From optimal transportation to optimal teleportation

  • G. Wolansky

    Department of Mathematics, Technion, Haifa 32000, Israel


The object of this paper is to study estimates of for small . Here is the Wasserstein metric on positive measures, , is a probability measure and a signed, neutral measure (). In [16] we proved uniform (in ) estimates for provided can be controlled in terms of , for any smooth function .

In this paper we extend the results to the case where such a control fails. This is the case where, e.g., has a disconnected support, or the dimension of (to be defined) is larger or equal to .

In the latter case we get such an estimate provided for . If we get a log-Lipschitz estimate.

As an application we obtain Hölder estimates in for curves of probability measures which are absolutely continuous in the total variation norm.

In case the support of is disconnected (corresponding to ) we obtain sharp estimates for (“optimal teleportation”):

where is expressed in terms of optimal transport on a metric graph, determined only by the relative distances between the connected components of the support of , and the weights of the measure in each connected component of this support.

Cite this article

G. Wolansky, From optimal transportation to optimal teleportation. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 7, pp. 1669–1685

DOI 10.1016/J.ANIHPC.2016.12.003