CAREER: Probability on Groups and Semigroups of Probabilities

概率是一个数学领域，在科学、工程、统计学、经济学、哲学以及每一门学科中都有广泛的应用，其中不确定性和随机性起着至关重要的作用。在概率学中，我们研究随机游走，它被用来模拟各种物理和经济现象，也与数学的其他领域有重要联系。在这个项目中，我们探索了一些关于随机游动的长期存在的谜团，特别是对理解路径依赖感兴趣：对于哪些随机游动，随机波动在开始时会对长期产生持久的影响？另一个目标是探索新发现的与信息论和经济学问题的联系：从长远来看，哪些信息来源更有价值？信息是如何定价的？这个项目的教育部分将包括为本科生提供新的研究机会，目的是通过让他们接触现代问题和技术来吸引有前途的学生进入这个领域。自20世纪60年代和70年代在弗斯滕伯格的基础工作中引入弗斯滕伯格-泊松边界以来，在一个群体上随机行走的芙丝汀堡-泊松边界一直是密集研究的对象。到目前为止，这种文献已经成熟，与邻近领域建立了许多联系。然而，一些令人信服的基本问题仍然悬而未决。例如：哪些组允许具有非平凡边界的有限支持的随机游动？在这个项目中，我们希望利用最近的突破来推进这一领域的最新进展，特别是研究所谓的“稳定性猜想”和“缺口猜想”。在拓扑群作用领域，邻近性和强邻近性的概念一直是大量研究的对象。最近，人们证明了一个有限生成的群是几乎幂零的当且仅当它在紧空间上的所有近邻作用都有不动点。这在最近的活动和芙丝汀堡-泊松边界之间建立了一种仍然神秘的联系，我们计划探索这一点。最后，这个项目还将探索随机优势，这是经济学和统计学中比较分布的一个重要工具。我们将继续最近启动的研究议程，研究随机优势和卷积的相互作用，并将其应用于布莱克韦尔实验理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Probability is a field of mathematics that has wide applications in science, engineering, statistics, economics, philosophy, and every discipline in which uncertainty and randomness play a crucial role. Within probability, we study random walks, which are used to model various physical and economic phenomena, and also have important connections to other fields of mathematics. In this project we explore some long standing mysteries regarding random walks, and in particular are interested in understanding path-dependence: for which random walks do random fluctuations in the beginning have a lasting impact on the long term? A further objective is to explore newly discovered connections to questions in information theory and economics: Which sources of information are more valuable in the long term? And how is information priced? The educational component of this project will include novel research opportunities for undergraduate students, with the aim of attracting promising students to this field by exposing them to modern questions and techniques.The Furstenberg-Poisson boundary of a random walk on a group has been the object of intense study since its introduction in Furstenberg's foundational work in the 1960s and 1970s. This literature is by now mature, with many established connections to neighboring fields. Yet, some compelling basic questions remain open. For example: which groups admit finitely supported random walks with a non-trivial boundary? In this project we hope to leverage recent breakthroughs to advance the state of the art in this field, and in particular to study the so-called "stability conjecture" and "gap conjecture." In the field of topological group actions, the notions of proximality and strong proximality have been the object of a large research effort. Recently, it has been shown that a finitely generated group is virtually nilpotent if and only if all of its proximal actions on compact spaces have fixed points. This establishes a still mysterious connection between proximal actions and the Furstenberg-Poisson boundary, which we plan to explore. Finally, this project will also explore stochastic dominance, which is is an important tool for comparison of distributions in economics and statistics. We will pursue a recently initiated research agenda studying the interaction of stochastic dominance and convolution, with applications to the Blackwell theory of experiments.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.