We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) : i∂tu + Δu = ±|u|2u on R d, d ≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity scrit = 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 wellposedness 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.
Transactions of the American Mathematical Society Series B
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Bényi, Árpád; Oh, Tadahiro; and Pocovnicu, Oana, "On the Probabilistic Cauchy Theory of the Cubic Nonlinear Schrödinger Equation on Rd, d≥3" (2015). Mathematics. Paper 40.