  We study regular coverings of graphs and manifolds with a focus on properties
of the heat equation. In particular, we look at stochastic incompleteness, the
Feller property and uniform transience; and investigate the connection between
the validity of these properties on the base space and its covering. For both
graphs and manifolds, we prove the equivalence of stochastic incompleteness of
the base and that of its cover. Along the way we also give some new conditions
for the Feller property to hold on graphs.
