Schottky-Invariant p-Adic Diffusion Operators

A parametrised diffusion operator on the regular domain $\mathrm{\Omega }$ of a p-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions
on $\mathrm{\Omega }$ which are invariant under the Schottky group $\mathrm{\Gamma }$, where integration is against the measure defined by an invariant regular differential 1-form ω. It is proven that the space of Schottky invariant L${}^2$ -functions on $\mathrm{\Omega }$ outside the zeros of ω has an orthonormal basis consisting of $\mathrm{\Gamma }$-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 supported 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 càdlàg.