Asymptotics and ergodicity of hypoelliptic random processes

随机性已被用于模拟物理、生物、金融等领域的许多现象。从布朗运动的经典例子开始，布朗运动用于描述受热波动影响的粒子的运动，后来用于模拟股票价格随时间的价值，随机技术已经找到了许多应用。例如，随机性是用于分析大数据的算法的关键因素。这种分析中的一个主要问题是理解一个随机系统是否以及如何收敛到平衡状态。本奖项的研究将根据所使用的模型研究这种收敛性。该项目包括培训研究生，向本科生介绍研究，而研究结果将通过出版物和在会议上的发言来传播。该项目涉及概率、分析和几何相结合的问题。研究次椭圆扩散和随机游走的小偏差、迭代对数和大偏差等极限规律是研究的方向之一。这些问题与次椭圆条件下的Cameron-Martin-Girsanov型拟不变性、泛函不等式的应用以及次椭圆和奇异条件下概率律的光滑性密切相关。这些方法包括各种概率技术，如耦合和狄利克雷形式。特别是研究大偏差和小偏差，Onsager-Machlup泛函（可以看作是动力系统的拉格朗日的类比），以及具有奇异势的大粒子系统的收敛平衡。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。

Randomness has been used to model numerous phenomena in physics, biology, finance etc. Starting with the classical example of Brownian motion used to describe the motion of particles subject to thermal fluctuations and later to model the value of stock prices over time, stochastic techniques have found many applications. For example, randomness is a key ingredient in algorithms used to analyze large data. One of the major questions in such an analysis is understanding if and how a random system converges to an equilibrium. The research in this award will study such convergence depending on the models used. The project includes training graduate students, introducing undergraduate students to research, while the results will be disseminated through publications and presentations at conferences. The project concerns problems combining probability, analysis, geometry. One of the directions of research is to study limits laws such as small deviations, laws of iterated logarithm and large deviations for hypoelliptic diffusions and random walks. These questions are closely related to the Cameron-Martin-Girsanov type quasi-invariance in hypoelliptic settings, applications to functional inequalities, and smoothness of probability laws in hypoelliptic and singular settings. The methods include diverse probabilistic techniques such as coupling and Dirichlet forms. In particular, research concerns large and small 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 the 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.