Markov Processes in Geometric Environments

许多基本的马尔可夫过程在一个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Cayley图上的随机游走，复杂组合结构（如树或匹配）上的有限马尔可夫链都是主要的例子。这一建议侧重于这些过程的行为和底层几何结构的性质之间的关系。它涉及分析、几何和概率之间的接口问题，主要由群体及其行动发挥作用。势理论，即对调和函数的研究，更一般地说，对热方程解的研究，是许多这些考虑的中心。随机过程在科学和人类活动的许多方面起着重要作用。洗牌程序的研究是一个有趣的，但复杂的和数学上有趣的例子。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，再到金融市场。它们也被用作高效计算的关键工具。在这种情况下，这些随机过程的行为有很强的结构约束。这些约束是根据过程的环境来表示的，通常具有复杂的组合或几何性质。这一建议的重点是研究这些随机过程的基本性质，以及它们与环境的整体结构的关系。

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.