# Locality of the heat kernel on metric measure spaces

## TL;DR

This paper investigates the locality properties of heat kernels on metric measure spaces, establishing conditions under which the heat kernel and Wiener measure are local, and applies these results to compute heat kernel asymptotics on complex spaces.

## Contribution

It introduces the concept of manifold-like spaces and proves heat kernel locality results, generalizing known facts from manifolds and metric graphs to broader classes of spaces.

## Key findings

- Wiener measure associated to Brownian motion is local
- Locality of Wiener measure plus decay bounds implies heat kernel locality
- Heat kernel asymptotics computed for particles on metric graphs

## Abstract

We will discuss what it means for a general heat kernel on a metric measure space to be local. We show that the Wiener measure associated to Brownian motion is local. Next we show that locality of the Wiener measure plus a suitable decay bound of the heat kernel implies locality of the heat kernel. We define a class of metric spaces we call manifold-like that satisfy the prerequisites for these theorems. This class includes Riemannian manifolds, metric graphs, products and some quotients of these as well as a number of more singular spaces. There exists a natural Dirichlet form based on the Laplacian on manifold-like spaces and we show that the associated Wiener measure and heat kernel are both local. These results unify and generalise facts known for manifolds and metric graphs. They provide a useful tool for computing heat kernel asymptotics for a large class of metric spaces. As an application we compute the heat kernel asymptotics for two identical particles living on a metric graph.