CAREER: Probability on Groups and Semigroups of Probabilities

California Institute Of Technology, Pasadena CA

Investigators

Abstract

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.