Volume 24, Number 4, October–December 2018
|Page(s)||1489 - 1501|
|Published online||24 October 2018|
Comparison between W2 distance and Ḣ−1 norm, and Localization of Wasserstein distance★
Institut Élie Cartan de Lorraine, Campus Aiguillettes,
1 boulevard des Aiguillettes, BP 70239,
Vandœuvre-lès-Nancy Cedex, France
* Corresponding author: firstname.lastname@example.org
It is well known that the quadratic Wasserstein distance W2(⋅, ⋅) is formally equivalent, for infinitesimally small perturbations, to some weighted H−1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W2 distance exhibits some localization phenomenon: if μ and ν are measures on ℝn and ϕ: ℝn → ℝ+ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ ⋅ μ and ϕ ⋅ ν by an explicit multiple of W2(μ, ν).
Mathematics Subject Classification: 49Q20 / 28A75 / 46E35
Key words: Wasserstein distance / homogeneous Sobolev norm / localization
© EDP Sciences, SMAI 2018
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