Generalised diffusion on moduli spaces of $p$-adic Mumford curves

A construction of a pseudo-differential operator on non-archimedean local fields invariant under a finite group action is given together with the solution of the corresponding Cauchy problem. This construction is applied to parts of the Gerritzen-Herrlich Teichmuller space in order to obtain a self-adjoint operator whose spectrum can decide about certain properties of the reduction graph of the corresponding Mumford curves.