{
  "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 $L^{2}$ (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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.24232",
  "author_total": 3,
  "authors": [
    {
      "name": "F. -U. Caja-Lopez"
    },
    {
      "name": "Matias G. Delgadino"
    },
    {
      "name": "Jun Kitagawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/15tkybzfegrzz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.24232",
  "fulltext_url": null,
  "id": "15tkybzfegrzz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.24232",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7162473153",
  "pdf_url": "https://arxiv.org/pdf/2605.24232",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/15tkybzfegrzz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2605.24232",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.24232"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance",
  "title_text": "Stability of optimal transport maps and second variation of the 2-Monge-Kantorovich distance",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/15tkybzfegrzz",
  "venue": "arXiv",
  "versions": []
}