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We consider the defocusing nonlinear Schr\"odinger equation on $\mathbb{T}^2$with Wick ordered power nonlinearity, and prove almost sure globalwell-posedness with respect to the associated Gibbs measure. The heart of thematter is the uniqueness of the solution as limit of solutions to canonicallytruncated systems. The invariance of the Gibbs measure under the globaldynamics follows as a consequence.

Invariant Gibbs measures and global strong solutions for nonlinear Schr\"odinger equations in dimension two