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。除了基本的兴趣外，研究这类系统的行为还有助于理解热量如何通过介质流动，肿瘤如何生长，水波如何传播，以及交通拥堵的形成和溶解。这个项目专注于行为的一个特定方面，称为大偏差，它是关于在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，对大偏差的研究在理解亚稳态之间的漂移、分析相变和对数值算法进行基准测试方面也有应用。这项研究旨在增进对这些领域和其他领域的理解。该项目还为研究生提供了研究培训的机会。具体地说，该项目将研究非对称相互作用粒子系统和非线性随机偏微分方程组的有限维和样本路径大偏差。它们在不同的体制中表现出很大的偏差，包括短期、中长期和长期体制。这个项目的主要目标是：i)建立各种制度的短期大偏差原则；ii)研究短期制度中的大偏差如何收敛到长期制度中的大偏差(主要是猜测的)；以及iii)调查中长期制度中的大偏差。这些目标包括对各种物理现象的数学研究，包括孤波和激波之间的对应，孤波的自发产生，以及动力学对称破缺。这个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来实现，与量子多体系统的连接，以及可积偏微分方程组的逆散射变换。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，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.