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On a mixture of Brenier and Strassen theorems. (English) Zbl 1469.60127

The transportation problem as it is stated in modern or more technical literature looks somewhat different because of the development of Riemannian geometry and measure theory. The general notion of transport cost that encompasses many costs introduced in [N. Gozlan et al., J. Funct. Anal. 273, 3327–3405 (2017; Zbl 1406.60032)], and the transportation problem have had an important role in several classical problems of the geometric measure theory and mathematical physics and economics. A several authors studied the transportation problem [H. Djellout et al., Ann. Probab. 32, 2702–2732 (2004; Zbl 1061.60011); N. Gozlan et al., Ann. Probab. 41, No. 5, 3112–3139 (2013; Zbl 1283.60029); Rev. Mat. Iberoam. 30, No. 1, 133–163, (2014; Zbl 1296.60040); Probab. Theory Related Fields 160, No. 1–2, 47–94 (2014; Zbl 1332.60037); J. Lott and C. Villani, Ann. Math. (2) 169, No. 3, 903–991 (2009; Zbl 1178.53038); L. Rifford, Math. Control Relat. Fields 3, No. 4, 467–487 (2013; Zbl 1275.53034); L. Rizzi, Calc. Var. Partial Differ. Equ. 55, No. 3, Article ID 60, 20 p. (2016; Zbl 1352.53026); F. Baudoin and N. Garofalo, J. Eur. Math. Soc. (JEMS) 19, No. 1, 151–219 (2017; Zbl 1359.53018); M. Talagrand, Geom. Funct. Anal. 6, No. 3, 587–600 (1996; Zbl 0859.46030)].
The principal objective in this paper is to give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [Gozlan et al., Zbl 1406.60032]. Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. The authors also establish some connections with Caffarelli’s contraction theorem [L. A. Caffarelli, Comm. Math. Phys. 214, 547–563 (2000; Zbl 0978.60107)].

MSC:

60G42 Martingales with discrete parameter
60E15 Inequalities; stochastic orderings
49Q20 Variational problems in a geometric measure-theoretic setting
49J55 Existence of optimal solutions to problems involving randomness
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