Comparison between W₂ distance and H^-1 norm, and localisation of Wasserstein distance

2011/04/24 by Rémi Peyre, Peyre, Rémi · 1 citation
Mathematics · #28A75 #46E35 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1104.4631

openalex publication_date 2011/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well known that the quadratic Wasserstein distance W₂ (\mathord\boldsymbol⋅, \mathord\boldsymbol⋅) 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 localisation phenomenon: if μ and ν are measures on Rⁿ and φ\colon 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).

Citations

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