Random data theory for the cubic fourth-order nonlinear Schrödinger equation

We consider the cubic nonlinear fourth-order Schrödinger equation \begin{document}$ i \partial_t u - \Delta^2 + \mu \Delta = \pm |u|^2 u, \quad \geq 0 $\end{document} on $ \mathbb R^N, N\geq 5 with random initial data. prove almost sure local well-posedness below scaling critical regularity. also probabilistic small data global and scattering. Finally, we scattering a large probability for randomized dilated cubes.