Analysis and Geometry of Certain Markov Chains and Processes

小行星9802855 这项研究的重点是某些马尔可夫过程，在一个几何演变 环境分析了过程的长时间行为与过程的动态特性之间的关系， 几何学研究。在有限马尔可夫链的背景下，分析和几何工具 的发展，以获得有关遍历性的定量结果。Sobolev的角色 型不等式。对于有限群上的随机游动， 研究了群体的行走行为和结构。随机的行为 无限Cayley图上的行走及其与等周不等式的关系 研究了通过考虑花环，给出了一些奇异行为的例子 产品和研究的拉普拉斯变换的访问点的数量和其他 相关功能在黎曼流形上的布朗运动的背景下， 研究者将研究具有端点流形上的双边整体热核界。这 需要紧区域的命中时间估计和Dirichlet热核的边界 无界区域的概念。局部紧空间上高斯测度的若干问题 团体也将接受调查。 本研究关注某些随机过程的基本性质。一 一个熟悉而又典型的例子是一副牌必须被 搅拌均匀。类似于洗牌的随机过程被用于 计算机程序来执行没有有效的确定性算法的任务， 是已知的。了解这些随机算法的工作情况非常重要， 精确的数学研究。洗牌的不同概括导致了 研究不同类型数学结构上的随机过程。研究者 将研究这些过程的长期行为。

9802855 Saloff-Coste This research focuses on certain Markov processes that evolve in a geometric environment. The relations between the long time behavior of the process and the geometry are studied. In the context of finite Markov chains, analytic and geometric tools are developed to obtain quantitative results concerning ergodicity. The role of Sobolev type inequalities is investigated. For random walk on finite groups, the relation between the behavior of the walk and the structure of the group is studied. The behavior of random walk on infinite Cayley graphs and its relation with isoperimetric inequalities are investigated. Examples showing exotic behaviors are produced by considering wreath products and studying the Laplace transform of the number of visited points and other related functionals. In the context of Brownian motion on Riemannian manifolds, the investigator will study two-sided global heat kernel bounds on manifolds with ends. This requires hitting time estimates for compact regions and bounds on the Dirichlet heat kernel of unbounded regions. Some problems concerning Gaussian measures on locally compact groups will also be investigated. This research is concerned with basic properties of certain random processes. A familiar yet typical example is the question of how many times a deck of cards must be shuffled to be mixed well. Random processes similar to card shuffling are used in computer programs to perform tasks for which no efficient deterministic algorithm is known. Understanding how well these stochastic algorithms work is important and calls for precise mathematical studies. Different generalizations of card shuffling lead to the study of random processes on different types of mathematical structures. The investigator will study the long time behavior of these processes.