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.
Two fundamental phenomena 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. Both typical behaviors and rare events of the type that positive random variables take smaller values are studied in areas where such events are of fundamental importance. The major findings extended the understanding of six related areas and built a general small value theory in this active research field. This research have made an impact on diverse areas of probability, which is both a fundamental way of viewing the world and a core mathematical discipline. The small value theory 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. Much of the PI's instructional activities are focused on teaching probability theory to future scientists, engineers, economists and statisticians. The PI is writing a book for graduate students and technically oriented audience to learn tools to estimate small value probability which appears in many different contexts. The NSF support of Ph.D. students, including at least three women, allows us to build a stronger research graduate program in probability at the University of Delaware.
