  We study Gibbs measures invariant under the flow of the NLS on the unit disc
of $\R^2$. For that purpose, we construct the dynamics on a phase space of
limited Sobolev regularity and a wighted Wiener measure invariant by the NLS
flow. The density of the measure is integrable with respect to the Wiener
measure for sub cubic nonlinear interactions. The existence of the dynamics is
obtained in Bourgain spaces of low regularity. The key ingredient are bilinear
Strichartz estimates for the free evolution. The bilinear effect in our
analysis results from simple properties of the Bessel functions and estimates
on series of Bessel functions.

Analysis of PDEs;Dynamical Systems