Limit Theorems for Non-Stationary and Random Dynamical Systems

尽管人们对处于平衡状态的物理系统的统计特性了解很多，但许多感兴趣的系统不能实际地建模为处于平衡状态，这意味着观察一个事件或对系统进行某种测量的机会不会随着时间而改变。有些物理系统会受到随时间变化的外部影响；例如，大气受到季节性海洋温度的影响，电网的状态受到随时间变化的电力需求的影响。物理系统也经常受到随机效应的影响，要么是由于测量读数误差，要么是由于系统本身的噪声。该研究项目旨在利用概率和分析的最新进展，对非平衡和随机复杂系统进行良好的统计理解。更好地理解非平衡系统和随机系统的统计特性将导致改进的预测，并扩展我们对可以预期在自然界中成立的统计定律的知识。该项目旨在基于鞅理论和分析的最新发展开发新技术，以研究中心极限定理，几乎确定不变性原理（布朗运动的一种强逼近形式），大偏差和其他高级统计，用于模拟复杂物理系统的非平衡和随机动力系统。要研究的系统的例子包括台球系统（多项式和指数混合），间歇性模型，洛伦兹图和流，以及由统计力学产生的系统。随机动力学的统计特性可以通过对所有噪声实现（退火统计）或单个噪声实现（淬灭统计）进行抽样来研究。该项目还将发展我们对退火和淬火统计特性以及两者之间相互作用的理解。

Although much is understood about the statistical properties of physical systems in equilibrium, many systems of interest cannot be modeled realistically as being in equilibrium, meaning that the chance of observing an event or taking a certain measurement on the system does not change over time. Some physical systems are subject to external influences that vary over time; for example, the atmosphere is affected by the seasonal ocean temperatures, and the state of an electrical grid is affected by the time-varying demand for power. Physical systems are often also subject to random effects, either through measurement reading errors or noise in the system itself. This research project aims to develop a good statistical understanding of non-equilibrium and random complex systems using recent advances in probability and analysis. A better understanding of the statistical properties of non-equilibrium systems and random systems will lead to improved predictions and expand our knowledge of the statistical laws that can be expected to hold in nature. The project aims to develop new techniques based on recent developments in martingale theory and analysis to investigate central limit theorems, almost sure invariance principles (a strong form of approximation by Brownian motion), large deviations, and other advanced statistics for a wide class of non-equilibrium and random dynamical systems that are used to model complex physical systems. Examples of the systems to be studied include billiard systems (polynomially and exponentially mixing), models of intermittency, Lorenz-like maps and flows, and systems arising from statistical mechanics. Statistical properties of random dynamics can be investigated by sampling over all noise realizations (annealed statistics) or along individual realizations of the noise (quenched statistics). The project will also develop our understanding of annealed and quenched statistical properties and the interplay between the two.