Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

— 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 nonlinearity is almost surely locally well-posed in L(R) 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 H(R) for any s > 0, and when d = 3 we prove global well-posedness in H(R) for any s > 1/6, which is a supercritical regime. Furthermore, we also obtain almost sure global well-posedness results with scattering for supercritical NLS on R without potential.