Wasserstein geometry and Ricci curvature bounds for Poisson spaces

We study the geometry of Poisson point processes from the point of view of optimal transport and Ricci lower bounds. We construct a Riemannian structure on the space of point processes and the associated distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒲</mml:mi> </mml:math> that corresponds to the Benamou–Brenier variational formula. Our main tool is a non-local continuity equation formulated with the difference operator. The closure of the domain of the relative entropy is a complete geodesic space, when endowed with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒲</mml:mi> </mml:math> . The geometry of this non-local infinite-dimensional space is analogous to that of spaces with positive Ricci curvature. Among others: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has an entropic Ricci curvature bounded from below by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> ; (c) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒲</mml:mi> </mml:math> satisfies an HWI inequality.