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.

该项目主要涉及在随机部分微分方程（SPDE）快速增长的领域中新数学方法的发展。特别关注开发新型技术，这些技术旨在分析从数学海洋学和气候模型，随机水文学，地统计学，经典宇宙学到统计和数学物理学的应用领域自然出现的特定SPDES。尽管作为项目的一部分研究的许多问题都是围绕SPDE中值得注意的问题的重点，但是成功解决这些问题的问题也可能有助于研究其他复杂的系统。期望开发的思想和技术具有足够的新颖性来开放新的研究领域，解决许多长期的开放问题并促进SPDES理论的进一步适用性。该项目将涉及研究生，PI将继续组织会议，并计划在研究领域合着一本教科书。该项目将在独立，现代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。这些连接中的一些是直接由应用领域中的问题进行的。这些问题包括从非线性统计反问题到噪声反应扩散方程的爆炸现象的分析。其他人则旨在为物理观察到的现象（例如间歇性和空隙动力学）提供数学解释。然而，其他人则与研究全新现象有关，例如宏观和微观多纹状体。预计这项研究将对SPDE，爆炸现象，物理多重术和相关多重随机领域的结构产生新的见解。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响审查标准通过评估来进行评估的。