Random Walks and Diffusions and Their Geometries

随机游走和扩散及其几何

基本信息

批准号：
1707589
负责人：
Laurent Saloff-Coste
金额：
$ 30万
依托单位：
Cornell University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1707589&HistoricalAwards=false
关键词：
Random Walks Diffusions Their Geometries

项目摘要

Randomness plays a role in many aspects of science and human activities. A familiar yet complex and mathematically interesting example is card shuffling, which serves as a model for many important phenomena involving the idea of mixing. More generally, randomness is used in modeling a wide range of systems, from polymers and DNA to image restoration and recognition, communication and social networks, and the behavior of financial markets. Random processes are also used as important tools for efficient computations and simulation. In all these applications, strong structural constraints associated with the complex combinatorial or geometric structure underlying the problem determine the behavior of the process. This research project is concerned with the fundamental properties of such stochastic processes and their relationship to global structures.The project focusses on Markov processes that are defined in terms of a related geometric or algebraic structure. The long-term and global properties of these processes are determined by and often reflect the global structure of the underlying space. The project involves questions at the interface between analysis, geometry, and probability with a major role played by groups and their actions. Partial differential equations and potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation, are also at the center of many of these considerations. Brownian motion on a Riemannian manifold and random walks on Cayley graphs of finitely generated groups provide key examples.

随机性在科学和人类活动的许多方面发挥着作用。一个熟悉但复杂且数学上有趣的例子是洗牌，它是涉及混合概念的许多重要现象的模型。更一般地说，随机性用于对各种系统进行建模，从聚合物和 DNA 到图像恢复和识别、通信和社交网络以及金融市场的行为。随机过程也被用作高效计算和模拟的重要工具。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过程的行为。该研究项目关注此类随机过程的基本属性及其与全局结构的关系。该项目重点关注根据相关几何或代数结构定义的马尔可夫过程。这些过程的长期和全局属性由底层空间的全局结构决定，并且通常反映了底层空间的全局结构。该项目涉及分析、几何和概率之间的接口问题，其中团体及其行动发挥着主要作用。偏微分方程和势论，即调和函数的研究，更一般地说，热方程解的研究，也是许多这些考虑的核心。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。