A geometric study of Wasserstein spaces: isometric rigidity in negative curvature

Jérôme Bertrand (1) , Benoît Kloeckner (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 IF - Institut Fourier

This article deals with the space of probability measures (with finite second order moments) over a CAT(0) space. The Wasserstein metric turns this space of measures into a geodesic space called Wasserstein space. We are interested in the geometric properties of this space. In this paper, we study its isometry group. The main result is that the isometry group of the Wasserstein space of a CAT(0) space X coincides with that of X when X is negatively curved. This result is known to be false when X is the Euclidean space.

Mots clés: metric geometry; Hadamard spaces; optimal transport

Domaines: Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]

Jérôme Bertrand, Benoît Kloeckner. A geometric study of Wasserstein spaces: isometric rigidity in negative curvature. International Mathematics Research Notices, 2016, 2016 (5), pp.1368-1386. ⟨hal-00974554v2⟩