Schottky-Invariant $p$-Adic Diffusion Operators

TL;DR

This paper constructs a $p$-adic diffusion operator invariant under a Schottky group, analyzes its spectral properties, and demonstrates its connection to a Markov process on the orbit space, extending $p$-adic harmonic analysis.

Contribution

It introduces a new class of Schottky-invariant $p$-adic diffusion operators and characterizes their eigenfunctions and eigenvalues, linking them to stochastic processes.

Findings

Eigenfunctions form an orthonormal basis of invariant extensions of Kozyrev wavelets.

Eigenvalues are explicitly calculated.

The heat equation solution defines a Markov process with cadlag paths.

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.