Asymptotics and ergodicity of hypoelliptic random processes

随机性已被用来模拟物理学，生物学，金融等众多现象，从经典的布朗运动的例子开始，用于描述粒子的运动受到热波动，后来模拟股票价格随时间的价值，随机技术已经发现了许多应用。例如，随机性是用于分析大数据的算法中的关键成分。在这样的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一个均衡。该奖项的研究将根据所使用的模型来研究这种收敛。该项目包括培训研究生，介绍本科生进行研究，而研究结果将通过出版物和在会议上的介绍加以传播。 该项目涉及的问题结合概率，分析，几何。研究亚椭圆扩散和随机游动的小偏差、重对数律和大偏差等极限律是研究的方向之一。这些问题密切相关的Cameron-Martin-Girsanov型准不变性在hypoelliptical设置，应用功能的不平等，光滑的概率定律在hypoelliptical和奇异设置。该方法包括不同的概率技术，如耦合和狄利克雷形式。特别是，研究关注大的和小的偏差，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。除了这些问题的理论意义外，一些答案还有实际用途。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。

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.