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

自然界中，波浪无处不在。当我们观察当我们把鹅卵石扔进湖中时形成的涟漪，这个膨胀的环称为波包；或者当我们观察当光波通过棱镜或水滴形成的彩虹时，注意到白光变成不同颜色的空间分离。模拟波传播现象的偏微分方程组(PDE)在理解此类物理和自然事件以及量子力学、光纤、铁磁、大气和水波等许多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，实际上是非线性波相互作用的副产品，因为它们在时间上传播。能够理解和描述这种模型在某些噪声条件下的动力学行为，或者给出一个初始的统计集合，并准确地描述这些模型中固有的随机性是如何传播的，这是研究自然界时准确预测波浪现象的基础。这个项目的目的是利用分析和概率方法回答关于长期动态和随机性在这方面的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互动，并促进了对美国研究生和初级研究人员的培训，从而从根本上为其STEM工作人员做出了贡献。确定性方法和概率方法在非线性偏微分方程中的相互作用自然相互促进，当它们结合在一起时，有助于对波动现象的深入理解，从而打开了向不同方向推进研究的新范式的大门。首席调查员研究了几个处于当前研究前沿的项目。这些问题分为两个相互关联的方向，主要目的是：(1)利用适当的定量准不变性、修正的能量和随机结构的稳定性理论，从概率的观点研究能量亚临界区域中色散流动的非平衡长期动力学；(2)在平衡统计力学的背景下，建立概率临界三维非线性薛定谔方程(也称为构造量子场论中的模型)的Gibbs测度的不变性；(3)建立二维环上双曲Sine-Gordon方程的适当的概率局域理论及其相关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。这项研究在色散方程和波动方程社区之间架起了桥梁，这些社区专门研究随机方程，并有助于从根本上理解非线性波动现象中随机性的传播。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。

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.