Applications of Rough Differential Systems: Theoretical Physics, Data Analysis, and Numerics

随机分析是一个术语，通常包括伊藤型随机积分和Malliavin演算技术。后者应被视为一种方法来定义在路径水平上的分析，并导致深入和有用的结果有关随机微分方程。该项目还涉及规则性结构理论，这是最近的突破，允许定义以前无法实现的广泛的奇异随机系统。 正则结构理论也给出了一个几乎确定性的观点随机微积分，而不是传统的方法，这是非常概率的本质。目前的项目应被视为对上述领域的贡献。粗糙路径，规则性结构和Malliavin演算技术的组合将被用来分析一些物理相关的模型，如连续抛物安德森模型在粗糙的环境。得益于这些基本技术，PI还将研究二维签名的代数和几何方面，并考虑到一些数据分析应用。该项目为研究生和本科生提供研究培训机会。该项目涉及理论物理和图像处理中的相关模型。一个显着的部分的建议是专门的本地化性能的抛物线安德森模型，并特别强调的签名方法也放在随机场。一个更具体的列表中的项目利害关系如下：（1）重整化抛物安德森模型的时刻。(2)非高斯过程的水平集。(3)连续聚合物测度的路径限制。(4)2-d签名和故障识别。PI计划开发概率，几何和分析方法和想法，这将导致对上述系统的定性和定量的更深入的理解。主要技术将结合联合收割机粗糙的路径，规律性结构和Malliavin演算技术，以及几何，分析和统计学习的基本工具。拟议的努力将有足够的新奇，以开辟新的研究领域。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Stochastic analysis is a term which usually encompasses both stochastic integration of Ito type and Malliavin calculus techniques. The latter should be seen as a way to define an analysis at the path level, and leads to deep and useful results concerning stochastic differential equations. This project is also concerned with the theory of regularity structures, which is a recent breakthrough allowing to define a wide range of singular stochastic systems previously out of reach. The regularity structures theory also gives an almost deterministic point of view on stochastic calculus, as opposed to the traditional approach which is very probabilistic in essence. The current project should be seen as a contribution in the areas mentioned above. A combination of rough paths, regularity structures and Malliavin calculus techniques will be used to analyze some physically relevant models such as the continuous parabolic Anderson model in rough environments. Thanks to those fundamental techniques, the PI will also study algebraic and geometric aspects of 2-d signatures with some data analysis applications in mind. The project provides research training opportunities for graduate and undergraduate students. This project is concerned with relevant models in theoretical physics and image processing. A significant portion of the proposal is devoted to localization properties of the parabolic Anderson model, and a special emphasis is also put on the signature method for random fields. A more specific list of the projects at stakes is the following: (1) Moments of renormalized parabolic Anderson models. (2) Level sets for non Gaussian processes. (3) Path confinement for the continuous polymer measure. (4) 2-d signatures and failure identification. The PI plans to develop probabilistic, geometric, and analytic methods and ideas that will lead to a deeper understanding of the qualitative and quantitative of the aforementioned systems. The main techniques will combine rough paths, regularity structures and Malliavin calculus techniques, together with fundamental tools in geometry, analysis and statistical learning. The proposed efforts will have sufficient novelty to open new research areas. They will also further promote the applicability of the theoretical techniques alluded to above.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.