Markov Processes in Geometric Environments

几何环境中的马尔可夫过程

• 批准号: 0603886
• 负责人: Laurent Saloff-Coste
• 金额: $ 26.1万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-01 至 2011-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603886&HistoricalAwards=false
• 关键词: Markov; Processes; Geometric; Environments

Many basic Markov processes evolve on a state space carryinga related geometric structure. Brownian motion on a Riemannianmanifold, random walks on Cayley graphs of finitely generatedgroups and finite Markov chains on complex combinatorial structures such as trees or matchingsare all primary examples.This proposal focuses on the relationships between the behavior of such processes and the properties ofthe underlying geometric structure. It involves problems at the interface betweenanalysis, geometry and probability with a major role played bygroups and their actions. Potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation,is at the center of many of these considerations.Random processes play an important role in many aspects of science andhuman activity. The study of card shuffling procedures is an entertaining yet complex and mathematically interesting example.Various random processes are used to model complex phenomena,from polymer molecules, to DNA analysis, to image restoration, to financial markets. They are also used as crucial tools for efficient computations. In such cases, there are strong structural constraints underlying the behaviorof these stochastic processes. These constraints are expressed in terms of the environment of the process which often has a complex combinatorial or geometric nature.This proposal focuses on the study of thefundamental properties of such stochastic processesand on how they relate to the global structure of the environment.

许多基本马尔可夫过程在带有相关几何结构的状态空间上演化。黎曼流形上的布朗运动、有限生成群的凯莱图上的随机游动以及复杂组合结构（例如树或匹配）上的有限马尔可夫链都是主要示例。该提案重点关注此类过程的行为与基础几何结构的属性之间的关系。它涉及分析、几何和概率之间的接口问题，其中群体及其行为发挥着主要作用。势论，即调和函数的研究，更一般地说，热方程解的研究，是许多这些考虑的核心。随机过程在科学和人类活动的许多方面发挥着重要作用。洗牌过程的研究是一个有趣但复杂且在数学上有趣的例子。各种随机过程被用来模拟复杂的现象，从聚合物分子到 DNA 分析，到图像恢复，到金融市场。它们也被用作高效计算的重要工具。在这种情况下，这些随机过程的行为背后存在强大的结构约束。这些约束以过程的环境来表达，该过程的环境通常具有复杂的组合或几何性质。本提案重点研究此类随机过程的基本属性以及它们与环境的全局结构的关系。