AI 中文总结
本文在图论线性框架下，引入Hill算子与unraveling算子，推广Hill和Kac的工作，得到马尔可夫过程首达时间、首达时间矩、平均 recurrence 时间等的有理代数公式，整合了瞬态与稳态性质的关系。
AI 中文摘要
在先前的一篇论文中，我们利用图论线性框架证明了连续时间马尔可夫过程的瞬态性质——分裂概率与首达时间（FPT）分布的矩——可通过基础图的生成森林表示为转移率的有理代数函数，这与仅使用生成树表示稳态（s.s.）概率的相关有理公式形成对比。生物物理学家Terrell Hill概述了一种基于修改后马尔可夫过程的稳态概率计算平均首达时间和分裂概率的方法，从而将轨迹系综的计算转化为单条轨迹的计算。类似地，Mark Kac证明了马尔可夫过程返回某状态的平均 recurrence 时间可表示为该状态稳态概率的函数。在此，我们进一步探究瞬态性质与稳态性质、森林与树、系综与单轨迹计算之间的关系。我们通过在图G上引入Hill算子$H_u[G]$来形式化Hill的方法，并利用它计算从状态u出发的条件与无条件首达时间的所有矩，这些矩可表示为$H_u[G]$稳态概率的有理函数，其中稳态概率来自树，而“交换因子”来自森林。随后，我们将其与 unraveling 算子$U_v[G]$结合，计算 recurrence 时间分布的所有矩，将其表示为G及相关交换因子稳态概率的有理函数。令人惊讶的是，Hill算子被证明是unraveling算子的左逆，这表明线性框架图上的算子代数可能具有更广泛的意义。我们的结果将先前零散的发现整合并推广为马尔可夫过程首达时间的通用有理代数公式集合。
英文摘要
In a preceding paper, we used the graph-theoretic linear framework to show how transient properties of continuous-time Markov processes -- splitting probabilities and the moments of first-passage time (FPT) distributions -- could be expressed as rational algebraic functions of the transition rates, by using spanning forests of the underlying graph. This contrasts with the related rational formulas for steady-state (s.s.) probabilities, which use only spanning trees. The biophysicist Terrell Hill sketched a procedure for calculating mean FPTs and splitting probabilities in terms of the s.s. probabilities of a modified Markov process, thereby converting calculations using ensembles of trajectories to those using a single trajectory. Similarly, Mark Kac showed that the mean recurrence time to a state of a Markov process could be expressed in terms of the s.s. probability of that state. Here, we explore further the relationships between transient and s.s. properties, forests and trees, and ensemble and single-trajectory calculations. We formalise Hill's procedure by introducing a Hill operator, $H_u[G]$, on a graph, $G$, and use it to calculate all moments of the conditional and unconditional FPTs from $u$ as rational functions of the s.s. probabilities of $H_u[G]$, which arise from trees and "exchange factors" which arise from forests. We then combine this with an unravelling operator, $U_v[G]$, to calculate all moments of the recurrence time distribution as rational functions of the s.s. probabilities of $G$ and related exchange factors. Surprisingly, the Hill operator turns out to be a left-inverse to the unravelling operator, suggesting that the algebra of operators on linear framework graphs may be of broader interest. Our results integrate and generalise previously disparate findings into a common repertoire of rational algebraic formulas for FPTs of Markov processes.
Comments58 pages, 5 figures
