Stochastic Differential Equations, Heat Kernel Analysis, and Random Matrix Theory

随机过程主宰着我们的世界：从股票价格的波动，到计算机或生物网络中嵌入系统的大规模行为，再到量子系统的核心行为。 大多数科学和工程问题部分归结为从结构中分离随机噪声，或者有时利用随机噪声放大结构。 20世纪世纪科学的伟大成就之一是发展了许多强大的工具来理解和分析各种系统中的随机噪声。 其中一套工具是随机分析，它对最初出现在物理学中的思想进行了严格的数学处理，为描述我们世界的状态方程添加了随机噪声。 这些所谓的随机微分方程在推进我们在许多领域的知识中发挥了基础性作用：经济学，系统工程，生物动力学，以及在数学中，应用于微分几何和量子信息理论等领域。本项目解决与随机微分方程，热核分析和随机矩阵理论相关的问题。中心主题是理解具有一定随机性的微分方程如何影响随机矩阵特征值的演化。这些想法与几何学有关，因为李群（几何对象的连续对称群）上的热流可以用这样的矩阵随机微分方程来表征。 在此，提出了10个研究项目，这些项目产生这些主题和应用程序之间的联系。 这些问题通常可以归结为研究几何激励的矩阵值随机微分方程的高维渐近性态。 特别值得注意的是有关布朗运动的精细（边）结构的高维李群从经典紧家庭的问题。 预期的研究完成后，将解决几个有趣的悬而未决的问题，并对随机微分方程理论和随机矩阵理论做出重大贡献。该奖项反映了NSF的法定使命，并被认为值得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来支持。

Random processes dominate our world: from the fluctuations of stock prices, to the large-scale behavior of queueing systems in computer or biological networks, to the core behavior of quantum systems. Most science and engineering problems in part boil down to separating random noise from structure, or sometimes utilizing random noise to amplify structure. One of the great triumphs of 20th Century science was the development of many robust tools for understanding and analyzing random noise in many kinds of systems. One such set of tools is stochastic analysis, which presents a rigorous mathematical treatment of ideas that first appeared in physics, adding random noise to the equations of state that describe our world. These so-called stochastic differential equations have playing a fundamental role in advancing our knowledge in many area: economics, systems engineering, biological dynamics, and within mathematics, with applications to fields like differential geometry and quantum information theory.This project addresses questions relating stochastic differential equations, heat kernel analysis, and random matrix theory. The central theme is understanding how differential equations with some randomness affect the evolution of eigenvalues of random matrices. These ideas connect with geometry, since the flow of heat on Lie groups (groups of continuous symmetries of geometric objects) can be characterized by such matrix stochastic differential equations. Herein, ten research projects are proposed which yield connections between these topics and applications to others. Generally, these problems can be described as studying the large-dimension asymptotic behavior of geometrically-motivated matrix-valued stochastic differential equations. Of particular note are questions related to the fine (edge) structure of Brownian motion on high dimensional Lie groups from the classical compact families. The intended research, upon completion, will settle several interesting open questions and present a major contribution both to the theory of stochastic differential equations and random matrix theory.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.