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In this paper, we study the probabilistic local well-posedness of the cubicSchr\"odinger equation (cubic NLS): $$ (i\partial_{t} + \Delta) u = \pm |u|^{2}u \text{ on } [0,T) \times \mathbb{R}^{d}, $$ with initial data being aunit-scale Wiener randomization of a given function $f$. We prove that asolution exists almost-surely locally in time provided $f\inH^{S}_{x}(\mathbb{R}^{d})$ with $S>\max\big(\frac{d-3}{4},\frac{d-4}{2}\big)$for $d\geq 3$. In particular, we establish that the local well-posedness holdsfor any $S>0$ when $d=3$. We also show that, under appropriate smallnessconditions for the initial data, solutions are global in time and scatter. Thesolutions are constructed as a sum of an explicit multilinear expansion of theflow in terms of the random initial data and of an additional smootherremainder term with deterministically subcritical regularity.  We develop the framework of directional space-time norms to control the(probabilistic) multilinear expansion and the (deterministic) remainder termand to obtain improved bilinear probabilistic-deterministic Strichartzestimates.

Higher order expansion for the probabilistic local well-posedness theory for a cubic nonlinear Schr\"odinger equation