From optimal transportation to optimal teleportation

Wolansky, G.

doi:10.1016/j.anihpc.2016.12.003

Wolansky, G.

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1669-1685.

Résumé

The object of this paper is to study estimates of $ϵ^{- q} W_{p} (μ + ϵ ν, μ)$ 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 ( $\int d ν = 0$ ). In [16] we proved uniform (in ϵ) estimates for $q = 1$ provided $\int ϕ d ν$ can be controlled in terms of $\int | \nabla ϕ |^{p / (p - 1)} d μ$ , 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 $1 / p + 1 / d \neq 1$ 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 $d = \infty$ ) we obtain sharp estimates for $q = 1 / p$ (“optimal teleportation”):

\lim_{ϵ \to 0} ϵ^{- 1 / p} W_{p} (μ, μ + ϵ ν) = {‖ ν ‖}_{μ}

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.

MR Zbl

DOI : 10.1016/j.anihpc.2016.12.003

Mots clés : Optimal transport, Monge–Kantorovich, Wasserstein metric

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     author = {Wolansky, G.},
     title = {From optimal transportation to optimal teleportation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1669--1685},
     publisher = {Elsevier},
     volume = {34},
     number = {7},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.12.003},
     mrnumber = {3724752},
     zbl = {1379.49042},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.12.003/}
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Wolansky, G. From optimal transportation to optimal teleportation. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 7, pp. 1669-1685. doi : 10.1016/j.anihpc.2016.12.003. http://www.numdam.org/articles/10.1016/j.anihpc.2016.12.003/

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