CAREER: Probability on Groups and Semigroups of Probabilities

概率论是数学的一个领域，在科学、工程、统计学、经济学、哲学以及每一个不确定性和随机性起关键作用的学科中都有广泛的应用。在概率论中，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的联系。在这个项目中，我们探索了一些关于随机漫步的长期谜团，特别是对理解路径依赖感兴趣：对于哪些随机漫步，开始时的随机波动会对长期产生持久的影响？进一步的目标是探索信息理论和经济学中新发现的问题之间的联系：从长远来看，哪些信息来源更有价值？信息是如何定价的？该项目的教育部分将包括为本科生提供新颖的研究机会，目的是通过让他们接触现代问题和技术，吸引有前途的学生进入这一领域。自20世纪60年代和70年代Furstenberg的基础工作中引入群体随机游走的Furstenberg- poisson边界以来，它一直是深入研究的对象。这方面的文献现已成熟，与邻近领域有许多既定的联系。然而，一些令人信服的基本问题仍未解决。例如：哪些组承认具有非平凡边界的有限支持随机漫步？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技术，特别是研究所谓的“稳定性猜想”和“间隙猜想”。在拓扑群体行为领域，近性和强近性的概念一直是大量研究的对象。最近证明了一个有限生成群是虚零的，当且仅当它在紧空间上的所有近端作用都有不动点。这在近端作用和弗斯滕伯格-泊松边界之间建立了一个仍然神秘的联系，我们计划对此进行探索。最后，本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重要工具。我们将继续进行最近发起的一项研究议程，研究随机优势和卷积的相互作用，并将其应用于布莱克威尔实验理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.