Schottky invariant diffusion on the transcendent p-adic upper half plane

Abstract

The transcendent part of the Drinfeld p-adic upper half plane is shown to be a Polish space. Using Radon measures associated with regular differential 1-forms invariant under Schottky groups allows to construct self-adjoint diffusion operators as Laplacian integral operators with kernel functions determined by the p-adic absolute value on the complex p-adic numbers. Their spectra are explicitly calculated and the corresponding Cauchy problems for their associated heat equations are found to be uniquely solvable and to determine Markov processes having paths which are cadlag. The heat kernels are shown to have explicitly given distribution functions, as well as boundary value problems associated with the heat equations under Dirchlet and von Neumann conditions are solved.