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We consider the cubic Schr\"odinger equation on the euclidean space perturbedby a short-range potential $V$. The presence of a negative simple eigenvaluefor $-\Delta+V$ gives rise to a curve of small and localized nonlinear groundstates that yield some time-periodic solutions known to be asymptoticallystable in the energy space. We study the persistence of these coherent statesunder rough perturbations. We shall construct a large measure set of smallscaling-supercritical solutions below the energy space that display someasymptotic stability behavior. The main difficulty is the need to handle theinteractions of localized and dispersive terms in the modulation equations. Todo so, we use a critical-weighted strategy to combine probabilistic nonlinearestimates in critical spaces based on $U^p, V^q$ (controlling higher orderterms) with some local energy decay estimates (controlling lower order terms).We also revisit in the perturbed setting the analysis of B\'enyi, Oh andPocovnicu on the probabilistic global well-posedness and scattering for smallsupercritical initial data. We use a distorted Fourier transform andsemiclassical functional calculus to generalize probabilistic and bilinearStrichartz estimates.

Asymptotic stability of small ground states for NLS under random perturbations