Scattering for the cubic Schr{\"o}dinger equation in 3D with randomized radial initial data

We obtain almost-sure scattering for the cubic defocusing Schrödinger equation in the Euclidean space R, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of Bényi, Oh and Pocovnicu [1]. It also extends the results of Dodson, Lührmann and Mendelson [19] on the energy-critical equation in R, to the energy-subcritical equation in R. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect that makes it subcritical, it still has infinite energy. In the present work, we first develop a stability theory from the deterministic scattering results below the energy space, due to Colliander, Keel, Staffilani, Takaoka and Tao. Then, we propose a globalization argument in which we set up the I-method with a Morawetz bootstrap in a stochastic setting. To our knowledge, this is the first almost-sure scattering result for an energy-subcritical Schrödinger equation outside the small data regime.