Chaotic CLT using Poisson Fourth Moment Theorem and Graph Theory

In this article, we consider the parabolic Anderson model and the hyperbolic Anderson model in dimension 1, driven by a time-independent Lévy noise, which is either white, or colored with integrable coloration kernel $κ$. Using a Poisson version of the Fourth Moment Theorem due to Döbler, Vidotto and Zheng (2018), we prove that the spatial integral of the solution satisfies a central limit theorem. We prove that this result also holds for the hyperbolic Anderson model in dimension $d=2$, if in addition $κ$ is bounded. To estimate the fourth cumulant of the projection of the spatial integral on a fixed chaos, we develop a novel technique using graph theory, which allows us to handle some intricate integrals involving the modified contractions.