AMC-SS: Statistical Analysis and Modeling of Complex Physical Systems

该项目将研究具有一定程度双曲性的动力系统的主要统计性质，如大偏差估计、极值统计和局部中心极限定理。研究耦合非一致双曲映象的动力学和统计性质。非平稳随机游走模型和相关的福克-普朗克方程将被开发来描述时间不均匀的系统，如货币兑换市场和高度湍流。此外，还将利用时空模式的特征来研究磁性材料中的外延和磁区增长。复杂系统--从天气系统和湍流流体到金融市场和动物种群--通常最好地根据它们的统计特性来理解，也就是试图提供对某些类型行为的概率的估计。对这些系统中的许多系统缺乏良好的统计理解，因为通常的建模假设，例如独立性，不能满足。该提案旨在加深我们对广泛复杂系统的统计知识，增强我们预测它们的行为和评估某些事件的风险的能力。此外，该提案将开发描述金融市场和动荡的数学模型，并改进磁性材料中无序图案的统计特征。

The project will investigate key statistical properties such as large deviation estimates, extreme value statistics and the local central limit theorem for dynamical systems with some degree of hyperbolicity. The dynamics and statistical properties of coupled non-uniformly hyperbolic maps will be studied. Non-stationary random walk models, and related Fokker-Planck equations, will be developed to describe time-inhomogeneous systems such as currency-exchange markets and highly turbulent flows. In addition, characterizations of spatio-temporal patterns will be exploited to study epitaxial and domain growth in magnetic materials.Complex systems - examples range from weather systems and turbulent fluids to financial markets and animal populations - are often best understood in terms of their statistical properties, that is, by trying to provide estimates of the probability of certain types of behavior. A good statistical understanding of many of these systems is lacking because the usual modeling assumptions, for example independence, are not satisfied. The proposal aims to deepen our knowledge of the statistics of a broad range of complex systems, to enhance our ability to predict their behavior and assess the risk of certain events. In addition the proposal will develop mathematical models to describe financial markets and turbulence as well as provide improved statistical characterizations of disordered patterns in magnetic materials.