On the use of normal forms in the propagation of random waves

We consider the evolution of the correlations between the Fourier coeficients of a solution of the Kamdostev-Petviavshvili II equation when these coefficients are initially independent random variables. We use the structure of normal forms of the equation to prove that those correlations remain small until times of order $\varepsilon^{-5/3}$ or $\varepsilon^{-2}$ depending on the quantity considered.