Heat kernel estimates and applications

热核估计和应用

• 批准号: 1004771
• 负责人: Laurent Saloff-Coste
• 金额: $ 32.95万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1004771&HistoricalAwards=false
• 关键词: Heat; kernel; estimates; applications

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.

许多基本的马尔可夫过程-包括群上的随机游动和黎曼流形或李群上的扩散-可以被看作是由一个潜在的几何结构定义的。这个项目研究过程的属性和底层几何属性之间的复杂关系。从技术上讲，这是通过获得对过程的过渡核（热核）的精确估计来完成的。其目的是获得更好的理解的大规模属性的过程，基于基础的几何，但在某些情况下，探索的几何与相关的马尔可夫过程的帮助下。马尔可夫过程在现代科学活动中发挥着基础作用，从物理学到生物学到金融，在那里他们模拟复杂的现象。它们也在计算机模拟中起着基本的作用。这个建议通过将过程的性质与它所演化的空间的几何关系来研究马尔可夫过程的基本性质。热如何在一大块合金中扩散？这又是如何取决于工件的形状和合金的可能变化的性质的呢？通过观察温度可以发现合金的什么性质？从精神上讲，这些是所考虑的一些问题。