A large probability averaging Theorem for the defocousing NLS

Abstract

We consider the nonlinear Schroedinger equation on the one dimensional torus, with a defocousing polynomial nonlinearity and study the dynamics corresponding to initial data in a set of large measure 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 ${\beta }^{2+\varsigma }$, $\beta$ being the inverse of the temperature and $\varsigma$ a positive number (we prove $\varsigma =1/10$). The proof is obtained by adapting to the context of Gibbs measure for PDEs some tools of Hamiltonian perturbation theory.