In this paper, we investigate the asymptotic behavior of the spatial average of the solution to the parabolic Anderson model with time-independent noise in dimension $d\ge 1$, as the domain of the integral becomes large. We consider three cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index $H\in (\frac{1}{4},\frac{1}{2})$ in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

在本文中，我们研究了维度上具有时间无关噪声的抛物线 Anderson 模型解的空间平均值的渐近行为$d\ge \mathrm{1个}$，随着积分域变大。我们考虑三种情况：（a）噪声具有可积协方差函数的情况；(b) 噪声的协方差由 Riesz 核给出的情况；(c) 粗糙噪声的情况，即具有指数的分数噪声$H\in (\frac{\mathrm{1个}}{\mathrm{4个}},\frac{\mathrm{1个}}{\mathrm{2个}})$在维度d = 1 中。在每种情况下，我们确定空间积分方差的数量级，我们通过估计其与标准正态分布的总变异距离来证明归一化空间积分的定量中心极限定理，并且我们给出相应的功能极限结果。