We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS)
:
i
∂
t
u
+
Δ
u
=
±
|
u
|
2
u
: i \partial _t u + \Delta u = \pm |u|^{2}u
on
R
d
\mathbb {R}^d
,
d
≥
3
d \geq 3
, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity
s
c
r
i
t
=
d
−
2
2
s_\mathrm {crit} = \frac {d-2}{2}
. More precisely, given a function on
R
d
\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
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
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.