2022/12/12 by T. Alodat, Alodat, T., Q. T. Le Gia +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Applications (stat.AP) #Differential Equations and Numerical Methods #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2212.05690
openalex publication_date 2022/12/12 · openalex created_date 2022/12/26 · openalex updated_date 2026/07/28
This paper develops a two-stage stochastic model to investigate evolution of random fields on the unit sphere \bS2 in \R3. The model is defined by a time-fractional stochastic diffusion equation on \bS2 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 \bS2 as an initial condition. In the second stage, the model becomes an inhomogeneous problem driven by a time-delayed Brownian motion on \bS2. 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.