A Geometric Study of Wasserstein Spaces: Isometric Rigidity in Negative Curvature

Abstract

Given a metric space |$X$|⁠, one defines its Wasserstein space |${\mathscr {W}_2}(X)$| as a set of sufficiently decaying probability measures on |$X$| endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of |${\mathscr {W}_2}(X)$| when |$X$| is a simply connected, non-positively curved metric spaces by considering its isometry group. When |$X$| is Euclidean, the second-named author proved that this isometry group is larger than the isometry group of |$X$|⁠. In contrast, we prove here a rigidity result: when |$X$| is negatively curved, any isometry of |${\mathscr {W}_2}(X)$| comes from an isometry of |$X$|⁠.