In this survey we discuss heat kernel estimates of self-similar type
on metric spaces with doubling measures. We characterize the tail functions
from heat kernel estimates in both non-local and local cases. In the local
case we also specify the domain of the energy form as a certain Besov space,
and identify the walk dimension in terms of the critical Besov exponent. The
techniques used include self-improvement of heat kernel upper bound and the
maximum principle for weak solutions. All proofs are completely analytic