  We consider the cubic Schr\"odinger equation on the euclidean space perturbed
by a short-range potential $V$. The presence of a negative simple eigenvalue
for $-\Delta+V$ gives rise to a curve of small and localized nonlinear ground
states that yield some time-periodic solutions known to be asymptotically
stable in the energy space. We study the persistence of these coherent states
under rough perturbations. We shall construct a large measure set of small
scaling-supercritical solutions below the energy space that display some
asymptotic stability behavior. The main difficulty is the need to handle the
interactions of localized and dispersive terms in the modulation equations. To
do so, we use a critical-weighted strategy to combine probabilistic nonlinear
estimates in critical spaces based on $U^p, V^q$ (controlling higher order
terms) with some local energy decay estimates (controlling lower order terms).
We also revisit in the perturbed setting the analysis of B\'enyi, Oh and
Pocovnicu on the probabilistic global well-posedness and scattering for small
supercritical initial data. We use a distorted Fourier transform and
semiclassical functional calculus to generalize probabilistic and bilinear
Strichartz estimates.

Analysis of PDEs