  In this paper we study the regularity of non-linear parabolic PDEs and
stochastic PDEs on metric measure spaces admitting heat kernels. In particular
we consider mild function solutions to abstract Cauchy problems and show that
the unique solution is H\"older continuous in time with values in a suitable
fractional Sobolev space. As this analysis is done via a-priori estimates, we
can apply this result to stochastic PDEs on metric measure spaces and solve the
equation in a pathwise sense for almost all paths. The main example of noise
term is of fractional Brownian type and the metric measure spaces can be
classical as well as given by various fractal structures. The whole approach is
low dimensional and works for spectral dimensions less than 4.
