Probabilistic preservation of regularity for periodic nonlinear Schr\"odinger equations

For certain non linear evolution equations, existence of global in time flows for large data is a fundamental and difficult question. In general, for dispersive and wave equations high regularity of the data does not automatically guarantee the existence of a global flow. One first needs to prove a global result at a level of regularity that matches that of a conserved quantity. Then, preservation of regularity allows to prove that the global flow exists for all smoother data. This mechanism cannot be applied in the non deterministic setting, such as the global well-posedness on the statistical ensemble of an invariant (Gibbs) measure, first obtained by Bourgain. We present a new and general technique to prove that data smoother than those in the statistical ensemble give rise to global flows, despite the fact that the measures carried by such smoother data are no longer invariant. As a consequence we close an important gap in the existence of global solutions for certain nonlinear Schr\"odinger equations.