Collaborative Research: Asymptotic Geometry and Analysis of Stochastic Partial Differential Equations

合作研究：渐近几何与随机偏微分方程分析

• 批准号: 1855185
• 负责人: Yimin Xiao
• 金额: $ 19.64万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855185&HistoricalAwards=false
• 关键词: Collaborative; Research; Asymptotic; Geometry; Analysis

This project is concerned with the development of a geometric/analytic theory of random fields, primarily those that arise from stochastic partial differential equations [SPDEs, for short]. Special emphasis is placed on certain SPDEs and related random fields that play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, and mathematical physics. The investigators will develop probabilistic, geometric, and analytic tools that will lead to a deeper understanding of a large family of physically- and mathematically-interesting SPDEs and related random fields. The investigators believe that these tools will have sufficient novelty to open new research areas, solve a number of long-standing open problems in the theory of SPDEs and related random fields, and also further promote their applicability. In addition, the activities include a sustained program to train graduate students and postdoctoral scholars, and to develop their careers in the mathematical and statistical sciences.It is both significant and challenging to characterize the fine local and asymptotic structure of SPDEs and related random fields. In their past investigations, the investigators have established a series of results on the asymptotic behavior, intermittency, and macroscopic-scale multifractal properties of the solutions of SPDEs and related random fields, developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in Levy processes, the Brownian sheet, and the theory of parabolic stochastic partial differential equations. The investigators have developed ideas, based on probability theory and geometric measure theory, for the analysis of non-Markovian Gaussian and stable random fields. They have also introduced renewal-theoretic and coupling techniques for the asymptotic analysis of solutions to a large class of nonlinear SPDEs. They plan to continue their investigation of precise quantitative connections between SPDEs, random fields, potential theory, and the geometry of random fractals. They believe that further pursuit of these connections will ultimately yield novel insights into the structure of SPDEs,large-scale physical multifractals, and related 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.

该项目与随机场的几何/分析理论的发展有关，这主要是由随机偏微分方程（简称SPDE）引起的。特别重点是在纯数学和应用数学，数学海洋学，随机水文学，地统计学和数学物理学的各个领域中起着核心作用的某些SPD和相关随机领域。研究人员将开发概率，几何和分析工具，从而更深入地了解一个大型的身体和数学杂志的SPDES和相关的随机领域。研究人员认为，这些工具将具有足够的新颖性来开放新的研究领域，解决了SPDE和相关随机领域的许多长期开放问题，并进一步促进其适用性。此外，这些活动包括一项持续的计划，以培训研究生和博士后学者，并在数学和统计科学中发展其职业。它表征SPDES的局部局部和渐近结构以及相关随机领域的良好本地和渐近结构既重要又具有挑战性。 In their past investigations, the investigators have established a series of results on the asymptotic behavior, intermittency, and macroscopic-scale multifractal properties of the solutions of SPDEs and related random fields, developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in Levy processes, the Brownian sheet, and the theory of parabolic stochastic partial differential equations.研究人员根据概率理论和几何测量理论开发了思想，用于分析非马克维亚高斯和稳定的随机领域。他们还引入了更新的理论和耦合技术，以对大型非线性SPDE的溶液进行渐近分析。他们计划继续研究SPDE，随机场，潜在理论和随机分形的几何形状之间的精确定量连接。他们认为，进一步追求这些联系将最终产生对SPDES结构，大规模的物理多型和相关随机领域的结构的新颖见解。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响标准通过评估来进行评估的。