论文标题
某些反应网络的产品形式固定分布的代数方法
An algebraic approach to product-form stationary distributions for some reaction networks
论文作者
论文摘要
马尔可夫连锁店的产品形式固定分布的确切结果在不同领域引起了人们的关注。在随机反应网络（CRN）中，固定分布在产品形式的特殊情况下大多是已知的。但是，尚无对固定分布具有产品形式的网络类别的完整表征。我们在CRN框架中开发了一种代数方法来用于产品形式的固定分布。在某些关于保守的ergodic CRN状态空间线性和分解的假设下，这为产品形式固定分布提供了足够和必要的代数条件。相应地，我们获得了参数空间的一个半格式子集，该子集捕获了速率，在相应的假设下，CRN具有产品形式。我们将开发的理论用于CRN和一些统计力学模型，除了从应用概率中勾勒出其他模型中的相关性。
Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative ergodic CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.
