Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics
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(2020) 26:60
Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics Sergey G. Bobkov1 · Michel Ledoux2,3 Received: 8 December 2019 / Revised: 26 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line. Keywords Transport distances · Fourier analytic inequalities · non-Euclidean metrics Mathematics Subject Classification Primary 60E · 60F
1 Introduction The Kantorovich transport distance between two (Borel) probability measures μ and ν on a separable metric space (E, ρ) is defined as W(μ, ν) = inf λ
ρ(x, y) dλ(x, y),
(1.1)
Communicated by Massimo Fornasier. Research was partially supported by the Simons Foundation and NSF Grant DMS-1855575.
B
Sergey G. Bobkov [email protected] Michel Ledoux [email protected]
1
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
2
Institut de Mathématiques de Toulouse, Université de Toulouse – Paul-Sabatier, 31062 Toulouse, France
3
Institut Universitaire de France, Paris, France 0123456789().: V,-vol
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where the infimum is running over all measures λ on the product space E × E with marginals μ and ν. It describes the minimal cost needed to pay in order to transport one measure to the other one, given that it costs ρ(x, y) to move a particle x to a particle y. Under mild moment assumptions, W may be used to metrize the topology of weak convergence in the space of probability distributions on E. This metric and related functionals also appear in a natural way in many mathematical areas and concrete problems. It is therefore not surprising that the literature on the optimal transport is rather rich and intensive, reflecting various models and focusing on specific families of probability distributions (see for example [3,14,21] which provide numerous references on the subject). Nevertheless, often it is not easy to compute or even to estimate the distance W(μ, ν). One exceptional case is the real line E = R with the canonical Euclidean metric ρ(x, y) = |x − y|, when (1.1) is reduced to the well-known formula W(μ, ν) =
∞
−∞
|Fμ (x) − Fν (x)| d x
(1.2)
in terms of the distribution functions Fμ and Fν associated to μ and ν. The case of the Euclidean space E = Rd of dimension d ≥ 2 turns out already to be quite non-trivial. In analogy with the Esseen’s Fourier analytic inequality which serves as a traditional approach to the central limit theorem with respect to the Kolmogorov distance (cf. e.g. [4]), one general upper bound on W has been recently considered in [7] for the class of compactly supported measures on Rd in terms of their Fourier-Stieltjes transforms√(characteristic functions). To describe this bound, in th
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