Analysis and Geometry of Random Fields Related to Stochastic Partial Differential Equations and Random Matrices

该项目致力于发展由随机偏微分方程组(SPDEs)和随机矩阵产生的随机场的解析和几何理论。特别强调的是矢量值和矩阵值随机场，它们在纯数学和应用数学、数学物理、天文学、生物成像、数学海洋学和统计学的各个领域中发挥着核心作用。首席调查员(PI)将开发概率、分析和几何工具来研究向量值和矩阵值随机场，这将导致对由SPDEs和随机矩阵系统产生的随机场有更深的理解。这些工具将具有足够的新颖性，以开辟新的研究领域，解决SPDEs理论、随机矩阵和相关随机域中的一些公开问题。此外，拟议的活动还将有助于培养研究生并发展他们在数学和统计科学方面的职业生涯。刻画向量值和矩阵值随机场的精细解析和几何结构具有重要意义和挑战性。在过去的研究中，PI已经建立了一系列关于高斯和更一般的无限可分随机场的结果，以及SPDEs的解。PI与他的合作者一起发展了关于高斯随机场、加性Lévy过程的分形几何和位势理论，以及SPDEs的解，并用它们解决了非马尔科夫高斯和稳定随机场、Lévy过程和SPDEs理论中的几个未决问题。PI计划继续研究向量值和矩阵值随机场、SPDEs、位势理论和随机分形几何之间的精确定量联系。这项拟议的研究最终将对理解向量值和矩阵值随机场、SPDEs和随机矩阵产生新的见解。预期的结果不仅将有助于随机场、SPDEs和随机矩阵的理论，还将促进它们在数学、数学物理和其他科学领域的适用性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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 an analytic and geometric theory of random fields that arise from stochastic partial differential equations (SPDEs) and random matrices. Special emphasis is placed on vector-valued and matrix-valued random fields that play a central role in various areas of pure and applied mathematics, mathematical physics, astronomy, bio-imaging, mathematical oceanography, and statistics. The Principal Inestigator (PI) will develop probabilistic, analytic, and geometric tools for studying vector-valued and matrix-valued random fields that will lead to a deeper understanding of random fields that arise from systems of SPDEs and random matrices. These tools will have sufficient novelty to open new research areas, solve a number of open problems in the theory of SPDEs, random matrices, and related random fields. Moreover, the proposed activities will also help to train graduate students and to develop their careers in the mathematical and statistical sciences. It is significant and challenging to characterize the fine analytic and geometric structures of vector-valued and matrix-valued random fields. In the past investigations, the PI has established a series of results on Gaussian and, more generally, infinitely divisible random fields, and the solutions of SPDEs. Together with his collaborators, the PI has developed fractal geometry and potential theory for Gaussian random fields, additive Lévy processes, solutions to SPDEs, and used them to resolve several outstanding open problems in non-Markovian Gaussian and stable random fields, Lévy processes, and the theory of SPDEs. The PI plans to continue his investigation of precise quantitative connections between vector-valued and matrix-valued random fields, SPDEs, potential theory, and the geometry of random fractals. The proposed research will ultimately yield novel insights into the understanding of vector-valued and matrix-valued random fields, SPDEs, and random matrices. The expected results will not only contribute to the theories of random fields, SPDEs, and random matrices but also promote their applicability in mathematics, mathematical physics, and in other scientific areas.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.