This project seeks to improve our understanding of certain key statistics of non-uniformly hyperbolic dynamical systems: return time statistics, extreme value statistics, dynamical Borel-Cantelli lemmas and large deviations. While there has been progress in describing these statistical properties for uniformly hyperbolic systems, uniform hyperbolicity seldom holds for physical systems and models based on non-uniform hyperbolicity are more realistic. Return time statistics quantify probabilities of a physical system transforming into or returning to a state in a certain period of time. Extreme value statistics give limit laws for successive maxima of a time series of observations on a system. Borel-Cantelli lemmas are a fundamental tool in establishing the almost-sure behavior of stochastic processes. The aim is to extend our understanding of these statistical properties to non-uniformly hyperbolic systems including those with polynomial correlation decay rate. A further goal is to obtain large deviation estimates for functions on dynamical systems with discontinuities (or singularities) or which give rise to multiplicative cocycles, such as matrix valued functions.

Some physical systems are so complex that, although they may be modeled by mathematical equations, they are best described from the viewpoint of statistics. For example, knowing the probabilities of a hurricane located offshore affecting certain towns on the coast is useful information and more certain information usually cannot be given. However, it is a major problem that the classical statistical assumption of complete randomness (independence of successive observations) is seldom satisfied for realistic models of physical systems since successive observations are highly correlated. Extreme value theory estimates the likelihood of observing an event of a certain magnitude, while large deviations estimates the probability of an outlier or rare event. This project aims to improve predictions of complex physical systems by providing an understanding of extreme value theory, large deviations and other statistics for a wide class of physically realistic mathematical models. The PI will also continue to work towards improving science education at all levels and by informing the public through outreach activities.

Project Report

Intellectual Merit. Statistics and probability are key tools in predicting the behavior of complex physical systems. Although complex systems may be modeled by deterministic rules, the large number of interactions involved and the sensitivity of behavior to slight changes in initial conditions means that often it is best to describe the likely behavior of a complex system from the viewpoint of probability and statistics. We developed new techniques to investigate large deviations, extreme value theory and return time statistics for a wide class of nonlinear ordinary differential equations and mappings that are used to model complex physical systems. This work enhances our conceptual understanding of physical systems and our ability to predict their behavior. Extreme value theory estimates the probability of observing an event of a certain magnitude in a given time period, return time statistics quantifies the probability of a system returning to or entering a certain state in a time-period while large deviations estimates the probability of an outlying or rare event. Knowledge of large deviations and extreme values is used practically to estimate risk- the early development of the theory was to estimate risk in the insurance industry. Large deviations is also used theoretically, for example to develop the foundations of statistical mechanics. The notion of recurrence plays a central role in mathematics and physics. Quantifying recurrence has important applications to number theory and statistical physics, amongst other areas. In particular we developed an extreme value theory of geometric Lorenz systems, which are a model of weather systems. We also investigated the recurrence properties of billiard systems, which are canonical models of statistical mechanics. The project demonstrated that the limit laws from probability theory are valid for many chaotic models that are used in applications. Broader Impact. This research was at the interface of applications and of interest to scientists modeling complex systems of biological or physical importance. The results of this work have been published in mathematical journals and also journals read by physicists, statisticians and probabilists to broaden their impact on the general scientific community. In particular the most significant findings of interest to statisticians were conveyed at a week-long joint workshop with statisticians on Extremes and Rare Events in 2014 at the Center for International Research in Mathematics, Luminy, France. Other aspects of this research were presented at a satellite meeting of the International Congress of Mathematicians in Daejeon, Korea. The investigator also lectured on facets of this research at two-week long summer schools for graduate students held in May 2013 and May 2014 at the University of Houston to roughly 70 graduate students from around the US. These lectures focused on introducing probabilistic techniques to the next generation of researchers. The investigator initiated the mathematical statistics option in the PhD program at the University of Houston, and has developed and taught courses on statistics, probability, ergodic theory and nonlinear dynamics. These activities have been enabled by this funding.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Bruce P. Palka
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University of Houston
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