Heat kernel estimates and applications

Cornell University, Ithaca NY

Investigators

Abstract

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.