We consider the Schrödinger equation with any power non-linearity on the three-dimensional torus. We construct non-trivial 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 slow growth control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new combination of the fluctuation-dissipation method and some features of the Gibbs measures theory for Hamiltonian PDEs, the strategy of the paper applies to other contexts.