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

Wai-TongLouis, Fan, Jinsu Kim, Chaojie Yuan
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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.
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马尔可夫链的边界诱导缓慢混合及其在随机反应网络中的应用
维数为 $d$ 的非负象限上的马尔可夫链常用于模拟随机反应网络中 $d$ 个实体(如 $d$ 化学物种)的随机动态。无限状态空间带来了技术挑战,象限的边界会对这些马尔可夫链的长期行为产生巨大影响。例如,由于消亡或缺乏实体,边界会减慢遍历马尔可夫链向其静态分布的收敛速度。在本文中,我们对一类随机反应网络和非负象限上更一般的马尔可夫链的收敛速度进行了量化。我们为此类马尔可夫链的混合时间表现出幂律下限建立了一般标准。对于象限边界面上的所有初始状态 $x$,该下界为阶$|x|^\theta$,其中$\theta$是马尔可夫链在象限边界附近的局部行为特征。更好地理解这些下限是如何产生的,将有助于深入了解化学反应网络的结构是如何促成缓慢混合的。
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