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In this paper, we study the local well-posedness of the cubic Schr\"odingerequation: \[ (i \partial_t - \mathscr{L}) u = \pm |u|^2 u \quad \text{ on } I\times \mathbb{R}^d, \] with randomized initial data, and $\mathscr{L}$ beingan operator of degree $\sigma \geq 2$. Using estimates in directional spaces,we improve and extend known results for the standard Schr\"odinger equation(i.e. $\mathscr{L} = \Delta$) to any dimension and obtain results under naturalassumptions for general $\mathscr{L}$.

Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators