Random walks, diffusions, semigroups, and associated geometries

随机过程在科学和其他人类活动的许多方面发挥着重要作用。洗牌过程的研究提供了一个有趣而复杂的数学有趣的例子。它也是许多重要混合现象的模型。我们使用随机性来理解复杂的现象，从聚合物分子的行为和DNA分析，到图像恢复和识别，通信和社交网络，以及金融市场的行为。随机过程也被用作有效计算的重要工具。在所有这些情况下，相关随机过程的行为背后都有很强的结构约束。这些约束是根据过程的环境来表达的，过程的环境通常具有复杂的组合或几何结构。这个建议的重点是研究这些随机过程的基本性质，以及它们如何与环境的全局结构相联系。许多基本的马尔可夫过程在一个具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，Cayley图上的随机游动和复杂组合结构上的有限马尔可夫链，如树或匹配，都是重要的例子。这个建议的重点是这样的过程的行为和基本的几何结构的属性之间的关系。它涉及的问题之间的接口分析，几何和概率的主要作用发挥了群体和他们的行动。潜在理论，即，调和函数的研究，以及更一般地，热方程的解的研究，也是许多这些考虑的中心。

Random processes play an important role in many aspects of science and other human activities. The study of card shuffling procedures provides an entertaining yet complex and mathematically interesting example. It also serves as a model for the many important mixing phenomena. We use randomness to understand complex phenomena, from the behavior of polymer molecules and DNA analysis, to image restoration and recognition, communication and social networks, and the behavior of financial markets. Random processes are also used as important tools for efficient computations. In all these cases, there are strong structural constraints underlying the behavior of the relevant random process. These constraints are expressed in terms of the environment of the process which often has a complex combinatorial or geometric structure. This proposal focuses on the study of the fundamental properties of such stochastic processes and on how they relate to the global structure of their environment.Many basic Markov processes evolve on a state space carrying a related geometric structure. Brownian motion on a Riemannian manifold, random walks on Cayley graphs of finitely generated groups and finite Markov chains on complex combinatorial structures such as trees or matchings are significant examples. This proposal focuses on the relationships between the behavior of such processes and the properties of the underlying geometric structure. It involves problems at the interface between analysis, geometry and probability with a major role played by groups and their actions. Potential theory, i.e., the study of harmonic functions and, more generally, of solutions of the heat equation, is also at the center of many of these considerations.