Evolution of time-fractional stochastic hyperbolic diffusion equations on the unit sphere

Abstract

This work examines the temporal evolution of a two-stage stochastic model for spherical random fields. The model uses a time-fractional stochastic hyperbolic diffusion equation, which describes the evolution of spherical random fields on the unit sphere in time. The diffusion operator incorporates a time-fractional derivative in the Caputo sense. In the first stage of the model, a homogeneous problem is considered, with an isotropic Gaussian random field on the unit sphere serving as the initial condition. In the second stage, the model transitions to an inhomogeneous problem driven by a time-delayed Brownian motion on the unit sphere. We used the Laplace transformation method to derive the solution of the model. We prove that the derived solution satisfies the time-fractional equation by establishing a novel result that provides a rigorous foundation for taking fractional integrals of stochastic integrals. The solution to the model is expressed through a real-valued expansion of spherical harmonics. To obtain an approximation, the expansion of the solution is truncated at a certain degree $L\geq1$. The analysis of truncation errors reveals their convergence behavior, showing that convergence rates are affected by the decay of the angular power spectra of the driving noise and the initial condition. In addition, we investigate the sample properties of the stochastic solution, demonstrating that, under some conditions, there exists a local H\"{o}lder continuous modification of the solution.

To illustrate the theoretical findings, numerical examples and simulations inspired by the cosmic microwave background (CMB) are presented.