This research centers on estimating the probability of rare events of small ball type. The two major objectives are to develop new methods of estimating small ball probabilities for Gaussian and closely related random processes, and to systematically study the existing techniques and applications which are spread over various topics. The recent completion of the connection between small ball probabilities and metric entropy problems allows applications of tools and results from functional analysis to provide one of the most powerful methods in the lower bound estimate for Gaussian processes. In turn, it suggests many further questions connected to applications in probability theory and approximation theory. The PI intends to extend this study to other random processes such as Gaussian chaos and stable processes. This research should lead to significant new knowledge about small ball probabilities and provide basic tools for the study of random processes. The primary focus of this research is a better understanding of rare random phenomena related to Gaussian processes and others which serve as models in many applications. These types of problems often arise in estimating the chances for rare events to occur in areas where such events are of fundamental importance, such as weather, economic indices, epidemics etc. This research should improve our understanding of rare random events and provide basic tools for the study of our random environment.
