Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance

Abstract

It is well known that the quadratic Wasserstein distance W 2 (⋅, ⋅) 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 W 2 distance exhibits some localization phenomenon: if μ and ν are measures on ℝ n and ϕ : ℝ n → 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 W 2 ( μ , ν ).