In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

在本文中，我们提出了由一般高斯乘法噪声驱动的随机分数热方程的定量中心极限定理，包括时空白噪声和具有空间协方差的白噪声的情况，由 Riesz 核或有界可积函数。我们表明，在半径的球的空间平均ř收敛，如- [R趋于无穷大，合适的再归一化后，朝在总偏差距离高斯限制。我们还提供了一个功能中心极限定理。因此，我们将最近证明的随机热方程的类似结果扩展到分数拉普拉斯算子的情况和一般噪声的情况。