  We consider a general class of $L^2$-valued stochastic processes that arise
primarily as solutions of parabolic SPDEs on p.c.f. fractals. Using a
Kolmogorov-type continuity theorem, conditions are found under which these
processes admit versions which are function-valued and jointly continuous in
space and time, and the associated H\"older exponents are computed. We apply
this theorem to the solutions of SPDEs in the theories of both da
Prato--Zabczyk and Walsh. We conclude by discussing a version of the parabolic
Anderson model on these fractals and demonstrate a weak form of intermittency.
