Diffusions and jump processes on groups and manifolds

群和流形上的扩散和跳跃过程

• 批准号: 2343868
• 负责人: Laurent Saloff-Coste
• 金额: $ 37万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2343868&HistoricalAwards=false
• 关键词: Diffusions; jump; processes; groups; manifolds

Modeling scientific experiments or human activities often involves randomness. Card shuffling procedures provide a familiar, yet complex and mathematically interesting example that serves as a model for many mixing phenomena. Randomness is used to understand image restoration and recognition, communication and social networks, the behavior of financial markets, as well as in the analysis of large data sets in general. It is an important tool in the study of efficient computations and scientific simulations. In all these applications, strong structural constraints associated with the complex combinatorial or geometric structure underlying the problem determine the behavior. This project is concerned with the fundamental properties of basic stochastic processes and how the behavior of these processes relates to the global geometric structure of their different environments. Postdoctoral associates, graduate students, and undergraduate students will be mentored and trained as part of this project. The funded research focusses on random processes that are defined by a related geometric or algebraic structure (e.g., Riemannian manifolds and groups). The global behaviors of these processes are determined by this underlying structure. In some cases, these behaviors can provide information on the underlying space and its structure. These explorations are at the interface between analysis, geometry, and probability, with the notion of group structure playing a key part. Partial differential equations and potential theory, i.e. the study of harmonic functions and solutions of the heat equation, are also central. Brownian motion on Riemannian Manifolds and random walks on Cayley graphs of finitely generated groups provide key examples. The notion of stable-like processes on nilpotent groups is also studied.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.

对科学实验或人类活动进行建模通常涉及随机性。洗牌过程提供了一个熟悉但复杂且数学上有趣的示例，可以作为许多混合现象的模型。随机性用于理解图像恢复和识别、通信和社交网络、金融市场的行为以及一般大型数据集的分析。它是研究高效计算和科学模拟的重要工具。在所有这些应用中，与问题背后的复杂组合或几何结构相关的强结构约束决定了行为。该项目关注基本随机过程的基本属性以及这些过程的行为如何与其不同环境的全局几何结构相关。作为该项目的一部分，博士后、研究生和本科生将接受指导和培训。 资助的研究重点是由相关几何或代数结构（例如黎曼流形和群）定义的随机过程。这些进程的全局行为是由这个底层结构决定的。在某些情况下，这些行为可以提供有关底层空间及其结构的信息。这些探索处于分析、几何和概率之间的界面，其中群结构的概念起着关键作用。偏微分方程和势论，即调和函数和热方程解的研究，也是核心。黎曼流形上的布朗运动和有限生成群的凯莱图上的随机游走提供了关键的例子。还研究了幂零群上的类稳定过程的概念。该奖项反映了 NSF 的法定使命，并且通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。