Probabilistic Methods in Analysis, Geometry, and Beyond

该项目开发了几个研究方向，结合概率，几何和分析，许多问题的物理动机。研究的主要对象是具有一定程度退化的随机系统，类似于受约束的运动。这种在高维或无限维中的简并扩散在不同的领域中有许多应用，包括量子场论（QFT），湍流，化学动力学和大数据环境。虽然它们在模拟许多这样的现象时很有用，但环境的退化和高维性质带来了数学挑战。一些项目是为研究生和可能的本科生。研究方向之一是研究亚椭圆情形下的Cameron-Martin-Girsanov型拟不变性及其在函数不等式、亚椭圆情形下的概率律光滑性和奇异情形下的应用。这是密切相关的渐近行为，这样的扩散，即，大偏差，Onsager-Machlup功能，可以被看作是一个模拟的拉格朗日动力系统，收敛到平衡的大粒子系统与奇异的潜力。退化（缺乏椭圆性）和高维都必须处理来自不同领域的新技术，如概率，遍历理论和次黎曼几何。虽然许多这些设置在应用中自然出现，但它们的数学分析并不容易。这类问题的答案除了理论意义外，有些还具有实际用途。例如，收敛到平衡态的速度，它对粒子数和其他参数的依赖性，或速度函数的显式形式都有许多应用。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。

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.