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——包括群上的随机游走和黎曼流形或李群上的扩散——可以被视为由基础几何结构定义。该项目研究过程属性与基础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转换内核（热内核）的精确估计来完成的。目的是基于基础几何更好地理解该过程的大规模特性，但在某些情况下，还可以借助相关的马尔可夫过程来探索几何。马尔可夫过程在现代中发挥着基础作用从物理学到生物学再到金融的科学活动，他们模拟复杂的现象。它们还在计算机模拟中发挥着基本作用。该提案通过将过程的属性与其演化的空间的几何形状相关联来研究马尔可夫过程的基本属性。热量如何在大块合金中扩散？这如何取决于工件的形状以及合金的不同性质？通过观察温度可以发现合金的哪些性质？从本质上讲，这些是需要考虑的一些问题。