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In this article, we investigate the asymptotic behaviour of the spatialintegral of the solution to the parabolic Anderson model with time independentnoise in dimension $d\geq 1$, as the domain of the integral becomes large. Weconsider 3 cases: (a) the case when the noise has an integrable covariancefunction; (b) the case when the covariance of the noise is given by the Rieszkernel; (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 identifythe order of magnitude of the variance of the spatial integral, we prove aquantitative central limit theorem for the normalized spatial integral byestimating its total variation distance to a standard normal distribution, andwe give the corresponding functional limit result.

Central limit theorems for heat equation with time-independent noise: the regular and rough cases