Analysis and Geometry of Certain Markov Chains and Processes

9802855 saloff - cost本研究的重点是在几何环境中进化的某些马尔可夫过程。研究了该工艺的长时间性能与几何形状的关系。在有限马尔可夫链的背景下，开发了解析和几何工具来获得关于遍历性的定量结果。研究了Sobolev型不等式的作用。对于有限群上的随机行走，研究了行走行为与群结构的关系。研究了无限Cayley图上的随机游走行为及其与等周不等式的关系。通过考虑环积，研究访问点数和其他相关泛函的拉普拉斯变换，产生了具有奇异行为的实例。在黎曼流形上的布朗运动的背景下，研究者将研究有端流形上的双面全局热核界。这需要对紧区域和无界区域的狄利克雷热核的边界进行时间估计。本文还研究了局部紧群上的高斯测度问题。本研究涉及某些随机过程的基本性质。一个熟悉但典型的例子是，一副牌必须洗多少次才能混合得很好。在计算机程序中使用类似洗牌的随机过程来执行没有有效确定性算法的任务。了解这些随机算法的工作原理是很重要的，这需要精确的数学研究。对洗牌的不同概括导致对不同类型数学结构的随机过程的研究。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.