# From optimal transportation to optimal teleportation

### G. Wolansky

Department of Mathematics, Technion, Haifa 32000, Israel

## Abstract

The object of this paper is to study estimates of $ϵ_{−q}W_{p}(μ+ϵν,μ)$ for small $ϵ>0$. Here $W_{p}$ is the Wasserstein metric on positive measures, $p>1$, $μ$ is a probability measure and $ν$ a signed, neutral measure ($∫dν=0$). In [16] we proved uniform (in $ϵ$) estimates for $q=1$ provided $∫ϕdν$ can be controlled in terms of $∫∣∇ϕ∣_{p/(p−1)}dμ$, for any smooth function $ϕ$.

In this paper we extend the results to the case where such a control fails. This is the case where, e.g., $μ$ has a disconnected support, or the dimension $d$ of $μ$ (to be defined) is larger or equal to $p/(p−1)$.

In the latter case we get such an estimate provided $1/p+1/d=1$ for $q=min(1,1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate.

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

In case the support of $μ$ is disconnected (corresponding to $d=∞$) we obtain sharp estimates for $q=1/p$ (“optimal teleportation”):

where $∥ν∥_{μ}$ 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 $μ$, and the weights of the measure $ν$ 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