High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study the realized power variations for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by the space-time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. We use the underlying explicit kernels and spectral/harmonic analysis, yielding temporal central limit theorems for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space-time white noise. On the other hand, it builds on and complements Allouba’s earlier works on time fractional SPIDEs and their gradient.

高阶和​​分数 PDE 在理论和建模许多现象中已经变得突出。在本文中，我们研究了由一到三维空间中的时空白噪声驱动的四阶时间分数阶随机偏积分微分方程 (SPIDE) 的已实现功率变化及其梯度，在时间上有无限二次变化和维度相关的高斯渐近分布。我们使用基础显式内核和频谱/谐波分析，得出时间分数 SPIDE 及其梯度的时间中心极限定理。一方面，这项工作建立在最近对由时空白噪声驱动的一般高斯过程和随机热方程的变化进行精细分析的工作的基础上。另一方面，