# Random spherical hyperbolic diffusion

## TL;DR

This paper develops a series solution for hyperbolic diffusion on the sphere, analyzes its properties, and applies it to cosmic microwave background data, providing insights into the stochastic modeling of such phenomena.

## Contribution

It derives an exact series solution for hyperbolic diffusion on the sphere with random initial conditions, including convergence analysis and applications to cosmic microwave background data.

## Key findings

- Series expansion converges with quantifiable error bounds
- Sample H"older continuity relates to angular power spectrum decay
- Numerical results support theoretical convergence and applicability

## Abstract

The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample H\"older continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to cosmic microwave background data are presented to illustrate the theoretical results.