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

该项目将研究具有一定双曲度的动力系统的关键统计性质，如大偏差估计、极值统计和局部中心极限定理。本文将研究耦合非均匀双曲映射的动力学和统计性质。非平稳随机游走模型，以及相关的福克-普朗克方程，将被用来描述时间非齐次系统，如货币交易市场和高度湍流。此外，时空模式的特征将被用于研究磁性材料的外延和畴生长。复杂系统——从天气系统和湍流到金融市场和动物种群——的例子通常是通过它们的统计特性来理解的，也就是说，通过试图提供某些类型行为的概率估计。由于通常的建模假设（例如独立性）没有得到满足，因此缺乏对这些系统的良好统计理解。该提案旨在加深我们对大范围复杂系统的统计知识，提高我们预测其行为和评估某些事件风险的能力。此外，该提案将开发数学模型来描述金融市场和动荡，并提供磁性材料中无序模式的改进统计特征。

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.