Markov Processes in Geometric Environments

许多基本的马尔可夫过程都是在具有相关几何结构的状态空间上演化的.黎曼流形上的布朗运动、群生成的Cayley图上的随机游动以及树或匹配等复杂组合结构上的有限Markov链都是主要的例子，本文着重讨论这些过程的行为与其几何结构的性质之间的关系。它涉及的问题之间的接口分析，几何和概率的主要作用所发挥的群体和他们的行动。潜在理论，即，调和函数的研究，更一般地说，热方程的解的研究，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的作用。洗牌过程的研究是一个有趣而复杂的数学有趣的例子。各种随机过程被用来模拟复杂的现象，从聚合物分子，DNA分析，图像恢复，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有很强的结构约束。这些约束是用过程所处的环境来表示的，而过程往往具有复杂的组合或几何性质，本项研究的重点是研究这种随机过程的基本性质，以及它们与环境的全局结构之间的关系.

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.