Analysis and Geometry of Random Fields Related to Stochastic Partial Differential Equations and Random Matrices

与随机偏微分方程和随机矩阵相关的随机场的分析和几何

• 批准号: 2153846
• 负责人: Yimin Xiao
• 金额: $ 22.05万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2153846&HistoricalAwards=false
• 关键词: Analysis; Geometry; Random; Fields; Related

This project is concerned with the development of an analytic and geometric theory of random fields that arise from stochastic partial differential equations (SPDEs) and random matrices. Special emphasis is placed on vector-valued and matrix-valued random fields that play a central role in various areas of pure and applied mathematics, mathematical physics, astronomy, bio-imaging, mathematical oceanography, and statistics. The Principal Inestigator (PI) will develop probabilistic, analytic, and geometric tools for studying vector-valued and matrix-valued random fields that will lead to a deeper understanding of random fields that arise from systems of SPDEs and random matrices. These tools will have sufficient novelty to open new research areas, solve a number of open problems in the theory of SPDEs, random matrices, and related random fields. Moreover, the proposed activities will also help to train graduate students and to develop their careers in the mathematical and statistical sciences. It is significant and challenging to characterize the fine analytic and geometric structures of vector-valued and matrix-valued random fields. In the past investigations, the PI has established a series of results on Gaussian and, more generally, infinitely divisible random fields, and the solutions of SPDEs. Together with his collaborators, the PI has developed fractal geometry and potential theory for Gaussian random fields, additive Lévy processes, solutions to SPDEs, and used them to resolve several outstanding open problems in non-Markovian Gaussian and stable random fields, Lévy processes, and the theory of SPDEs. The PI plans to continue his investigation of precise quantitative connections between vector-valued and matrix-valued random fields, SPDEs, potential theory, and the geometry of random fractals. The proposed research will ultimately yield novel insights into the understanding of vector-valued and matrix-valued random fields, SPDEs, and random matrices. The expected results will not only contribute to the theories of random fields, SPDEs, and random matrices but also promote their applicability in mathematics, mathematical physics, and in other scientific areas.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.

该项目涉及由随机部分微分方程（SPDE）和随机物质产生的随机场的分析和几何理论的发展。特别重点是在纯数学和应用数学，数学物理学，天文学，生物成像，数学海洋学和统计学的各个领域中起着核心作用的矢量值和基质值随机字段。主要的入侵器（PI）将开发概率，分析和几何工具，用于研究矢量值和矩阵值的随机场，这将导致对SPDES和随机物品系统产生的随机场有更深入的了解。这些工具将具有足够的新颖性，可以打开新的研究领域，解决SPDE，随机物品和相关随机领域的许多开放问题。此外，拟议的活动还将有助于培训研究生并在数学和统计科学中发展职业。表征矢量值和基质值随机场的精细​​分析和几何结构是巨大的挑战。在过去的研究中，PI已经建立了一系列有关高斯和更普遍的可分裂的随机场以及SPDES解决方案的结果。 PI与他的合作者一起，为高斯随机领域，添加剂Lévy过程，SPDES解决方案开发了分形的几何形状和潜在理论，并用它们来解决非马克维亚高斯和稳定的随机领域，lévy流程和SPDES理论中的几个出色的开放问题。 PI计划继续调查矢量值和基质值随机场，SPDE，潜在理论和随机分形的几何形状之间的精确定量连接。拟议的研究最终将对对矢量值和基质值的随机场，SPDE和随机物品的理解产生新的见解。预期的结果不仅将有助于随机领域，SPDE和随机物品的理论，而且还会促进其在数学，数学物理学和其他科学领域的适用性。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估被认为是宝贵的支持。