Collaborative Research: Asymptotic Geometry and Analysis of Stochastic Partial Differential Equations

该项目涉及随机场几何/分析理论的发展，主要是由随机偏微分方程[简称 SPDE] 产生的随机场。特别强调某些 SPDE 和相关的随机场，它们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的各个领域中发挥着核心作用。研究人员将开发概率、几何和分析工具，从而更深入地了解大量物理和数学上有趣的 SPDE 以及相关的随机场。研究人员相信，这些工具将具有足够的新颖性来开辟新的研究领域，解决SPDE理论和相关随机场中的一些长期存在的开放性问题，并进一步提升其适用性。此外，这些活动还包括培养研究生和博士后学者的持续计划，并发展他们在数学和统计科学领域的职业生涯。 表征 SPDE 和相关随机场的精细​​局部和渐近结构既具有重要意义，又具有挑战性。 在过去的研究中，研究人员建立了一系列关于SPDE和相关随机场解的渐近行为、间歇性和宏观尺度多重分形性质的结果，发展了加性Levy过程和布朗片的势理论，并用它们解决了Levy过程、布朗片和抛物型随机偏微分方程理论中的几个突出的开放问题。研究人员基于概率论和几何测度理论，提出了用于分析非马尔可夫高斯场和稳定随机场的想法。他们还引入了更新理论和耦合技术，用于对一大类非线性 SPDE 的解进行渐近分析。他们计划继续研究 SPDE、随机场、势论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产生对 SPDE、大规模物理多重分形和相关随机场结构的新颖见解。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with the development of a geometric/analytic theory of random fields, primarily those that arise from stochastic partial differential equations [SPDEs, for short]. Special emphasis is placed on certain SPDEs and related random fields that play a central role in various areas of pure and applied mathematics, mathematical oceanography, stochastic hydrology, geostatistics, and mathematical physics. The investigators will develop probabilistic, geometric, and analytic tools that will lead to a deeper understanding of a large family of physically- and mathematically-interesting SPDEs and related random fields. The investigators believe that these tools will have sufficient novelty to open new research areas, solve a number of long-standing open problems in the theory of SPDEs and related random fields, and also further promote their applicability. In addition, the activities include a sustained program to train graduate students and postdoctoral scholars, and to develop their careers in the mathematical and statistical sciences.It is both significant and challenging to characterize the fine local and asymptotic structure of SPDEs and related random fields. In their past investigations, the investigators have established a series of results on the asymptotic behavior, intermittency, and macroscopic-scale multifractal properties of the solutions of SPDEs and related random fields, developed potential theories for additive Levy processes and the Brownian sheet, and used them to resolve several outstanding open problems in Levy processes, the Brownian sheet, and the theory of parabolic stochastic partial differential equations. The investigators have developed ideas, based on probability theory and geometric measure theory, for the analysis of non-Markovian Gaussian and stable random fields. They have also 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 SPDEs, random fields, potential theory, and the geometry of random fractals. They believe that further pursuit of these connections will ultimately yield novel insights into the structure of SPDEs,large-scale physical multifractals, and related random fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.