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Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability. (English) Zbl 1454.90028

Summary: In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance \(W_2^2(\mu,\nu)\) between two probability measures \(\mu\) and \(\nu\) with finite second order moments on \(\mathbb{R}^d\) is the composition of a martingale coupling with an optimal transport map \(\mathcal{T}\). We check the existence of an optimal coupling in which this map gives the unique optimal coupling between \(\mu\) and \(\mathcal{T}\#\mu\). Next, we give a direct proof that \(\sigma\mapsto W_2^2(\sigma,\nu)\) is differentiable at \(\mu\) in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between \(\mu\) and \(\nu\) and this coupling is given by a map. It was known combining results by L. Ambrosio et al. [Gradient flows in metric spaces and in the space of probability measures. Basel: Birkhäuser (2005; Zbl 1090.35002)] and L. Ambrosio and W. Gangbo [Commun. Pure Appl. Math. 61, No. 1, 18–53 (2008; Zbl 1132.37028)] that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of W. Gangbo and A. Tudorascu [J. Math. Pures Appl. (9) 125, 119–174 (2019; Zbl 1419.35234)]. Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function \(F\) on \(L^2(\Omega,\mathbb{P};\mathbb{R}^d)\) is enough for the Fréchet differential at \(X\) to be a measurable function of \(X\).

MSC:

90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
60G42 Martingales with discrete parameter
60E15 Inequalities; stochastic orderings
58B10 Differentiability questions for infinite-dimensional manifolds
49J50 Fréchet and Gateaux differentiability in optimization
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References:

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