  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.
