On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ^d, d ≥ 3

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i∂_t u+Δu = ±|u|2u on ℝ^d, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity s_{crit} = (d−2)/2. More precisely, given a function on 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 = 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.