Comparison between $W_2$ distance and ${\dot{H}}^{-1}$ norm, and localisation of Wasserstein distance

Abstract

It is well known that the quadratic Wasserstein distance $W_2(\cdot ,\cdot )$ 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 localisation phenomenon: if $\mu$ and $\nu$ are measures on ${\mathbf{R}}^n$ and $\phi :{\mathbf{R}}^n\to {\mathbf{R}}_{+}$ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between $\phi \cdot \mu$ and $\phi \cdot \nu$ by an explicit multiple of $W_2(\mu ,\nu )$.

Added a new section about application to the localisation of Wasserstein distance

References in corpus (1)

• Mass transport and uniform rectifiability