In this talk, we consider the Cauchy problem for the fractional NLS with cubic nonlinearity (FNLS), posed on the one-dimensional torus T, subject to a gaussian random initial data with negative regularity. Exploiting the structure of the Liouville equation for the transport of the gaussian measure, we can show global-in-time bounds for the L^p norm of the density of the evolved measure. These bounds rely exclusively on the probabilistic local well posedness theory for FNLS. We can then use Bourgain’s invariant measure argument to extend these bounds to the solution of FNLS emanating from almost every initial data. In a certain range of the parameters of this problem, the global well posedness holds for initial data which is rougher than what is allowed by the deterministic local well posedness theory. 

This is joint work with J. Forlano (UCLA).