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 $0<p<\infty$ 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 $0<p<\infty$.
Categories: math.FA math-ph math.MG math.MP