Intermittency and Random Fractals

间歇性和随机分形

• 批准号: 1307470
• 负责人: Davar Khoshnevisan
• 金额: $ 39万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1307470&HistoricalAwards=false
• 关键词: Intermittency; Random; Fractals

This proposal is concerned with the development of an analytic/geometric theory of random fields, primarily those that arise from stochastic PDEs [SPDEs]. Special emphasis is placed on two extremal universality classes of SPDEs that are driven by fully non-linear multiplicative noise. The Investigators have developed ideas, based in geometric-measure theory, for the analysis of non-Markovian Gaussian and stable random fields. And they have 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 random fields, potential theory, stochastic PDEs, and the geometry of random fractals. And they believe that further pursuit of these connections will ultimately yield novel insights into the structure of random fields, physical multifractals, and related stochastic PDEs.Stochastic PDEs and random fields play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, mathematical physics and other scientific areas. It is significant and challenging to characterize the fine local and asymptotic structures of SPDEs and related random fields. The Investigators believe that the proposed research will have sufficient novelty to solve a number of long-standing open problems in the theory of stochastic PDEs and related random fields, and also further promote their applicability in various scientific areas. Moreover, the proposed activities will also help to train graduate students and to develop their careers in the mathematical and statistical sciences.

该建议涉及随机场的分析/几何理论的发展，主要是由随机PDE引起的[SPDE]。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。 研究人员已经开发了基于几何量化理论的思想，用于分析非马克维亚高斯和稳定的随机场。他们引入了更新的理论和耦合技术，用于对大型非线性SPDE的解决方案的渐近分析。他们计划继续研究随机场，潜在理论，随机PDES和随机分形的几何形状之间的精确定量连接。他们认为，对这些联系的进一步追求最终将对随机领域，物理多种形式和相关随机PDE的结构产生新颖的见解。构成的PDE和随机领域在纯数学和应用数学，数学海洋学，随机水平，随机水文，地理学，数学物理学和其他科学的各个领域中起着核心作用。 表征SPDE和相关随机场的精细​​局部和渐近结构是重要的，具有挑战性的。研究人员认为，拟议的研究将具有足够的新颖性来解决随机PDE和相关随机领域的许多长期开放问题，并进一步促进其在各个科学领域的适用性。 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。