Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schr\"odinger equation

We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{3} < s < 1$. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schr\"odinger equation, which is inspired by certain function spaces used in the study of the Schr\"odinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in $H^s_x(\mathbb{R}^4)$ with $\frac{1}{2} < s < 1$.