Spherically Restricted Random Hyperbolic Diffusion

TL;DR

This paper analyzes solutions to hyperbolic diffusion equations with random initial conditions in three-dimensional space, focusing on their restrictions to the sphere, approximation methods, convergence rates, and dependence properties, supported by numerical validation.

Contribution

It introduces a framework for studying spherical restrictions of hyperbolic diffusion solutions with random initial data, including approximation techniques and convergence analysis based on spectral measures.

Findings

Derived bounds for mean-square convergence rates.

Established conditions linking spectral decay to solution smoothness.

Numerical results confirm theoretical predictions.

Abstract

This paper investigates solutions of hyperbolic diffusion equations in $\mathbb{R}^3$ with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere $S^2$ are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the mean-square convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the H\"{o}lder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.