Statistical properties of dynamical systems: large deviations, extremes, return time statistics and dynamical Borel-Cantelli lemmas.

本项目旨在提高我们对非均匀双曲动力系统的某些关键统计的理解：返回时间统计、极值统计、动态Borel-Cantelli引理和大偏差。虽然在描述一致双曲系统的这些统计性质方面已经取得了进展，但一致双曲性很少适用于物理系统，基于非一致双曲性的模型更现实。返回时间统计量化了物理系统在一定时间内转换到或返回到某一状态的概率。极值统计给出了系统上观测值的时间序列连续最大值的极限律。Borel-Cantelli引理是建立随机过程几乎确定行为的基本工具。目的是将我们对这些统计性质的理解扩展到非均匀双曲系统，包括那些多项式相关衰减率的系统。进一步的目标是获得具有不连续（或奇点）或产生乘法环的动力系统（如矩阵值函数）上的函数的大偏差估计。有些物理系统非常复杂，尽管可以用数学方程来建模，但从统计学的角度来描述它们是最好的。例如，了解一场位于近海的飓风影响沿海某些城镇的概率是有用的信息，但通常无法提供更确定的信息。然而，一个主要问题是，由于连续观测高度相关，物理系统的实际模型很少满足经典的完全随机（连续观测独立）的统计假设。极值理论估计观察到某一量级事件的可能性，而大偏差估计异常或罕见事件的概率。该项目旨在通过为广泛的物理现实数学模型提供对极值理论、大偏差和其他统计数据的理解，提高对复杂物理系统的预测。PI还将继续努力改善各级科学教育，并通过外展活动向公众宣传。

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.