New Challenges in the Study of Propagation of Randomness for Nonlinear Evolution Equations

波浪在自然界中无处不在。当我们把一颗鹅卵石扔进湖中时，我们观察到它所形成的涟漪，这个膨胀的环被称为波包；或者当我们看到当光波穿过棱镜或水滴时形成的彩虹时，注意到白光在空间上分离成不同的颜色。偏微分方程（PDE）模拟波传播现象在理解这些物理和自然事件以及量子力学、光纤、铁磁、大气和水波以及许多其他物理模型方面发挥了重要作用。在这些情况下，波动现象永远不会太平滑或太简单，事实上，随着时间的推移，它们是非线性波动相互作用的副产品。能够理解和描述这些模型在某些噪声条件下或给定初始统计集合的动态行为，并精确描述这些模型中固有的随机性如何传播，是在研究自然世界时准确预测波动现象的基础。该项目旨在利用分析和概率方法回答有关长期动态和随机性传播的几个核心问题。该项目的工作及其与科学的联系促进了跨学科的互动，并促进了美国研究生和初级研究人员的培训，从而从根本上为其STEM劳动力做出了贡献。非线性偏微分方程中确定性方法和概率方法的相互作用自然地相互促进，当结合起来有助于对波动现象的深刻理解，这为推动研究向各个方向发展的新范式打开了大门。首席研究员研究当前研究前沿的几个项目。这些问题分为两个相互关联的方向，其主要目的是：(1)利用适当的定量准不变性、修正能量和随机结构的稳定性理论，从概率角度研究能量亚临界状态下色散流动的非平衡长时间动力学；(2)在平衡统计力学的背景下，建立了概率临界三维非线性Schrödinger方程（也称为构造量子场论模型）的Gibbs测度的不变性；(3)建立了二维环面上双曲正弦-戈登方程的概率局部理论及其相关Gibbs测度的不变性；(4)发展了非线性波动方程和非高斯数据的随机张量理论。该研究是专门研究随机方程的色散方程和波动方程社区之间的桥梁，有助于从根本上理解非线性波动现象中的随机性传播。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Waves are everywhere in nature. We observe them when we look at the ripples that form when we throw a pebble in a lake, the expanding ring called a wave-packet; or when we look at a rainbow that is formed when light wave passes through a prism or water droplet and note the spatial separation of white light into different colors. Partial differential equations (PDE) modeling wave propagation phenomena have played a fundamental role in understanding such physical and natural events as well as quantum mechanics, fiber optics, ferromagnetism, atmospheric and water waves, and many other physical models. In these cases, wave phenomena are never too smooth or too simple, and in fact the byproduct of nonlinear wave interactions as they propagate in time. Being able to understand and describe the dynamical behavior of such models under certain noisy conditions or given an initial statistical ensemble and have a precise description of how the inherent randomness built in these models propagates, is fundamental to accurately predict wave phenomena when studying the natural world. This project is aimed at answering several central questions about long-time dynamics and the propagation of randomness in this context using analytical and probabilistic methodologies. The work of the project, and its connections to science, promotes interdisciplinary interactions and fosters the training of graduate students and junior researchers in the United States thus fundamentally contributing to its STEM workforce. The interplay of deterministic methods in nonlinear PDE and probabilistic ones naturally feed off each other and when combined contribute to a deep understanding of wave phenomena, which opens the door to new paradigms that move research forward in various directions. The Principal Investigator studies several projects at the forefront of current research. The problems, grouped in two interrelated directions, aim broadly at: (1) studying the out of equilibrium long-time dynamics of dispersive flows from a probabilistic viewpoint in energy subcritical regimes by means of suitable quantitative quasi-invariance, modified energies and stability theory of random structures; (2) establishing the invariance of Gibbs measures for the probabilistically critical three-dimensional nonlinear Schrödinger equation (also known as a model in constructive quantum field theory) in the context of equilibrium statistical mechanics; (3) establishing a suitable probabilistic local theory of the hyperbolic sine-Gordon equation on 2D tori and the invariance of its associated Gibbs measure; and (4) the development of the random tensor theory for the nonlinear wave equations and for non-Gaussian data. The research bridges between the dispersive and wave equations communities that specialize in stochastic equations and contributes to understanding in a fundamental way propagation of randomness in nonlinear wave phenomena.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.