A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schr\"odinger equations

We consider the defocusing nonlinear Schr"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in ${\mathbb{R}}^2$. Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.