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.

翻译：本文发展了一个两阶段随机模型，用于研究三维欧氏空间$\R^3$中单位球面$\bS^2$上随机场的演化。该模型由$\bS^2$上的时间分数阶随机扩散方程定义，扩散算子受控于黎曼-刘维尔意义下的时间分数阶导数。在第一阶段，模型以各向同性的高斯随机场作为初始条件的齐次问题为特征。在第二阶段，模型变为由$\bS^2$上时间延迟的布朗运动驱动的非齐次问题。模型的解以复球谐函数展开的形式给出。通过在度$L\geq1$处截断解展开得到解的近似。推导了截断误差随$L$变化的收敛速率以及均方误差随时间变化的规律。研究表明，收敛速率不仅取决于驱动噪声和初始条件的角功率谱衰减特性，还依赖于分数阶导数的阶数。我们研究了随机解的样本性质，并证明该解是具有各向同性赫尔德连续性的随机场。最后，受宇宙微波背景辐射（CMB）启发的数值算例和模拟验证了理论结果。