Transport of Gaussian measures by the flow of the nonlinear Schrödinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schrödinger equation. As a consequence, we prove that a family of natural Gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case we only get local in time quasi-invariance because of a blow-up obstruction. Our results extend as well to generic odd power nonlinearities.