds based on local boundary behavior.

Insights into how reaction network structure influences mixing speed.

Abstract

Markov chains on the non-negative quadrant of dimension $d$ are often used to model the stochastic dynamics of the number of $d$ entities, such as $d$ 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 towards its stationary distribution due to the extinction or the lack of an entity. In this paper, we quantify this slow-down for a class of stochastic reaction networks and for more general Markov chains on the non-negative quadrant. We establish general criteria for such a Markov chain to exhibit a power-law lower bound for its mixing time. The lower bound is of order $|x|^\theta$ for all initial state $x$ on a boundary face of the quadrant, where $\theta$ is characterized by the local behavior of the Markov chain near the boundary of the quadrant. A better understanding of how these lower bounds arise leads to insights into how the structure of chemical reaction networks contributes to slow-mixing.