Asymptotics and ergodicity of hypoelliptic random processes

亚椭圆随机过程的渐近性和遍历性

• 批准号: 2246549
• 负责人: Maria Gordina
• 金额: $ 33万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-05-15 至 2026-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246549&HistoricalAwards=false
• 关键词: Asymptotics; ergodicity; hypoelliptic; random; processes

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.

随机性已经被用来模拟物理学、生物学、金融学等领域的大量现象。从经典的布朗运动例子开始，布朗运动被用来描述受热波动影响的粒子的运动，后来又被用来模拟股票价格随时间的变化，随机技术得到了许多应用。例如，随机性是用于分析大数据的算法中的一个关键因素。这种分析中的一个主要问题是理解随机系统是否以及如何收敛到均衡。本奖项的研究将根据所使用的模型来研究这种趋同。该项目包括培训研究生，向本科生介绍研究，同时将通过出版物和在会议上的陈述来传播结果。该项目涉及结合概率、分析、几何的问题。其中一个研究方向是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规律。这些问题与次椭圆环境下的Cameron-Martin-Girsanov型拟不变性、泛函不等式的应用以及次椭圆和奇异环境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Dirichlet形式。特别是，研究涉及大偏差和小偏差，可被视为动力系统拉格朗日模拟的Onsager-Machlup泛函，以及具有奇异势的大粒子系统的收敛到平衡。无论是退化(缺乏椭圆性)还是高维，都必须用来自不同领域的新技术来处理，如概率、遍历理论和次黎曼几何。虽然这些设置中的许多都是在应用程序中自然产生的，但它们的数学分析并不容易。除了这些问题的理论意义外，一些答案还具有实际用途。例如，收敛到平衡的速度，它对粒子数量和其他参数的依赖，或速率函数的显式形式有许多应用。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。