RUI: Boundary and entropy of random walks on groups

RUI：群体随机游走的边界和熵

• 批准号: 2246727
• 负责人: Behrang Forghani
• 金额: $ 21.65万
• 依托单位: College of Charleston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246727&HistoricalAwards=false
• 关键词: RUI; Boundary; entropy; random; walks

Applications of random walks emerge in different branches of science and our daily life. In game theory, random walks can model shuffling a deck of cards to find optimal shuffle iterations. In finance, the unpredictability of price changes of stocks can be modeled by random walks. The project will apply rigorous mathematical tools for studying the long-term behavior of random walks. In particular, the following two problems will be investigated: first, how geometric or algebraic features can capture the long-term behavior of random walks. Second, how different quantities can describe the long-term behavior of a random walk. The research will engage undergraduate and graduate students. This engagement with students exposes them to contemporary areas of mathematics not part of the regular curriculum. The work will expand and refine existing results on random walk invariants such as boundaries and asymptotic entropies. These arise from the interaction between random walks on groups and their algebraic or geometric structures. The project will improve understanding of the Poisson boundary, which provides a representation of bounded harmonic functions on a group for a given probability measure. The current work will leverage recent developments, such as the theory of pivotal times, to identify the Poisson boundary of random walks on hyperbolic-like groups. Another goal is to investigate connections between quotients of the Poisson boundaries with subgroups of a given group. The area of research has strong connections to geometric group theory, ergodic theory, and information theory. This project is jointly funded by the DMS Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

随机步行的应用在科学的不同分支和我们的日常生活中出现。在游戏理论中，随机步行可以模拟洗牌，以找到最佳的迭代式迭代。在金融中，股票价格变化的不可预测性可以通过随机步行来建模。该项目将应用严格的数学工具来研究随机步行的长期行为。特别是，将研究以下两个问题：首先，几何或代数特征如何捕获随机步行的长期行为。其次，不同数量如何描述随机步行的长期行为。这项研究将吸引本科生和研究生。与学生的这种互动使他们接触了当代数学领域，而不是常规课程的一部分。这项工作将扩大并完善现有的结果，以随机步行不变性（例如边界和渐近熵）。这些是由随机步行对组及其代数或几何结构之间的相互作用引起的。该项目将改善对泊松边界的理解，该边界为给定概率度量提供了界限的谐波功能的表示。当前的工作将利用最近的发展，例如关键时期的理论，以确定在类似双曲线的群体上随机步行的泊松边界。另一个目标是研究泊松边界的商与给定组的子组之间的联系。研究领域与几何群体理论，厄运理论和信息理论有着密切的联系。该项目由DMS概率计划共同资助和启发竞争性研究的既定计划（EPSCOR）。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。