Diffusion operators on p-adic analytic manifolds

2025/10/26 by Bradley, Patrick Erik
#11G07 #35P10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.22563

Abstract

Kernel functions for Laplacian integral operators are constructed on p-adic analytic manifolds using charts and transition maps from an atlas with connected nerve complex. In the compact case, an operator of Vladimirov-Taibleson type parametrised by a real parameter s is defined. Its kernel function uses a geodetic-like distance function on the nerve complex of its atlas. The L²-spectrum of this operator is established, and it is shown that it gives rise to a Feller semigroup. In this way, the Cauchy problem for the corresponding heat equation is solved in the positive by a transition function of a Markov process. The existence of a heat kernel function and a Green function in the case s > 1 is proven. As an application, it is shown how to express the number of points on the reduction curve defined over the residue field of an elliptic curve with good reduction in terms of the eigenvalues of a Vladimirov-Taibleson-like operator. This provides for an alternative way of counting points on elliptic curves defined over finite fields.

Citations

• Vladimirov-Pearson Operators on ζ-regular Ultrametric Cantor Sets
• Boundary Value Problems for p-Adic Elliptic Parisi-Zúñiga Diffusion
• Schottky invariant diffusion on the transcendent p-adic upper half plane
• Time-Varying Energy Landscapes and Temperature paths: Dynamic Transition Rates in locally Ultrametric Complex Systems
• A non-autonomous p-Adic diffusion equation on time changing graphs
• Brownian Motion in the p-Adic Integers is a Limit of Discrete Time Random Walks
• The spectrum of the Vladimirov sub-Laplacian on the compact Engel group
• Schottky-Invariant p-Adic Diffusion Operators
• Brownian Motion in a Vector Space over a Local Field is a Scaling Limit
• Heat Equations and Hearing the Genus on p-adic Mumford Curves via Automorphic Forms
• Components and Exit Times of Brownian Motion in two or more p-Adic Dimensions
• 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

Related

• Boundary Value Problems for p-Adic Elliptic Parisi-Zúñiga Diffusion
• Schottky invariant diffusion on the transcendent p-adic upper half plane
• A non-autonomous p-Adic diffusion equation on time changing graphs
• Schottky-Invariant p-Adic Diffusion Operators
• Taibleson Operators, p-adic Parabolic Equations and Ultrametric Diffusion