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Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we provestochastic versions of Strichartz estimates for Schr\"odinger with harmonicpotential. As a consequence, we show that the nonlinear Schr\"odinger equationwith quadratic potential and any polynomial non-linearity is almost surelylocally well-posed in $L^{2}(\R^{d})$ for any $d\geq 2$. Then, we show that wecan combine this result with the high-low frequency decomposition method ofBourgain 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 almostsure global well-posedness results with scattering for NLS on $\R^{d}$ withoutpotential. We prove scattering results for $L^2-$supercritical equations and$L^2-$subcritical equations with initial conditions in $L^2$ without additionaldecay or regularity assumption.

Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator