Gaussian Methods and Small Value Problems

概率论中的两个基本现象是典型的行为，如期望值和大数定律，以及罕见的事件，如极大或极小的值。这项研究的中心是发展高斯方法，用于研究正随机量取较小值的典型行为和罕见事件。主要目标是扩大对相关主题的理解，并在对各种技术和应用进行系统研究的基础上建立一般理论。等周型高斯不等式提供了独立(复杂)结构和独立(更简单)结构之间的比较，在某些(可能性限制)情况下，它变成了一个等式。它们已经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。最近关于高斯和密切相关的随机过程的几种新技术的发展拓宽了我们对小偏差概率及其与相关概率主题的联系的理解，如高斯随机矩阵、不相交指数和随机分配。反过来，它提出了许多与概率论和几何泛函分析中的应用有关的更深层次的问题。在降低尾概率、布朗追踪模型和首次退出时间方面的成功应用将扩展到对随机多项式的实零点和高斯混沌的详细研究，这一研究对概率的各个领域具有更广泛的影响，这既是一种基本的世界观，也是一门核心的数学学科。高斯过程理论在概率论中具有基础性的重要意义。它的发展集中在现有方法在各种领域的应用，以及在当前推动知识进步的兴趣所推动的新技术和问题上。这项拟议的研究是研究人员长期研究计划的关键一步，该计划旨在系统地开发适用于密切相关的随机过程的新的高斯方法。这项研究应该会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 and laws of large numbers, and rare events such as extremely big or small values. This research centers on developing Gaussian methods 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 related topics and build a general 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, the Brownian pursuit models and the first exit times will be expanded to a detailed study of real zeros of random polynomials 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 processes is of fundamental importance in probability. Its development is centered on applications of the existing methods to a variety of fields and new techniques and problems motivated by current interests ofadvancing 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 should improve our understanding of important random events and provide basic tools for the study of our random environment.