Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS

We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schr\"odinger equation. As an example, we also show that the energy-critical cubic nonlinear Schr\"odinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.