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

基本信息

批准号：
1608575
负责人：
Davar Khoshnevisan
金额：
$ 19万
依托单位：
University of Utah
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 2019-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608575&HistoricalAwards=false
关键词：
Collaborative Research Fractals Multifractals Stochastic

项目摘要

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.

随机部分微分方程（SPDE）用于建模广泛的自然现象，并在纯数学和应用数学，数学海洋学，随机水文学，地理统计和数学物理学的各个领域中起着核心作用。该研究项目涉及随机领域的分析/几何理论的发展，主要是由SPDE引起的。该项目旨在开发概率，分析和几何工具，从而使对物理上相关的随机领域有更深入的了解。可以预料，这些工具将具有足够的新颖性来开放新的研究领域，解决SPDE理论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。该项目涉及研究生。这是巨大的，具有挑战性的，要表征SPDE和相关随机领域的精细局部和渐近结构。在过去的工作中，研究人员开发了基于几何量学理论的思想，用于分析非马克维亚高斯和稳定的随机领域，并引入了更新的理论和耦合技术，以针对大型非线性SPDE对解决方案的渐近分析。该研究项目继续研究随机场，潜在理论，SPDE和随机分形的几何形状之间的精确定量连接。特别强调由完全非线性乘法噪声驱动的两个极端普遍性类别。预计对这些连接的进一步追求将产生对随机场，物理多重分子和相关SPDE的结构的新见解。