Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schr\"odinger with harmonic potential. As a consequence, we show that the nonlinear Schr\"odinger equation with quadratic potential and any polynomial non-linearity is almost surely locally well-posed in $L^{2}(\R^{d})$ for any $d\geq 2$. 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^{s}(\R^{2}) for any $s>0$, and when $d=3$ we prove global well-posedness in \H^{s}(\R^{3}) 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 $\R^{d}$ without potential. We prove scattering results for $L^2-$supercritical equations and $L^2-$subcritical equations with initial conditions in $L^2$ without additional decay or regularity assumption.