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

On the probabilistic Cauchy theory of the cubic nonlinear Schr\"odinger equation on $\mathbb R^d$, $d \geq 3$