#### Document Type

Article

#### Publication Date

5-26-2015

#### Abstract

We consider the Cauchy problem of the cubic nonlinear Schr**ö**dinger equation (NLS) :* i*∂_{t}*u *+ Δ*u* = ±|*u*|^{2}*u* on R* ^{ d}*,

*d*≥ 3, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity

*crit =*

^{s}*d-2/2.*More precisely, given a function on R

*, 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}*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.

#### Publication Title

Transactions of the American Mathematical Society Series B

#### Volume

2

#### First Page

1

#### Last Page

50

#### Required Publisher's Statement

© Copyright 2015, American Mathematical Society

#### Recommended Citation

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.

http://cedar.wwu.edu/math_facpubs/40