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都在观察它。这可能很简单，就像我们看彩虹一样：色散导致白光在空间上分离成不同的颜色。或者，当我们把一块鹅卵石扔进湖里时，我们会看到水波的涟漪：不断膨胀的环被称为“波包”，我们注意到波以不同的速度传播，最长的传播最快，最短的传播最慢。但在量子力学、等离子体、光纤、铁磁性、大气和水波等许多环境中也会出现波动现象。由于自然界中的波在传播时以非线性方式相互作用，并且具有不同的特性，如幅度、长度、振荡、速度和位置，因此了解它们在某些噪声条件下或在某些介质中进行测量时的行为是很重要的，因为在某些介质中，小误差是不可避免的。理解通过光缆发送信号的最有效方式，或者能够在温度接近绝对零度时预测气体的性质(玻色-爱因斯坦凝聚体)是自然界中两种截然不同的现象，但它们都是同一非线性模型解的两个方面。在研究自然界时，能够在给定初始统计集合的情况下理解和描述此类模型解的动力学行为，并准确描述这些模型中固有的随机性是如何传播的，这是准确预测波动现象的基础。这个项目的目的是回答几个关于长时间动力学和在非线性、色散和波动现象背景下的随机性传播的中心问题。同时，该项目的工作旨在培养美国研究生和初级研究人员。在过去的几年里，确定性方法和概率方法在非线性发展方程研究中的协同作用，从根本上加深了我们对解的动力学的理解，并打开了新范式的大门，这些新范式推动了研究在不同方向上向前发展。为了应对当前研究前沿的重要挑战，旨在定量了解一般波动现象的动力学特性，首席研究员(PI)采用了一种创新的方法，其基础是整合了分析、概率、统计力学、动力系统、组合学和解析数论的方法和思想，并结合了最近受到奇异随机抛物方程惊人进展启发的新方法的推动。作为该项目的一部分，PI将在非线性发展方程的三个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期动力学，非线性Hartree方程的Gibbs测度的不变性--源于N体薛定谔方程的平均场极限--以及环面上的非线性波动和薛定谔方程，以及一种新的概率拟线性双曲理论的发展。要研究的问题的好处是，它们按不同的难度等级进行评分，每个问题都导致独立的部分进展和更深的理解。一些问题将作为这个项目的一部分，为研究生和博士后研究员带来极好的研究问题。此外，PI的工作将导致新的研究生主题课程的开发，从而丰富新一代研究人员的发展。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。

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.