Heat kernel estimates and applications

许多基本的马尔可夫过程——包括群上的随机游走和黎曼流形或李群上的扩散——可以被看作是由底层几何结构定义的。该项目研究了过程属性和底层几何属性之间的复杂关系。从技术上讲，这是通过获得过程的过渡核（热核）的精确估计来完成的。目的是更好地理解基于底层几何的过程的大规模性质，但在某些情况下，也可以在相关马尔可夫过程的帮助下探索几何。马尔可夫过程在现代科学活动中发挥着重要作用，从物理学到生物学再到金融学，它们都是复杂现象的模型。它们在计算机模拟中也起着基本的作用。本文通过将马尔可夫过程的性质与其所处空间的几何形状联系起来，研究了马尔可夫过程的基本性质。热是如何在一大块合金中扩散的？这又如何取决于工件的形状和合金的不同性质呢？通过观察温度可以发现合金的什么性质？在精神上，这些是我们考虑的一些问题。

Many fundamental Markov processes --- including random walk on group and diffusion on Riemannian manifold or Lie group --- can be viewed as defined by an underlying geometric structure. This project studies the intricate relationships between the properties of the process and the properties of the underlying geometry. Technically, this is done by obtaining precise estimates on the transition kernel of the process (the heat kernel). The aim is to gain a better understanding of the large scale properties of the process, based on the underlying geometry but also, in some cases, to explore the geometry with the help of the associated Markov process.Markov processes play a fundamental role in modern scientific activities, from physics to biology to finance, where they model complex phenomena. They also play a basic role in computer simulation.This proposal studies basic properties of Markov processes by relating the properties of the process to the geometry of the space in which it evolves. How does heat diffuses in a large piece of alloy? and how does this depends on the shape of the piece and the perhaps varying nature of the alloy? What can one discover about the nature of the alloy by observing temperatures? These are, in spirit, some of the questions that are considered.