#### Vol. 7, No. 4, 2014

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Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

### Aurélien Poiret, Didier Robert and Laurent Thomann

Vol. 7 (2014), No. 4, 997–1026
##### Abstract

Thanks to an approach inspired by Burq and Lebeau [Ann. Sci. Éc. Norm. Supér. (4) 6:6 (2013)], 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 ${L}^{2}\left({ℝ}^{d}\right)$ for any $d\ge 2$. 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 ${\mathsc{ℋ}}^{s}\left({ℝ}^{2}\right)$ for any $s>0$, and when $d=3$ we prove global well-posedness in ${\mathsc{ℋ}}^{s}\left({ℝ}^{3}\right)$ for any $s>\frac{1}{6}$, 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.

##### Keywords
harmonic oscillator, supercritical nonlinear Schrödinger equation, random initial conditions, scattering, global solutions
##### Mathematical Subject Classification 2010
Primary: 35P05, 35Q55, 35R60