Dynamics of singular stochastic nonlinear dispersive PDEs

弥散在自然界中无处不在。最著名的色散例子是彩虹，色散效应将白光在空间上分成不同波长(不同颜色)的分量。非线性色散偏微分方程组(PDE)，如非线性薛定谔方程(NLS)和非线性波动方程(NLW)，自然地出现在描述现实世界中波动现象的模型中。在过去的三十年里，确定性的非线性色散偏微分方程组的研究取得了长足的发展，其中以Kenig，Bourain和Tao等人为首的调和分析起到了基础性的作用。近年来，确定性分析与概率论的结合在该领域发挥着越来越重要的作用。这种概率观点使我们能够超越确定性分析的局限。更重要的是，了解随机扰动在实际中的作用也是至关重要的，因为这种随机扰动是普遍存在的。本研究的主要目的是发展新的数学思想和技术来澄清随机非线性色散偏微分方程研究中长期存在的基本问题，并以随机NLS和随机NLW为主要例子。在奇异随机抛物型偏微分方程组领域，由海尔和古比内利及其合作者领导的研究已经取得了重大进展。这使得惊人的理论成为可能，这些理论正在改变这一领域的研究格局。然而，他们的新理论是为处理抛物型问题而设计的，对于如何使其适用于解色散方程，尚不是一个先验的清楚。尽管最近取得了一些令人振奋的进展，但我们对随机色散偏微分方程组的理解仍然很不令人满意。在这些拟议的项目中，首席研究员(PI)将研究随机色散偏微分方程组领域的几个公开问题。更具体地说，PI将专注于研究周期域中随机NLS和NLW的不变度量以及局部和全局时间解的性质。PI计划通过将离散偏微分方程、随机分析、概率论和调和分析的工具与最近的进展相结合来解决这些问题。

Dispersion exists ubiquitously in nature. The most famous example of dispersion is seen in a rainbow, where dispersion effect separates the white light spatially into components of different wavelengths (different colours). Nonlinear dispersive partial differential equations (PDEs), such as nonlinear Schrodinger equations (NLS) and nonlinear wave equations (NLW), appear naturally in models describing wave phenomena in the real world. In the past thirty years, the study of deterministic nonlinear dispersive PDEs has seen significant development, in which harmonic analysis has played a fundamental role, led by Kenig, Bourgain and Tao, among others. In recent years, a combination of deterministic analysis with probability theory has played an increasingly important role in the field. This probabilistic perspective allows us to go beyond the limits of deterministic analysis. More importantly, it is also essential to understand the effect of stochastic perturbation in practice since such stochastic perturbation is ubiquitous.The main objective of this research is to develop novel mathematical ideas and techniques to clarify long-standing fundamental questions in the study of stochastic nonlinear dispersive PDEs, with primary examples given by stochastic NLS and stochastic NLW. In the field of singular stochastic parabolic PDEs, significant progress has been taking place led by Hairer and Gubinelli with their collaborators. This has enabled striking theories which are changing the landscape of the study in this field. However, their new theories are designed to handle parabolic problems, and it is not a priori clear on how to adapt them to solve dispersive equations. Despite some exciting recent progress, our understanding of stochastic dispersive PDEs is still very far from satisfactory. In these proposed projects, the principal investigator (PI) will study several open problems in the field of stochastic dispersive PDEs. More specifically, the PI will focus on studying the properties of invariant measures and the local and global-in-time solutions to stochastic NLS and NLW in periodic domains. The PI plans to address these problems by combining tools from dispersive PDEs, stochastic analysis, probability theory and harmonic analysis with recent progress.