From optimal transportation to optimal teleportation

Abstract

The object of this paper is to study estimates of $\epsilon^{-q}W_p(\mu+\epsilon\nu, \mu)$ for small $\epsilon>0$. 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 [W1] we proved uniform (in $\epsilon$) estimates for $q=1$ provided $\int \phi d\nu$ can be controlled in terms of the $\int|\nabla\phi|^{p/(p-1)}d\mu$, for any smooth function $\phi$. In this paper we extend the results to the case where such a control fails. This is the case where if, e.g. $\mu$ has a disconnected support, or if the dimension of $\mu$ , $d$ (to be defined) is larger or equal $p/(p-1)$. In the later case we get such an estimate provided $1/p+1/d\not=1$ 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 H\"{o}lder 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"): $$ \lim_{\epsilon\rightarrow 0}\epsilon^{-1/p}W_p(\mu, \mu+\epsilon\nu) = \|\nu\|_{\mu}$$ where $\|\nu\|_{\mu}$ 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.