Squared quadratic Wasserstein distance: Optimal couplings and Lions differentiability

In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance W2 2 (τ; v) between two probability measures τ and v with nite second order moments on Rd is the composition of a martingale coupling with an optimal transport map T . We check the existence of an optimal coupling in which this map gives the unique optimal coupling between τ and T #τ. Next, we give a direct proof that → 7! W2 2 (→; v) is differentiable at τ in the Lions (Cours au College de France. 2008) sense i there is a unique optimal coupling between τ and v and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savare (Lectures in Mathematics ETH Zurich. Birkhauser 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 Frechet differentiability of a law invariant function F on L2(; P;Rd) is enough for the Frechet dierential at X to be a measurable function of X.