On the exotic isometry flow of the quadratic Wasserstein space over the real line

Abstract

Kloeckner discovered that the quadratic Wasserstein space over the real line (denoted by W2(R)) is quite peculiar, as its isometry group contains an exotic isometry flow. His result implies that it can happen that an isometry Φ fixes all Dirac measures, but still, Φ is not the identity of W2(R). This is the only known example of this surprising and counterintuitive phenomenon. Kloeckner also proved that the image of each finitely supported measure under these isometries (and thus under all isometry) is a finitely supported measure. Recently we showed that the exotic isometry flow can be represented as a unitary group on L2((0,1)). In this paper, we calculate the generator of this group, and we show that every exotic isometry (and thus every isometry) maps the set of all absolutely continuous measures belonging to W2(R) onto itself.