Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation

We prove a new smoothing type property for solutions of the 1d quintic Schr"odinger 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 because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities.