Almost sure scattering for the defocusing cubic nonlinear Schr\"odinger equation on $\mathbb{R}^3\times\mathbb{T}$

We consider the Cauchy problem for the defocusing cubic nonlinear Schr\"odinger equation (NLS) on the waveguide manifold $\mathbb{R}^3\times\mathbb{T}$ and establish almost sure scattering for random initial data, where no symmetry conditions are imposed and the result is available for arbitrarily rough data $f\in H^s$ with $s\in\mathbb{R}$. The main new ingredient is a layer-by-layer refinement of the newly established randomization introduced by Shen-Soffer-Wu \cite{ShenSofferWu21}, which enables us to also obtain strongly smoothing effect from the randomization for the forcing term along the periodic direction. It is worth noting that such smoothing effect generally can not hold for purely compact manifolds, which is on the contrary available for the present model thanks to the mixed type nature of the underlying domain. As a byproduct, by assuming that the initial data are periodically trivial, we also obtain the almost sure scattering for the defocusing cubic NLS on $\mathbb{R}^3$ which parallels the ones by Camps \cite{Camps} and Shen-Soffer-Wu \cite{Shen2022}. To our knowledge, the paper also gives the first almost sure well-posedness result for NLS on product spaces.