Gaussian Methods and Small Value Problems

University Of Delaware, Newark DE

Investigators

Abstract

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 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 should improve our understanding of important random events and provide basic tools for the study of our random environment.