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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都发挥着作用。一个熟悉但复杂且数学上有趣的例子是洗牌，它是许多涉及混合思想的重要现象的模型。更一般地说，随机性被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。随机过程也被用作有效计算和模拟的重要工具。在所有这些应用中，与问题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。本研究计画主要探讨随机过程的基本性质及其与整体结构的关系，并以相关几何或代数结构定义的马尔可夫过程为研究对象。这些过程的长期和全局性质由底层空间的全局结构决定，并往往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，其中小组及其行动发挥了重要作用。偏微分方程和势能理论，即，调和函数的研究，以及更一般地，热方程的解的研究，也是许多这些考虑的中心。黎曼流形上的布朗运动和Cayley图上的随机游动都提供了关键的例子。