| Abstract
| - In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W22( μ, ν) between two probability measures μ and ν with finite second order moments on ℝ d is 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 between μ and 𝛵# μ. Next, we give a direct proof that σ ↦ W22( σ, ν) is differentiable at μ in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between μ and ν and 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 function F on L2(Ω, ℙ; ℝ d) is enough for the Fréchet differential at X to be a measurable function of X.
|
