Random data Cauchy theory for the fourth order nonlinear Schr\"{o}dinger equation with cubic nonlinearity

We consider the Cauchy problem for the fourth order nonlinear Schr"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$on$$\mathbb{R}$ ^d$,$d \ge 3$, with random initial data, where $\mathrm{\partial }$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , , \dots , , \frac{\partial}{\partial x_d}$or$|\nabla |= $\mathcal{F}$^{-1}[|\xi | $\mathcal{F}$]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s({\mathbb{R}}^d)$ with $\max (\frac{d-5}{2},\frac{d-5}{6})<s$, whose lower bound is below the scale critical regularity $s_c=\frac{d-3}{2}$.