  We consider the defocusing nonlinear Schr\"odinger equations on the
two-dimensional compact Riemannian manifold without boundary or a bounded
domain in $\R^2$. 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.
