Schottky-Invariant $p$-Adic Diffusion Operators

Abstract

A parametrised diffusion operator on the regular domain $\hat{I}\text{c◯}$ of a $p$-adic Schottky group is constructed. It is defined as an integral operator on the complex-valued functions on $\hat{I}\text{c◯}$ which are invariant under the Schottky group $\hat{I}“$, where integration is against the measure defined by an invariant regular differential 1-form $\ddot{I}‰$. It is proven that the space of Schottky invariant $L^2$-functions on $\hat{I}\text{c◯}$ outside the zeros of $\ddot{I}‰$ has an orthonormal basis consiting of $\hat{I}“$-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 $\ddot{I}‰$, and gives rise to a strong Markov process on the corresponding orbit space for the Schottky group whose paths are cà dlà g.

25 pages, introduction contains clarifications, typos corrected, statements modified and proofs corrected. Eigenvalue calculation now fixed. This is the accepted version in JFAA