On isometric embeddings of Wasserstein spaces -- the discrete case

The aim of this short paper is to offer a complete characterization of all (not necessarily surjective) isometric embeddings of the Wasserstein space ${\mathcal{W}}_p(\mathcal{X})$, where $\mathcal{X}$ is a countable discrete metric space and $0<p<\mathrm{\infty }$ is any parameter value. Roughly speaking, we will prove that any isometric embedding can be described by a special kind of $\mathcal{X}\times (0,1]$-indexed family of nonnegative finite measures. Our result implies that a typical non-surjective isometric embedding of ${\mathcal{W}}_p(\mathcal{X})$ splits mass and does not preserve the shape of measures. In order to stress that the lack of surjectivity is what makes things challenging, we will prove alternatively that ${\mathcal{W}}_p(\mathcal{X})$ is isometrically rigid for all $0<p<\mathrm{\infty }$.