A large probability averaging theorem for the defocusing NLS

We consider the nonlinear Schrödinger equation on the one dimensional torus, with a defocusing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measures with respect to the Gibbs measure. We prove that along the corresponding solutions the modulus of the Fourier coefficients is approximately constant for times of order β2+ζ, β being the inverse of the temperature and ζ a positive number (we prove ζ = 1/10). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.