Small Value Theory in Probability

概率论中的两个基本现象是典型的行为，如期望值，大数定律和中心极限定理，以及罕见的事件，如极大或极小的值。本研究的重点是发展研究正随机量取较小值的典型行为和罕见事件的方法和理论。主要目标是扩展对五个相关领域的理解，并在系统研究各种技术和应用的基础上建立一个通用的小价值理论。等周型高斯不等式提供了依赖（复杂）结构和独立（简单）结构之间的比较，在某些（可能性限制）情况下，独立结构变为相等。它们已被用作各种问题的基本工具，并在深入理解随机现象方面发挥了至关重要的作用。近年来，高斯过程和相关随机过程的一些新技术的发展拓宽了我们对小偏差概率及其与高斯随机矩阵、不相交指数和随机分配等概率相关主题的联系的理解，同时也提出了许多与概率论和几何泛函分析中的应用有关的问题。这些在降低尾概率、随机函数的零点和首次退出时间等方面的成功应用将扩展到布朗追踪模型和高斯混沌的详细研究中，这一研究将对概率的各个领域产生更广泛的影响，概率既是观察世界的基本方式，也是数学的核心学科。高斯（钟形曲线）过程理论在概率论和统计学中具有根本的重要性。它的发展集中在现有的方法应用到各种领域和新技术和问题的推动下，目前的利益，推进知识。拟议的研究是一个关键的一步，在调查员的长期研究计划，系统地开发新的高斯方法，面向应用程序密切相关的随机过程。这项研究有利于本科生和研究生的教育和研究。许多开放的问题和建议的研究结果可以作为学生的课程项目。它能激发学生学习和研究概率论的兴趣。这项研究将提高我们对重要随机事件的理解，并为研究我们的随机环境提供基本工具。

Two fundamental phenomenons in probability theory are typical behaviors such as expected values, laws of large numbers and central limit theorems, and rare events such as extremely big or small values. This research centers on developing methods and theory for the study of both typical behaviors and rare events of the type that positive random quantities take smaller values. The major objective is to extend the understanding of five related areas and build a general small value theory based on systematic study of various techniques and applications. The isoperimetric type Gaussian inequalities provide comparisons between dependent (complected) structure and independent (simpler) one which becomes an equality in certain (possibility limiting) cases. They have been used as basic tools in various problems and played a crucial role in deeper understanding of random phenomenon. The recent development of several new techniques for Gaussian and closely related random processes broadened our understanding of small deviation probabilities and their connections with related topics of probability such as Gaussian random matrices, non-intersection exponents and random assignments.In turn, it suggests many further questions connected to applications in probability theory and geometric functional analysis. The very successful applications to lower tail probabilities, zeros of random functions and the first exit times will be expanded to a detailed study of Brownian pursuit models and Gaussian chaos.This research has a broader impact on diverse areas of probability, which is both a fundamental way of viewing the world and a core mathematical discipline. The theory of Gaussian (bell curve) processes is of fundamental importance in probability and statistics. Its development is centered on applications of the existing methods to a variety of fields and new techniques and problems motivated by current interests of advancing knowledge. The proposed research is a key step in the investigator's long term research plan of systematically developing new Gaussian methods geared for applications to closely related random processes. This research benefits both undergraduate and graduate education and research. Many open problems and results from the proposed study can be used as students course projects. It can stimulate the interests of students in leaning and studying probability theory. This research should improve our understanding of important random events and provide basic tools for the study of our random environment.