Analysis and Geometry of Certain Markov Chains and Processes

9802855 Saloff-Coste这项研究的重点是在几何环境中演化的某些马尔可夫过程。研究了过程的长时间行为与几何形状的关系。在有限马氏链的背景下，发展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游动，研究了游动行为与群结构之间的关系。研究了无限Cayley图上随机游动的性质及其与等周不等式的关系。通过考虑花环乘积和研究访问点数的拉普拉斯变换和其他相关泛函，给出了显示奇异行为的例子。在黎曼流形上的布朗运动的背景下，研究者将研究具有末端的流形上的双边整体热核界。这需要紧致区域的命中时间估计和无界区域的Dirichlet热核的界。文中还讨论了局部紧群上的一些关于Gauss测度的问题。本研究主要研究某些随机过程的基本性质。一个熟悉而又典型的例子是，一副牌必须洗多少次才能很好地混合。在计算机程序中使用类似于洗牌的随机过程来执行没有已知的有效确定性算法的任务。了解这些随机算法的工作情况很重要，需要进行精确的数学研究。洗牌的不同概括导致了对不同类型数学结构上的随机过程的研究。研究人员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.