Geometry of Random Fields and Stochastic Partial Differential Equations

这个建议是关于发展一个系统的方法来研究随机场和随机偏微分方程（SPDE）的分析和几何性质。特别强调高斯、稳定和利维随机场，例如布朗单和加性利维过程，以及由高斯或利维噪声驱动的SPDE的解。所提到的例子是自然出现在纯数学和应用数学、数学海洋学、随机水文学、地质统计学、数学和统计物理学等各个领域的随机场。拟议的研究计划收集和开发概率，分析和几何工具，这将导致更深入地了解各种随机场的分析和几何。提出者相信这些工具将具有足够的新奇来解决随机场理论中一些长期存在的开放问题，并进一步促进其进一步的适用性。在过去的研究中，提出者已经发展了可加Levy过程和布朗单的潜在理论，并利用它们解决了Levy过程理论和布朗单分析中的几个悬而未决的问题。该提案者已经开发的想法，在几何测量理论的基础上，调查非马尔可夫高斯和稳定的随机场。他们还引入了更新理论技术来分析一类由奇异随机噪声驱动的抛物随机偏微分方程的解，并计划继续研究随机场、势理论、随机偏微分方程和随机分形几何之间的精确定量联系。他们相信，对这些联系的进一步追求最终将对随机场和相关随机偏微分方程的结构产生新的见解。

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.