Collaborative Research: Fractals, Multifractals, and Stochastic Partial Differential Equations

合作研究：分形、多重分形和随机偏微分方程

• 批准号: 1607089
• 负责人: Yimin Xiao
• 金额: $ 11万
• 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: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607089&HistoricalAwards=false
• 关键词: Collaborative; Research; Fractals; Multifractals; Stochastic

Stochastic partial differential equations (SPDE) are employed to model a wide range of natural phenomena and play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geo-statistics, and mathematical physics. This research project is concerned with the development of an analytic/geometric theory of random fields, primarily those that arise from SPDE. The project aims to develop probabilistic, analytic, and geometric tools that will lead to a deeper understanding of physically-relevant random fields. It is anticipated that these tools will have sufficient novelty to open new research areas, solve a number of long-standing open problems in the theory of SPDE and related random fields, and further promote their applicability. The project involves graduate students in the research.It is significant and challenging to characterize the fine local and asymptotic structures of SPDE and related random fields. In past work, the investigators developed ideas, based in geometric-measure theory, for the analysis of non-Markovian Gaussian and stable random fields, and they introduced renewal-theoretic and coupling techniques for the asymptotic analysis of solutions to a large class of nonlinear SPDE. This research project continues investigation of precise quantitative connections between random fields, potential theory, SPDE, and the geometry of random fractals. Special emphasis is placed on two extremal universality classes of SPDE that are driven by fully non-linear multiplicative noise. Further pursuit of these connections is expected to yield novel insights into the structure of random fields, physical multifractals, and related SPDE.

随机偏微分方程 (SPDE) 用于模拟各种自然现象，并在纯数学和应用数学、数学海洋学、随机水文学、地质统计学和数学物理学等各个领域发挥着核心作用。该研究项目涉及随机场解析/几何理论的发展，主要是由 SPDE 产生的随机场。 该项目旨在开发概率、分析和几何工具，以加深对物理相关随机场的理解。 预计这些工具将具有足够的新颖性，开辟新的研究领域，解决SPDE理论及相关随机场中的一些长期悬而未决的开放性问题，并进一步提升其适用性。该项目涉及研究生参与的研究。表征SPDE及相关随机场的精细​​局部渐近结构具有重要意义和挑战性。 在过去的工作中，研究人员基于几何测度理论提出了分析非马尔可夫高斯场和稳定随机场的想法，并引入了更新理论和耦合技术来对一大类非线性方程组的解进行渐近分析。 SPDE。该研究项目继续研究随机场、势论、SPDE 和随机分形几何之间的精确定量联系。特别强调由完全非线性乘性噪声驱动的 SPDE 的两个极值普遍性类别。 对这些联系的进一步探索有望产生对随机场、物理多重分形和相关 SPDE 结构的新见解。