Heat kernels on Euclidean complexes

TL;DR

This thesis develops heat kernel analysis on Euclidean polyhedral complexes, establishing bounds and asymptotic behaviors, especially when these complexes have an underlying group structure.

Contribution

It introduces a Dirichlet form and heat kernel on Euclidean complexes, providing uniform bounds and asymptotic equivalence with group heat kernels.

Findings

Established uniform small time heat kernel bounds.

Proved large time asymptotic equivalence with group heat kernels.

Derived a Poincare inequality for complexes with bounded geometry.

Abstract

In this thesis we describe a type of metric space called an Euclidean polyhedral complex. We define a Dirichlet form on it; this is used to give a corresponding heat kernel. We provide a uniform small time Poincare inequality for complexes with bounded geometry and use this to determine uniform small time heat kernel bounds via a theorem of Sturm. We then consider such complexes with an underlying finitely generated group structure. We use techniques of Saloff-Coste and Pittet to show a large time asymptotic equivalence for the heat kernel on the complex and the heat kernel on the group.