标题： 单位球上的时分数学随机双曲扩散方程的演化 显示英文标题

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

摘要： 本文研究了二维随机模型在时间上的演化，该模型用于球面随机场。 该模型使用一个时间分数阶随机双曲扩散方程，该方程描述了球面随机场在时间上在$\bS^2$上的演化。 扩散算子包含了一个以Caputo意义的时间分数阶导数。 在模型的第一阶段，考虑了一个齐次问题，其中在$\bS^2$上的各向同性高斯随机场作为初始条件。 在第二阶段，模型转换为由在$\bS^2$上的时间延迟布朗运动驱动的非齐次问题。 模型的解通过实球谐函数的级数表示。 为了获得近似解，将解的展开式截断到某个次数$L\geq1$。 截断误差的分析揭示了它们的收敛行为，表明收敛速率受到驱动噪声和初始条件的角功率谱衰减的影响。 此外，我们研究了随机解的样本性质，证明了在某些条件下，解存在局部Hölder连续的修正。 为了说明理论结果，给出了受宇宙微波背景（CMB）启发的数值示例和模拟。 显示英文摘要

摘要： This paper 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 $\bS^2$ 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 $\bS^2$ serving as the initial condition. In the second stage, the model transitions to an inhomogeneous problem driven by a time-delayed Brownian motion on $\bS^2$. The solution to the model is expressed through a series of real 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.