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

这个项目关注的是从随机偏微分方程（SPDE）和随机矩阵产生的随机场的分析和几何理论的发展。特别强调的是向量值和矩阵值的随机场，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发挥着核心作用。主要调查员（PI）将开发用于研究向量值和矩阵值随机场的概率，分析和几何工具，这将导致对SPDE和随机矩阵系统产生的随机场有更深入的了解。这些工具将有足够的新奇，以开辟新的研究领域，解决一些开放的问题，在理论的随机微分方程，随机矩阵，以及相关的随机领域。此外，拟议的活动还将有助于培训研究生，并发展他们在数学和统计科学方面的职业。刻画向量值和矩阵值随机场的精细解析结构和几何结构是一个重要而又具有挑战性的问题。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SPDE的解决方案。PI与他的合作者一起开发了高斯随机场的分形几何和势理论，添加剂Lévy过程，SPDE的解决方案，并使用它们来解决非马尔可夫高斯和稳定随机场，Lévy过程和SPDE理论中的几个悬而未决的问题。PI计划继续研究向量值和矩阵值随机场，SPDE，势理论和随机分形几何之间的精确定量联系。拟议的研究将最终产生新的见解向量值和矩阵值的随机场，SPDE和随机矩阵的理解。预期的结果不仅有助于随机场、随机微分方程和随机矩阵的理论，而且还促进了它们在数学、数学物理和其他科学领域的应用。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 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.