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arXiv · 2605.24232

Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance

F. -U. Caja-Lopez·Matias G. Delgadino·Jun Kitagawa
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Abstract

We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under H\"older regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,\alpha}$) stability estimates for optimal transport maps in terms of the 2-Monge-Kantorovich distance (respectively $L^{p}$ distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz $L^2$ stability estimate for the Brenier potentials in terms of the $L^2$ distance between the source and target densities. The proofs rely on a precise characterization of the linear response of the Brenier potential along smooth interpolations of the data, obtained by linearizing the Monge-Amp\`ere equation in divergence form. As a further application of this approach, we derive an explicit formula for the second variation of the quadratic Monge-Kantorovich distance.

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F. -U. Caja-Lopez, Matias G. Delgadino, Jun Kitagawa. 2026-05-22. Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance. https://arxiv.org/abs/2605.24232

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