We show that on Wiener chaoses, asymptotic normality always implies asymptotic smoothness of the density. In particular, this implies, that without further assumption, every sequence of Wiener chaoses converging in law to a Gaussian has automatically their densities, as well as all their derivatives, converging uniformly to those of a Gaussian.
We use optimal transport ideas to study the geometry of the space of discrete measures over a separable metric space, equipped with the Poisson measure. In particular, we construct a variational distance of this space, and establish non-local infinite dimensional analogues of results regarding the geometry of the Wasserstein space over a metric measure space with synthetic Ricci curvature bounded below: the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; the Poisson space has a Ricci curvature, in the entropic sense, bounded below by 1; the distance W satisfies an HWI inequality.