Markov processes serve as foundational models in many scientific disciplines, such as molecular dynamics, and their simulation forms a common basis for analysis. While simulations produce useful trajectories, obtaining macroscopic information directly from microstate data presents significant challenges. This paper addresses this gap by introducing the concept of membership functions being the macrostates themselves. We derive equations for the holding times of these macrostates and demonstrate their consistency with the classical definition. Furthermore, we discuss the application of the ISOKANN method for learning these quantities from simulation data. In addition, we present a novel method for extracting transition paths based on the ISOKANN results and demonstrate its efficacy by applying it to simulations of the mu-opioid receptor. With this approach we provide a new perspective on analyzing the macroscopic behaviour of Markov systems.

翻译：马尔可夫过程是许多科学学科（如分子动力学）中的基础模型，其模拟构成了分析的共同基础。尽管模拟能产生有用的轨迹，但从微观态数据直接获取宏观信息仍面临重大挑战。本文通过引入隶属函数作为宏观态本身的概念来填补这一空白。我们推导了这些宏观态保持时间的方程，并证明了其与经典定义的一致性。此外，我们讨论了利用ISOKANN方法从模拟数据中学习这些量的应用。同时，我们提出了一种基于ISOKANN结果提取转移路径的新方法，并通过将其应用于μ-阿片受体的模拟展示了其有效性。通过这种方法，我们为分析马尔可夫系统的宏观行为提供了新的视角。