Large Deviations in Large Non-equilibrium Systems

现代概率论研究的核心是研究复杂随机系统的行为。除了具有根本意义外，研究此类系统的行为还可以提高对热量如何流过介质、肿瘤如何生长、水波如何传播以及交通堵塞如何形成和消除的理解。该项目侧重于被称为大偏差的行为的一个特定方面，这是关于感兴趣的系统中发生的小概率事件。大偏差理论是概率论的支柱之一;理解它如何在不同类型的系统中表现出来是一项实际而重要的任务。此外，大偏差的研究可以应用于理解亚稳态之间的偏移，分析相变和基准数值算法。这项研究旨在促进对这些和其他领域的了解。本计画亦提供研究训练机会给研究生，具体而言，本计画将研究非对称相互作用粒子系统及非线性随机偏微分方程之有限维及样本路径大偏差。他们表现出很大的偏差，在各种制度，包括短期，中长时间，和长期制度。该项目的主要目标是：i）建立各种系统的短时大偏差原理; ii）研究短时区的大偏差如何收敛到长时区的大偏差（主要是自然的）; iii）研究中长时区的大偏差。这些目标涉及各种物理现象的数学研究，包括孤子波和冲击波之间的对应关系，孤子波的自发产生，以及动力学对称性破缺。该项目的目标将通过使用Feynman-Kac公式的随机分析和可积结构的组合、与量子多体系统的连接以及可积偏微分方程的逆散射变换来实现。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

Central to research in modern probability theory is the study of how complex random systems behave. Besides being of fundamental interest, studying the behavior of such systems improves understanding of how heat flows through a medium, how tumors grow, how water waves propagate, and how traffic jams form and dissolve. This project focuses on a particular aspect of the behavior known as large deviations, which is about small-probability events occurring in the systems of interest. The theory of large deviations is one of the pillars of probability theory; understanding how it manifests itself in different types of systems is a practical and important task. Further, the study of large deviations has applications to understanding excursions between meta-stable states, analyzing phase transitions, and benchmarking numerical algorithms. The research aims to advance understanding in these and other areas. The project also provides research training opportunities for graduate students.In concrete terms, this project will study the finite-dimensional and sample-path large deviations of asymmetric interacting particle systems and of nonlinear stochastic partial differential equations. They exhibit large deviations in various regimes, including the short-time, intermediately-long-time, and long-time regimes. The main goals of this project are: i) establishing the short-time large deviation principles of various systems; ii) studying how the large deviations in the short-time regime converge to the (mostly conjectural) large deviations in the long-time regime; and iii) investigating the large deviations in the intermediately-long-time regime. These goals involve the mathematical study of various physical phenomena, including the correspondences between soliton waves and shock waves, the spontaneous generation of soliton waves, and dynamical symmetry breaking. The goals of this project will be achieved by a combination of stochastic analysis and integrable structures using the Feynman-Kac formula, the connection to quantum-many-body systems, and the inverse scattering transform of integrable partial differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.