From optimal transportation to optimal teleportation

Abstract

The object of this paper is to study estimates of ${ϵ}^{-q}{W}_p(\mu +ϵ\nu ,\mu )$ for small $ϵ>0$. Here $W_p$ is the Wasserstein metric on positive measures, $p>1$, $\mu$ is a probability measure and $\nu$ a signed, neutral measure ($\int d\nu =0$). In [16] we proved uniform (in $ϵ$) estimates for $q=1$ provided $\int \varphi d\nu$ can be controlled in terms of $\int |\nabla \varphi {|}^{p/(p-1)}d\mu$, for any smooth function $\varphi$.

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

In the latter case we get such an estimate provided $1/p+1/d\ne 1$ 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 $\mu$ is disconnected (corresponding to $d=\infty$) we obtain sharp estimates for $q=1/p$ (“optimal teleportation”):

where $\Vert \nu {\Vert }_{\mu }$ 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 $\mu$, and the weights of the measure $\nu$ in each connected component of this support.