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

TL;DR

This paper constructs and analyzes Schottky invariant diffusion operators on the p-adic upper half plane, explicitly calculating their spectra and solving associated heat equations, revealing properties of related Markov processes.

Contribution

It introduces a new class of self-adjoint diffusion operators on the p-adic upper half plane and explicitly determines their spectral and heat kernel properties.

Findings

Spectra of diffusion operators are explicitly calculated.

Heat equations are uniquely solvable with explicit distribution functions.

Markov processes with cadlag paths are constructed from the heat kernels.

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.