Random data theory for the cubic fourth-order nonlinear Schr\"odinger equation

Abstract

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