Geometry of Random Fields and Stochastic Partial Differential Equations

随机场和随机偏微分方程的几何

• 批准号: 1006903
• 负责人: Davar Khoshnevisan
• 金额: $ 36万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006903&HistoricalAwards=false
• 关键词: Geometry; Random; Fields; Stochastic; Partial

This proposal is concerned with the development of a systematic approach to the study of the analytic and geometric properties of random fields and stochastic partial differential equations (SPDEs). Special emphasis is placed on Gaussian, stable, and Levy random fields such as the Brownian sheet and additive Levy processes, as well as the solutions of SPDEs that are driven by Gaussian or Levy noises. The mentioned examples are random fields that arise naturally in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, and mathematical as well as statistical physics. The proposed research plans to gather and develop probabilistic, analytic, and geometric tools that will lead to a deeper understanding of the analysis and geometry of various random fields. The Proposers believe that these tools will have sufficient novelty to solve a number of long-standing open problems in the theory of random fields, and also further promote their further applicability.In their past investigations, the Proposers have developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in the theory of Levy processes and the analysis of the Brownian sheet. The Proposers have developed ideas, based in geometric-measure theory, for investigating non-Markovian Gaussian and stable random fields. And they have introduced renewal-theoretic techniques for the asymptotic analysis of solutions to a large class of parabolic stochastic PDEs driven by singular random noises.The Proposers 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 and related stochastic PDEs.

这一建议涉及发展一种系统的方法来研究随机场和随机偏微分方程(SPDEs)的解析和几何性质。特别强调了高斯、稳定和Levy随机场，如布朗单过程和加性Levy过程，以及由高斯或Levy噪声驱动的SPDEs的解。上述例子是在纯数学和应用数学、数学海洋学、随机水文学、地质统计学、数学和统计物理学的各个领域中自然产生的随机场。拟议的研究计划收集和开发概率、解析和几何工具，这些工具将导致对各种随机场的分析和几何有更深的理解。在过去的研究中，他们发展了关于可加Levy过程和布朗单的潜在理论，并用它们解决了Levy过程理论和布朗单分析中的几个突出的公开问题。提出者以几何测量理论为基础，提出了研究非马尔可夫高斯和稳定随机场的想法。他们还引入了更新理论技术来分析一大类由奇异随机噪声驱动的抛物型随机偏微分方程解的渐近分析。他们相信，对这些联系的进一步追求最终将产生对随机场和相关随机偏微分方程的结构的新见解。