Strong Short Time Asymptotics and Convolution Approximation of the Heat Kernel

TL;DR

This paper presents a simplified proof of the short-time asymptotic expansion of heat kernels on compact Riemannian manifolds, demonstrating the effectiveness of convolution approximations and their applications to the cut locus.

Contribution

It introduces a concise proof of heat kernel asymptotics and shows how convolution of approximate kernels can effectively approximate the true heat kernel.

Findings

Exponential decay of the difference between approximate and true heat kernels.

Repeated convolution of approximate kernels converges to the true heat kernel.

Application of the method to derive asymptotics at the cut locus.

Abstract

We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this to show that repeated convolution of the approximate heat kernels can be used to approximate the heat kernel on all of $M$, which is related to expressing the heat kernel as a path integral. This scheme is then applied to obtain a short-time asymptotic expansion of the heat kernel at the cut locus.