Gaussian fluctuations for the parabolic Anderson model with L\'evy white noise

In this article, we consider the parabolic Anderson model driven by a L\'evy white noise with finite variance in dimension 1, and we study the asymptotic behaviour of the spatial average of the solution. The main result shows that, with appropriate normalization and centering, the spatial integral converges in distribution to the standard normal distribution, and gives an estimate for the rate of this convergence in the Wasserstein distance, the Kolmogorov distance, and the Fortet-Mourier distance. We also prove the functional limit theorem corresponding to this result.