Chaotic CLT using Poisson Fourth Moment Theorem and Graph Theory

Source abstract

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.