Long time well-posedness and wave turbulence limit for a quadratic non-lienar Schr{ö}dinger equation with well prepared random initial data

We prove a long time well-posedness result for a quadratic non-linear Schr{ö}dinger equation with random well prepared initial conditions in dimension d $\ge$ 1. In some regimes, we derive the Wave Kinetic equation associated with the equation in dimension d $\ge$ 5, almost up to the kinetic time to