Vladimirov-Pearson Operators on ζ-regular Ultrametric Cantor Sets

2025/04/29 by Patrick Erik Bradley, Bradley, Patrick Erik · 2 citations
Mathematics · #35P10 #47D07 #Algebraic and Geometric Analysis #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2504.20753

openalex publication_date 2025/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

A new operator for certain types of ultrametric Cantor sets is constructed using the measure coming from the spectral triple associated with the Cantor set, as well as its zeta function. Under certain mild conditions on that measure, it is shown that it is an integral operator similar to the Vladimirov-Taibleson operator on the p-adic integers. Its spectral properties are studied, and the Markov property and kernel representation of the heat kernel generated by this so-called Vladimirov-Pearson operator is shown, viewed as acting on a certain Sobolev space. A large class of these operators have a heat kernel and a Green function explicitly given by the ultrametric wavelets on the Cantor set, which are eigenfunctions of the operator.

Citations

• Time-Varying Energy Landscapes and Temperature paths: Dynamic Transition Rates in locally Ultrametric Complex Systems
• Approximating Diffusion on Finite Multi-Topology Systems Using Ultrametrics
• A non-autonomous p-Adic diffusion equation on time changing graphs
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• Schottky-Invariant p-Adic Diffusion Operators
• Brownian Motion in a Vector Space over a Local Field is a Scaling Limit
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• Reaction-diffusion Equations on Complex Networks and Turing Patterns, via p-Adic Analysis
• Linear and Nonlinear Heat Equations on a p-Adic Ball
• Non-Archimedean White Noise, Pseudodifferential Stochastic Equations, and Massive Euclidean Fields
• Noncommutative Riemannian Geometry and Diffusion on Ultrametric Cantor Sets
• Parabolic Equations and Markov Processes Over p-adic Fields

Cited by

• Diffusion operators on p-adic analytic manifolds
• Toward Holography on Biregular Trees

Related

• Dirichlet forms and ultrametric Cantor sets associated to higher-rank graphs
• Wavelets and spectral triples for higher-rank graphs
• The spectrum of the Vladimirov sub-Laplacian on the compact Engel group
• Embedding of self-similar ultrametric Cantor sets
• Scalar spectral measures associated with an Operator-Fractal