Continuous random field solutions to parabolic SPDEs on p.c.f. fractals

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.