Analysis of Stochastic Partial Differential Equations

随机偏微分方程的分析

• 批准号: 2245242
• 负责人: Davar Khoshnevisan
• 金额: $ 32.35万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2245242&HistoricalAwards=false
• 关键词: Analysis; Stochastic; Partial; Differential; Equations

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.

该项目主要关注快速发展的随机偏微分方程（SPDEs）领域新数学方法的开发。特别注意开发新的技术，旨在分析特定的家庭自然产生的SPDE的应用领域，从数学海洋学和气候模型，随机水文学，地质统计学，经典宇宙学，统计和数学物理。尽管作为项目一部分研究的许多问题都围绕着SPDE中值得注意的问题，但这些问题的成功解决也可能有助于研究其他复杂系统。所发展的思想和技术有望具有足够的新奇，以开辟新的研究领域，解决一些长期存在的开放问题，并促进SPDE理论的进一步应用。该项目将涉及研究生，PI将继续组织会议，并计划共同编写一本研究领域的教科书。该项目将在独立的，现代感兴趣的具体问题的背景下进行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激发的。这些问题的范围从非线性统计逆问题到分析噪声反应扩散方程的爆破现象。另一些则旨在为物理观察到的现象提供数学解释，如非线性和空隙动力学。还有一些则与研究全新的现象有关，如宏观和微观多重分形。这项研究预计将产生新的见解结构的SPDE，爆破现象，物理多重分形，以及相关的多重分形random fields.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。