  In this article, we first present the construction of Gibbs measures
associated to nonlinear Schr\"odinger equations with harmonic potential. Then
we show that the corresponding Cauchy problem is globally well-posed for rough
initial conditions in a statistical set (the support of the measures). Finally,
we prove that the Gibbs measures are indeed invariant by the flow of the
equation. As a byproduct of our analysis, we give a global well-posedness and
scattering result for the $L^2$ critical and super-critical NLS (without
harmonic potential).
