Schottky-Invariant p-Adic Diffusion Operators

A parametrised diffusion operator on the regular domain Ω of a p-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on Ω which are invariant under the Schottky group Γ, where integration is against the measure defined by an invariant regular differential 1-form ω. It is proven that the space of Schottky invariant L²-functions on Ω outside the zeros of ω has an orthonormal basis consiting of Γ-invariant extensions of Kozyrev wavelets which are eigenfunctions of the operator. The eigenvalues are calculated, and it is shown that the heat equation for this operator provides a unique solution for its Cauchy problem with Schottky-invariant continuous initial conditions supportes outside the zero set of ω, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are c\`adl\`ag.