Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914594377695232 |
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| author | Caja-Lopez, F. -U. Delgadino, Matias G. Kitagawa, Jun |
| author_facet | Caja-Lopez, F. -U. Delgadino, Matias G. Kitagawa, Jun |
| contents | We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,α}$) 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-Ampère 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24232 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance Caja-Lopez, F. -U. Delgadino, Matias G. Kitagawa, Jun Analysis of PDEs 49Q22, 35J96, 49K40 We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,α}$) 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-Ampère 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. |
| title | Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance |
| topic | Analysis of PDEs 49Q22, 35J96, 49K40 |
| url | https://arxiv.org/abs/2605.24232 |