发表机构
Institute of Theoretical Physics, Jagiellonian University; Doctoral School of Exact and Natural Sciences, Jagiellonian University; Center for Theoretical Physics, Polish Academy of Sciences（雅盖隆大学理论物理研究所; 雅盖隆大学精确与自然科学博士学院; 波兰科学院理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用微分几何研究马尔可夫动力学中概率单纯形的有限时间可达性问题，刻画Finsler度量并推导可达集边界，揭示其与Kullback-Leibler相对熵的关联。
AI 中文摘要
有限时间可达性问题——即两个概率向量能否在时间t内通过给定集合中的生成元相互转化——是马尔可夫动力学中的一个关键问题。我们利用微分几何的工具来解决归一化Kolmogorov生成元的这一问题。连接有序状态对的最短时间T定义了概率单纯形中的一个（非对称）拟距离，该距离区分了出射可达集和入射可达集，并捕捉了马尔可夫动力学的固有不可逆性。我们刻画了具有有界对角元素的归一化生成元所对应的Finsler度量，并给出了饱和速度极限的时间无关演化。推导了两个可达集的解析边界，并利用它们相应体积的比值来评估所获得的界限。关于均匀概率向量的不对称性被证明与Kullback-Leibler相对熵有关。
英文摘要
The finite-time reachability problem - whether two probability vectors can be transformed in a time t through a generator from a given set - is a key question in Markovian dynamics. We address this problem for normalized Kolmogorov generators with tools from differential geometry. The shortest time T connecting an ordered pair of states defines an (asymmetric) quasi-distance in the probability simplex, which distinguishes between outgoing and incoming reachable sets, and captures the inherent irreversibility of Markov dynamics. The Finsler metric corresponding to normalized generators with bounded diagonal entries is characterized, and time-independent evolutions that saturate speed limits are presented. Analytical boundaries of both reachable sets are derived and the ratio of their corresponding volumes assess the bounds obtained. The asymmetry with respect to the uniform probability vector is shown to be related to the Kullback-Leibler relative entropy.
Comments15+10 pages, 11 figures
