Locality of the centred Wasserstein tangent cone

Abstract.

Wasserstein geodesics in the space of measures are minimizing an integral cost, that is, a global compromise between local contributions. We show that a particular subset of geodesics is instead characterized by a local condition, which is linked to a decomposition of the initial measure in pieces of constant dimension''. This has already been observed in smooth cases by Lott, and justified using PDEs: our argument is almost only based on convex analysis, and does not need restriction on the measure.