Probabilistic Methods in Analysis, Geometry, and Beyond

本项目发展了概率论、几何和分析相结合的几个研究方向，其中许多问题是由物理驱动的。研究的主要对象是具有一定程度的类似于约束运动的退化的随机系统。这种高维或无限维的简并扩散在量子场论（QFT）、湍流、化学动力学和大数据环境等不同领域有许多应用。虽然它们对许多这类现象的建模很有用，但环境的退化和高维性质构成了数学上的挑战。一些项目是为研究生和本科生准备的。此外，研究结果将用于当地中学和大学一级的指导和教育活动。本文的研究方向之一是研究拟不变性在次椭圆和奇异情况下的Cameron-Martin-Girsanov型拟不变性及其在泛函不等式、概率律的光滑性等方面的应用。这与此类扩散的渐近行为密切相关，即大偏差，可视为动力系统拉格朗日的类似的Onsager-Machlup泛函，以及具有奇异势的大粒子系统收敛到平衡。简并性（缺乏椭圆性）和高维性都需要用到来自不同领域的新技术，如概率论、遍历理论和亚黎曼几何。虽然这些设置在应用程序中自然出现，但它们的数学分析并不容易。这些问题除了具有理论意义外，有些答案还有实际用途。例如，收敛到平衡的速率，它对粒子数量和其他参数的依赖，或者速率函数的显式形式都有许多应用。该奖项反映了美国国家科学基金会的法定使命，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。

This project develops several research directions combining probability, geometry and analysis, with many problems motivated by physics. The main object of study are random systems with a certain level of degeneracy similar to a constrained movement. Such degenerate diffusions in high or infinite dimensions have many applications in different fields including quantum field theory (QFT), turbulence, chemical dynamics, and large data environments. While they are useful in modelling many of such phenomena, degenerate and high-dimensional nature of the setting pose mathematical challenges. A number of the projects are for graduate and possibly undergraduate students. In addition, the results will be used for mentoring and educational activities at the local middle school and at the undergraduate level.One of the directions of research is to study Cameron-Martin-Girsanov type quasi-invariance in hypoelliptic settings, and its applications to functional inequalities, smoothness of probability laws in subelliptic and singular settings. This is closely related to asymptotic behavior of such diffusions, namely, large deviations, the Onsager–Machlup functional which can be viewed as an analog of the Lagrangian of a dynamical system, and convergence to equilibrium of a large particle system with singular potentials. Both degeneracy (lack of ellipticity) and high dimensions have to be dealt with new techniques coming from different fields such as probability, ergodic theory and sub-Riemannian geometry. While many of these settings arise naturally in applications, their mathematical analysis is not easy. In addition to theoretical significance of such questions, some answers have practical uses. For example, the rate of convergence to the equilibrium, its dependence on the number of particles and other parameters, or an explicit form of the rate function have many applications.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.