On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ${\mathbb{R}}^d$, $d\ge 3$

Abstract

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) on ${\mathbb{R}}^d$, $d\ge 3$, with random initial data and prove almost sure well-posedness results below the scaling critical regularity $s_{\text{crit}}=\frac{d-2}{2}$. More precisely, given a function on ${\mathbb{R}}^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove `conditional' almost sure global well-posedness for $d=4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d\ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.

52 pages. Minor modifications

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