Applications of non-Archimedean analysis (a p-adic analog of the symmetric stable process and the corresponding heat-like equation) to dynamics of complex systems like proteins or spin glasses. See http://www.worldscinet.com/brl/03/0303/S1793048008000836.html
Applications of fractional calculus (a branch of classical analysis dealing with fractional derivatives and integrals) to diffusion phenomena in disordered systems. See a survey paper with an interesting title "The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics":
Mathematically, this is (in the simplest case) a kind of a theory of parabolic equations with a fractional time derivative.