Propagation of Randomness in Nonlinear Evolution Equations

我们都熟悉色散和波动现象，因为我们在自然界中一直观察到它。它可以像我们看彩虹一样简单：色散导致白光在空间上分离成不同的颜色。或者，当我们把一颗鹅卵石扔进湖中时，看到的波纹：这个膨胀的环被称为“波包”，我们注意到波以不同的速度传播，最长的波传播得最快，最短的波传播得最慢。但波动现象也出现在量子力学、等离子体、光纤、铁磁性、大气和水波以及许多其他环境中。由于自然界中的波在传播过程中以非线性方式相互作用，并且随着时间的推移具有振幅、长度、振荡、速度和位置等不同的特性，因此了解它们在某些噪声条件下或在某些不可避免的小误差介质中进行测量时的行为是很重要的。理解通过光纤电缆发送信号的最有效方式或能够预测温度接近绝对零度时气体的性质（玻色-爱因斯坦凝聚）是两种非常不同的自然现象，但它们都是同一个非线性模型的解决方案的两个方面。能够理解和描述在给定初始统计集合的情况下这些模型的解的动力学行为，并精确描述这些模型中固有的随机性是如何传播的，这是在研究自然世界时准确预测波动现象的基础。本项目旨在回答关于非线性色散和波动现象背景下的长期动力学和随机性传播的几个核心问题。与此同时，该项目的工作旨在促进对美国研究生和初级研究人员的培训。过去几年来，在非线性演化方程研究中，确定性和概率方法之间的协同作用进一步加深了我们对基本解决方案动力学的理解，并为推动研究向各个方向发展的新范式打开了大门。为了解决当前研究前沿的重要挑战，旨在定量理解一般波动现象的动力学特性，首席研究员（PI）采用了一种基于分析，概率，统计力学，动力系统，组合学和解析数论，加上最近的新方法的推动，这些新方法的灵感来自于奇异随机抛物方程的惊人进展。作为该项目的一部分，PI将在非线性演化方程的三个前沿研究领域探索几个令人兴奋的方向，其中确定性和概率方法的相互作用是取得进展的关键。这些问题旨在从概率的角度研究色散流动的长时间动力学，以及非线性Hartree方程（由n体Schrödinger方程的平均场极限引起）和非线性波方程和Schrödinger方程在环面上的Gibbs测量的不变性；并在一个新的概率拟线性双曲理论的发展。所要研究的问题的优点是它们被划分为不同的难度等级，每一个都导致独立的部分进展和更深入的理解。作为该项目的一部分，一些问题将为研究生和博士后研究人员带来优秀的研究问题。此外，PI的工作将引导新的研究生专题课程的发展，从而丰富新一代研究人员的发展。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。

We are all familiar with dispersive and wave phenomena since we observe it all the time in nature. It could be as simple as when we look at a rainbow: dispersion causes the spatial separation of white light into different colors. Or when we look at the ripples that form when we throw a pebble in the lake: the expanding ring is called a “wave-packet” and we note that waves travel at different speeds, the longest going fastest and the shortest ones slowest. But wave phenomena also arise in quantum mechanics, plasmas, fiber optics, ferromagnetism, atmospheric and water waves and many other settings. Because waves in nature interact in a nonlinear fashion as they propagate and have different properties such as amplitude, length, oscillation, speed, and position over time, it is important to understand how they may behave under certain noisy conditions or when taking measurements in certain media where small errors are unavoidable. Understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero (a Bose-Einstein condensate) are two very different phenomena in nature but are both aspects of solutions to the same nonlinear model. Being able to understand and describe the dynamical behavior of solutions to such models 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 the context of nonlinear dispersive and wave phenomena. At the same time, the work of the project is designed to foster the training of graduate students and junior researchers in the U.S.The synergy between deterministic and probabilistic approaches in the study of nonlinear evolution equations in the last few years has furthered our understanding of the dynamics of solutions in fundamental ways and opened the door to new paradigms that have moved research forward in various directions. To address important challenges at the cutting edge of current research, aimed at a quantitative understanding of the dynamical properties of generic wave phenomena, the principal investigator (PI) adopts an innovative approach based on the integration of methods and ideas from analysis, probability, statistical mechanics, dynamical systems, combinatorics and analytic number theory coupled with the impetus of recent new methods that were inspired by the spectacular advances in singular stochastic parabolic equations. As part of this project, the PI will explore several exciting directions in three areas of research at the forefront of nonlinear evolution equations, where the interplay of deterministic and probabilistic approaches is the key to make progress. The problems aim at studying the long-time dynamics of dispersive flows from a probabilistic viewpoint, the invariance of Gibbs measures for the nonlinear Hartree equation - arising from the mean field limit for the N-body Schrödinger equation - and for the nonlinear wave and Schrödinger equations on tori; and at the development of a new probabilistic quasilinear hyperbolic theory. The problems to be studied have the advantage that they are graded at different levels of difficulty, each leading to independent partial progress and deeper understanding. Some of the questions that will be pursued as part of this project lead to excellent research problems for graduate doctoral students and postdoctoral fellows. Furthermore, the PI’s work will lead to the development of new graduate topics courses, thus enriching the development of the new generation of researchers.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.