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, 54506 Vandœuvre-lès-Nancy Cedex, France

^* Corresponding author: remi.peyre@univ-lorraine.fr

Received: 23 March 2017

Abstract

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).