Lifted and geometric differentiability of the squared quadratic Wasserstein distance.

Summary 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 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 μ and T#μ. Next, we prove that σ↦W22(σ,ν) is differentiable at μ in the Lions~\cite{Lions} and the geometric senses iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. Besides, we give a self-contained proof that mere Fréchet differentiability of a law invariant function F on L2(Ω,P.Rd) is enough for the Fréchet differential at X to be a measurable function of X.