Wasserstein geometry and Ricci curvature bounds for Poisson spaces

Abstract: Let $\varUpsilon$ be the configuration space over a complete and separable metric base space, endowed with the Poisson measure $\pi$. We study the geometry of $\varUpsilon$ from the point of view of optimal transport and Ricci-lower bounds. To do so, we define a formal Riemannian structure on $\mathscr{P}{1}(\varUpsilon)$, the space of probability measures over $\varUpsilon$ with finite first moment, and we construct an extended distance $\mathcal{W}$ on $\mathscr{P}{1}(\varUpsilon)$. The distance $\mathcal{W}$ corresponds, in our setting, to the Benamou--Brenier variational formulation of the Wasserstein distance. Our main technical tool is a non-local continuity equation defined via the difference operator on the Poisson space. We show that the closure of the domain of the relative entropy is a complete geodesic space, when endowed with $\mathcal{W}$. We 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. In particular, we obtain that: (a) the Ornstein--Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has a Ricci curvature, in the entropic sense, bounded below by $1$; (c) the distance $\mathcal{W}$ satisfies an HWI inequality.