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

Peyre, Rémi

ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1489-1501

It is well known that the quadratic Wasserstein distance $W_2(·,·)$ is formally equivalent, for infinitesimally small perturbations, to some weighted ${\mathrm{H}}^{-1}$ 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 $\mu$ and $\nu$ are measures on ${ℝ}^n$ and $\phi :{ℝ}^n\to {ℝ}_{+}$ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between $\phi ·\mu$ and $\phi ·\nu$ by an explicit multiple of $W_2(\mu ,\nu )$.

Classification : 49Q20, 28A75, 46E35
Keywords: Wasserstein distance, homogeneous Sobolev norm, localization

[1] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les inégalités de Sobolev logarithmiques, vol. 10 of Panoramas et Synthèses [Panoramas and Syntheses]. With a preface by Dominique Bakry and Michel Ledoux. Société Mathématique de France, Paris (2000) | MR | Zbl

[2] J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge−Kantorovich mass transfer problem. Numer. Math. 84 (2000) 375–393 | MR | Zbl | DOI

[3] H.P. Boas and E.J. Straube, Integral inequalities of Hardy and Poincaré type. Proc. Amer. Math. Soc. 103 (1988) 172–176 | MR | Zbl

[4] S.G. Bobkov, Spectral gap and concentration for some spherically symmetric probability measures. In Geometric aspects of functional analysis. In Vol. 1807 of Lect. Notes Math., Springer, Berlin (2003) 37–43 | MR | Zbl

[5] D. Cordero−Erausquin, R.J. Mccann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 (2001) 219–257 | MR | Zbl | DOI

[6] R. Hurri−Syrjänen, An improved Poincaré inequality. Proc. Amer. Math. Soc. 120 (1994) 213–222 | MR | Zbl

[7] G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. 86 (2006) 68–79 | MR | Zbl | DOI

[8] R.J. Mccann, A convexity principle for interacting gases. Adv. Math. 128 (1997) 153–179 | MR | Zbl | DOI

[9] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361–400 | MR | Zbl | DOI

[10] A. Pratelli, Equivalence between some definitions for the optimal mass transport problem and for the transport density on manifolds. Ann. Mat. Pura Appl. 184 (2005) 215–238 | Zbl | MR | DOI

[11] R.T. Seeley, Spherical harmonics. Amer. Math. Monthly 73 (1966) 115–121 | Zbl | MR | DOI

[12] X. Tolsa, Mass transport and uniform rectifiability. Geom. Funct. Anal. 22 (2012) 478–527 | Zbl | MR | DOI

[13] C. Villani, Topics in optimal transportation. In Vol. 58 of Graduate Studies in Mathematics. American Mathematical Society (2003) | Zbl | MR

[14] C. Villani, Optimal Transport: Old and New. In Vol. 338 of Grundlehren der Mathematischen Wissenschaften. Springer (2009) | Zbl | MR