  In this paper, our discussion mainly focuses on equations with energy
supercritical nonlinearities. We establish probabilistic global well-posedness
(GWP) results for the cubic Schr\"odinger equation with any fractional power of
the Laplacian in all dimensions. We consider both low and high regularities in
the radial setting, in dimension $\geq 2$. In the high regularity result, an
{\it Inviscid - Infinite dimensional (IID) limit} is employed while in the low
regularity global well-posedness result, we make use of the Skorokhod
representation theorem. The IID limit is presented in details as an independent
approach that applies to a wide range of Hamiltonian PDEs. Moreover we discuss
the adaptation to the periodic settings, in any dimension, for smooth
regularities.
