  Motivated by the recently proven presence of ultrametricity in physical
models (certain spin glasses) and the very recent study of Turing patterns on
locally ultrametric state spaces, first non-autonomous diffusion operators on
such spaces, where finitely many compact p-adic spaces are joined by a graph
structure, are studied, including their Dirichlet and van Neumann eigenvalues.
Secondly, the Cauchy problem for the heat equation associated with these
operators is solved, its solution approximated by Trotter products, and
thirdly, the corresponding Feller property as well as the Markov property (a
Hunt process) is established.
