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PreprintCC BY 4.0arXiv

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 Hölder regularity assumptions, we prove Lipschitz L2 (respectively C0,α) stability estimates for optimal transport maps in terms of the $2$-Monge--Kantorovich distance (respectively Lp distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz L2 stability estimate for the Brenier potentials in terms of the L2 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.

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Cite

BibTeXRIS
BibTeX
@misc{cajalopez2026stability,
  title = {{Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance}},
  author = {F. -U. Caja-Lopez and Matias G. Delgadino and Jun Kitagawa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.24232},
  eprint = {2605.24232},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24232},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance
AU  - F. -U. Caja-Lopez
AU  - Matias G. Delgadino
AU  - Jun Kitagawa
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2605.24232
UR  - https://arxiv.org/abs/2605.24232
AB  - 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^0,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.
ER  -