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

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