  We consider the Schr\"odinger equations with arbitrary (large) power
non-linearity on the three-dimensional torus. We construct non-trivial
probability measures supported on Sobolev spaces and show that the equations
are globally well-posed on the supports of these measures, respectively.
Moreover, these measures are invariant under the flows that are constructed.
Therefore, the constructed solutions are recurrent in time.\\ Also, we show
\textit{slow growth} control on the time evolution of the solutions. A
generalization to any dimension is given. Our proof relies on a new approach
combining the fluctuation-dissipation method and some features of the Gibbs
measures theory for Hamiltonian PDEs. The strategy of the paper applies to
other contexts
