Almost sure global well-posedness for the energy supercritical NLS on the unit ball of $\mathbb{R}^3$.

We present two almost sure global well-posedness results for the energy-supercritical nonlinear Schrodinger equations (NLS) on the unit ball of $\mathbb{R}^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions.This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinearities with critical and supercritical initial conditions, we prove the almost sure global well-posedness and the invariance of the measure under the solution flows in critical and supercritical spaces. This global result is built on a compactness argument and the Skorokhod representation theorem.