A non-autonomous p-Adic diffusion equation on time changing graphs

TL;DR

This paper studies non-autonomous p-adic diffusion equations on time-varying graphs, analyzing eigenvalues, solving the heat equation, and establishing the Markov process properties in ultrametric spaces.

Contribution

It introduces the first non-autonomous diffusion operators on ultrametric spaces connected by graphs, including spectral analysis and stochastic process properties.

Findings

Eigenvalues of the diffusion operators are characterized.

The heat equation solution is constructed and approximated.

The associated Markov process is proven to have the Feller property.

Abstract

Motivated by the recently proven presence of ultrametricity in physical models (certain spin glasses) and the very recent study of Turing patterns on locally ultrametric state spaces, first non-autonomous diffusion operators on such spaces, where finitely many compact p-adic spaces are joined by a graph structure, are studied, including their Dirichlet and van Neumann eigenvalues. Secondly, the Cauchy problem for the heat equation associated with these operators is solved, its solution approximated by Trotter products, and thirdly, the corresponding Feller property as well as the Markov property (a Hunt process) is established.