Schottky-Invariant $p$-Adic Diffusion Operators

Abstract

A parametrised diffusion operator on the regular domain $\Omega$ of a $p$-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on $\Omega$ which are invariant under the Schottky group $\Gamma$, where integration is against the measure defined by an invariant regular differential 1-form $\omega$. It is proven that the space of Schottky invariant $L^2$-functions on $\Omega$ outside the zeros of $\omega$ has an orthonormal basis consiting of $\Gamma$-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 supportes outside the zero set of $\omega$, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are c\`adl\`ag.