Singular evolution of Gaussian measures under energy-critical and supercritical nonlinear Schrödinger equations

We consider a family of nonlinear Schrödinger equations with odd-power nonlinearities and fractional dispersion in an energy-critical/supercritical regime. For a family of Gaussian measures supported on high-regularity spaces, we show that the flow immediately evolves these measures to ones that are mutually singular with respect to their initial distributions. In the cubic case, the threshold for singularity exactly corresponds to the energy scaling for the equation. In particular, we establish singular evolution of Gaussian measures under the flow of the energy-critical cubic NLS on the 4-dimensional torus.