On approximation for time-fractional stochastic diffusion equations on the unit sphere

TL;DR

This paper develops a two-stage stochastic model for time-fractional diffusion equations on the sphere, providing approximation methods, convergence rates, and sample properties, with applications to cosmic microwave background simulations.

Contribution

It introduces a novel two-stage stochastic framework for time-fractional diffusion on the sphere, including approximation and convergence analysis, and explores sample properties of the solution.

Findings

Convergence rates depend on angular power spectrum decay and fractional order.

Solution is an isotropic H"older continuous random field.

Numerical simulations illustrate theoretical results.

Abstract

This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere $\bS^2$ in $\R^3$. The model is defined by a time-fractional stochastic diffusion equation on $\bS^2$ governed by a diffusion operator with the time-fractional derivative defined in the Riemann-Liouville sense. In the first stage, the model is characterized by a homogeneous problem with an isotropic Gaussian random field on $\bS^2$ as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is given in the form of an expansion in terms of complex spherical harmonics. An approximation to the solution is given by truncating the expansion of the solution at degree $L\geq1$. The rate of convergence of the truncation errors as a function of $L$ and the mean square errors as a function of time are also derived. It is shown that the convergence rates depend not only on the decay of the angular power spectrum of the driving noise and the initial condition, but also on the order of the fractional derivative. We study sample properties of the stochastic solution and show that the solution is an isotropic H\"{o}lder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.