On a mixture of Brenier and Strassen Theorems

Abstract

We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405). Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem (Caffarelli, Comm. Math. Phys. 214 (2000) 547–563).

Weak monotone rearrangement on the line

“…An important tool in the proof of Theorem 2.1 is C-monotonicity, a concept which was introduced for the weak optimal transport problem in [6,13,8].…”

“…The problem (1.2), and in particular its one dimensional version, is investigated in [2,15,14,19,18,20,11,13,8,5]. The main purpose of this note is to give a complete geometric characterization of the optimizer µ * in one dimension.…”

Existence, duality, and cyclical monotonicity for weak transport costs

“…We note that related concepts already appeared in [6, Proposition 4.1] (where necessity of a 2-step optimality condition is established) and in [23] (necessity in the case of compactly supported measures and a quadratic cost criterion). To the best of our knowledge, our sufficient criterion is the first of its kind for weak transport costs.…”

“…We remark that the 2-step monotonicity principle for weak transport costs has already proved vital in [6] for the construction of a martingale counterpart to the Brenier theorem and the Benamou-Brenier formula. On the other hand, we conjecture that this monotonicity principle could be used in order to generalize [23] to non-quadratic costs.…”

“…Indeed, particular costs of the form (1.1) were already considered by Marton [30,29] and Talagrand [40,41]. Further papers directly related to [25] include [37,36,38,21,23]. Notably the weak transport problem (1.1) also yields a natural framework to investigate a number of related problems: it appears in the recursive formulation of the causal transport problem [7], in [1,2,15,6] it is used to investigate martingale optimal transport problems, in [3] it is applied to prove stability of pricing and hedging in mathematical finance, it appears in the characterization of optimal mechanism for the multiple good monopolist [19] and motivates the investigation of linear transfers in [17].…”