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 from simulations 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 the analysis of macroscopic behaviour of Markov systems.

翻译：马尔可夫过程是分子动力学等众多科学领域的基础模型，其模拟构成了分析的共同基础。虽然模拟能生成有用的轨迹，但直接从微观状态数据获取宏观信息仍面临重大挑战。本文通过引入隶属函数即宏观状态本身的概念来弥补这一空白。我们推导了这些宏观状态驻留时间的方程，并证明了其与经典定义的一致性。此外，我们讨论了应用ISOKANN方法从模拟数据中学习这些量的具体途径。进一步地，我们提出了一种基于ISOKANN结果从模拟中提取转移路径的新方法，并通过将其应用于{\mu}-阿片受体的模拟验证了该方法的有效性。这一研究为马尔可夫系统的宏观行为分析提供了新的视角。