Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Abstract. — Thanks to an approach inspired from Burq-Lebeau [6], we prove stochastic versions of Strichartz estimates for Schrödinger with harmonic potential. As a consequence, we show that the nonlinear Schrödinger equation with quadratic potential and any polynomial non-linearity is almost surely locally well-posed in L2(Rd) for any d ≥ 2. Then, we show that we can combine this result with the high-low frequency decomposition method of Bourgain to prove a.s. global well-posedness results for the cubic equation: when d = 2, we prove global well-posedness in Hs(R2) for any s> 0, and when d = 3 we prove global well-posedness in Hs(R3) for any s> 1/6, which is a supercritical regime. Furthermore, we also obtain almost sure global well-posedness results with scattering for NLS on Rd without potential. We prove scattering results for L2−supercritical equations and L2−subcritical equations with initial conditions in L2 without additional decay or regularity assumption. 1. Introduction an