Analysis of Stochastic Partial Differential Equations

本项目主要关注在快速发展的随机偏微分方程（SPDEs）领域发展新的数学方法。特别注意发展新技术，以分析从数学海洋学和气候模式、随机水文学、地质统计学、经典宇宙学到统计和数学物理学等应用领域中自然产生的特定SPDEs族。尽管作为项目的一部分所研究的许多问题都围绕着spde中值得注意的问题，但成功解决这些问题也可能有助于研究其他复杂系统。所开发的思想和技术有望具有足够的新颖性，以开辟新的研究领域，解决一些长期存在的开放性问题，并进一步促进spde理论的适用性。该项目将涉及研究生，PI将继续组织会议，并计划与人合著一本研究领域的教科书。该项目将在独立的、现代兴趣的具体问题的背景下，研究spde、随机场和随机分形之间的定量联系。其中一些联系是由应用领域的问题直接引起的。这些问题包括从非线性统计逆问题到噪声反应扩散方程爆破现象的分析。另一些则旨在为间歇性和空洞动力学等物理观察现象提供数学解释。而另一些则与研究全新的现象有关，比如宏观和微观的多重分形。这项研究有望对spde的结构、爆炸现象、物理多重分形以及相关的多重分形随机场产生新的见解。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。

This project is chiefly concerned with the development of new mathematical methods in the rapidly growing field of stochastic partial differential equations (SPDEs). Special attention is paid to developing novel techniques that are designed to analyze specific families of SPDEs that arise naturally in application areas that range from mathematical oceanography and climate models, stochastic hydrology, geostatistics, classical cosmology, to statistical and mathematical physics. Although many of the problems studied as part of the project are centered around noteworthy questions in SPDEs, a successful resolution of these problems will likely also help study other complex systems. The developed ideas and techniques are expected to have sufficient novelty to open new research areas, solve a number of long-standing open problems and promote further applicability of the theory of SPDEs. The project will involve graduate students, and the PI will continue organizing conferences and is planning to co-author a textbook in the area of research. The project will engage quantitative connections between SPDEs, random fields, and random fractals in the context of concrete questions of independent, modern interest. Some of these connections are motivated directly by questions in applications areas. These include problems that range from nonlinear statistical inverse problems to the analysis of the blowup phenomenon for noisy reaction-diffusion equations. Others are aimed at providing mathematical explanations for physically observed phenomena such as intermittency and void dynamics. Yet others are related to studying entirely new phenomena such as macroscopic and microscopic multifractality. This research is expected to yield novel insights into the structure of SPDEs, blowup phenomena, physical multifractals, and the related multifractal random fields.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.