Scattering for the cubic Schrödinger equation in 3D with randomized radial initial data

Abstract:

We obtain almost-sure scattering for the Schrödinger equation with a defocusing cubic nonlinearity in the Euclidean space $\mathbb {R}^3$, 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 [Trans. Amer. Math. Soc. Ser. B 2 (2015), pp. 1–50]. It also extends the results of Dodson, Lührmann and Mendelson [Adv. Math. 347 (2019), pp. 619–676] on the energy-critical equation in $\mathbb {R}^4$, to the energy-subcritical equation in $\mathbb {R}^3$. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect which 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.