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

This paper develops a two-stage stochastic model to investigate the evolution of random fields on the unit sphere S2 in R3. The model is defined by a time-fractional stochastic diffusion equation on S2 governed by a diffusion operator with a 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 S2 as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on S2. 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≥1. 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 Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.