Boundary Value Problems for p-Adic Elliptic Parisi-Zúñiga Diffusion
Patrick Erik Bradley
TL;DR
The work extends elliptic PDE-like machinery to compact $d$-dimensional $p$-adic domains by constructing coordinate-based Vladimirov-type Laplacians, forming Sobolev spaces, and defining a second-order elliptic divergence operator $P(\mathcal{L})$. It establishes a full spectral theory for $P(\mathcal{L})$, including an explicit $L^2$-basis and a degree-2 polynomial in the sub-Laplacians, and proves solvability of Poisson problems under boundary conditions. The paper further develops the associated Markov semigroup, proves the existence and convergence of the heat kernel, and derives a Green function, thereby providing diffusion-type dynamics on $p$-adic domains. These results yield a rigorous framework for boundary value problems, spectral analysis, and diffusion on $p$-adic spaces with potential applications to $p$-adic geometric objects and stochastic processes on non-Archimedean spaces.
Abstract
Elliptic integral-differential operators resembling the classical elliptic partial differential equations are defined over a compact d-dimensional p-adic domain together with associated Sobolev spaces relying on coordinate Vladimirov-type Laplacians dating back to an idea of Wilson Zúñiga-Galindo in his previous work. The associated Poisson equations under boundary conditions are solved and their $L_2$-spectra are determined. Under certain finiteness conditions, a Markov semigroup acting on the Sobolev spaces which are also Hilbert spaces can be associated with such an operator and the boundary condition. It is shown that this also has an explicitly given heat kernel as an $L_2$-function, which allows a Green function to be derived from it.