  It is well-known that stochastic processes on fractal spaces or in certain
random media exhibit anomalous heat kernel behaviour. One manifestation of such
irregular behaviour is the presence of fluctuations in the short- or long-time
asymptotics of the on-diagonal heat kernel. In this note we review some
examples for which such fluctuations are known to occur, including Brownian
motion on certain deterministic or random fractals, and simple random walks on
various examples of random graph trees, such as the incipient infinite cluster
of critical percolation on a regular tree and low-dimensional uniform spanning
trees. We also announce some new results that add the one-dimensional Bouchaud
trap model to this class of examples.

Probability;Analysis of PDEs