论文标题
在单位球上的时间分数随机扩散方程进行近似
On approximation for time-fractional stochastic diffusion equations on the unit sphere
论文作者
论文摘要
本文开发了一个两阶段的随机模型，以研究单位球体上随机字段的演变$ \ bs^2 $ in $ \ r^3 $。该模型是由$ \ bs^2 $上的时间分数随机扩散方程定义的，由扩散操作员控制的，具有在Riemann-Liouville Sense中定义的时间级衍生物。在第一阶段，该模型的特征是在$ \ bs^2 $上作为初始条件上的各向同性高斯随机场的均质问题。在第二阶段，该模型成为一个不均匀的问题，该问题是由$ \ bs^2 $的时间延迟的布朗尼运动驱动的。该模型的解决方案以复杂的球形谐波的膨胀形式给出。通过将解决方案的扩展为$ l \ geq1 $，给出了解决方案的近似值。还得出了截断误差的收敛速率作为$ l $的函数，而均方误差随时间的函数也得出了。结果表明，收敛速率不仅取决于驱动噪声和初始条件的角功率谱的衰减，还取决于分数衍生物的顺序。我们研究随机溶液的样品特性，并表明该溶液是各向同性Hölder连续随机场。给出了受宇宙微波背景（CMB）启发的数值示例和模拟，以说明理论发现。
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 Hölder continuous random field. Numerical examples and simulations inspired by the cosmic microwave background (CMB) are given to illustrate the theoretical findings.
