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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).

On a mixture of Brenier and Strassen Theorems