  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.
