Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability

Benjamin Jourdain,Aurélien Alfonsi

doi:10.1051/ps/2020013

Abstract

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distanceW22(μ,ν) between two probability measuresμandνwith finite second order moments on ℝdis the composition of a martingale coupling with an optimal transport map 𝛵. We check the existence of an optimal coupling in which this map gives the unique optimal coupling betweenμand 𝛵#μ. Next, we give a direct proof thatσ↦W22(σ,ν) is differentiable atμin the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling betweenμandνand this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant functionFonL2(Ω, ℙ; ℝd) is enough for the Fréchet differential atXto be a measurable function ofX.

Highlights

We are interested in the structure of optimal couplings for the squared quadratic Wasserstein distance W22(μ, ν) between μ and ν in the set P2(Rd) of probability measures with finite second order moments on Rd, and in the differentiability of W22(μ, ν) with respect to μ
According to [11], there exists only one W2optimal coupling π between μ and each ν ∈ P2(Rd) and this coupling is given by a map T (i.e. π = (Id, T )#μ where Id denotes the identity function on Rd) iff μ gives 0 mass to the c − c hypersurfaces of dimension d − 1
There exist measures ν ∈ P2(Rd) such that either the unique optimal coupling is not given by a map or there exist distinct optimal couplings

Summary

Introduction

We are interested in the structure of optimal couplings for the squared quadratic Wasserstein distance W22(μ, ν) between μ and ν in the set P2(Rd) of probability measures with finite second order moments on Rd, and in the differentiability of W22(μ, ν) with respect to μ. Keywords and phrases: Optimal transport, Wasserstein distance, differentiability, couplings of probability measures, convex order. There exist measures ν ∈ P2(Rd) such that either the unique optimal coupling (uniqueness holds in dimension d = 1 for instance) is not given by a map or there exist distinct optimal couplings. To prove the necessary condition, we use that the Frechet differentiability at X ∼ μ of the lift on an atomless probability space is enough for the Frechet derivative at X to be a.s. equal to a measurable function of X, a consequence of [10] that we show again using simple probabilistic arguments.

Structure of quadratic Wasserstein optimal couplings

Differentiability of the squared quadratic Wasserstein distance