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One-third Hölder stability of optimal transport maps from log-concave sources

  • Songbo Wang
  • Xiaozhen Wang
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Abstract

We prove quantitative stability of quadratic optimal transport maps from arbitrary log-concave sources to bounded targets. If the source ρ is isotropic on ℝd and the targets μ, ν are supported in a common ball of radius R, their Brenier maps satisfy lVert Tμ−TνrVertL2(ρ) leqslant 17d2/3R2/3W2(μ,ν)1/3. The one-third exponent is optimal for dgeqslant2. The proof combines a linear stability estimate for Brenier potentials in a weighted norm with a matching weighted reverse Poincaré inequality. The weight is chosen in terms of the negative log-density of the source to account for its tails. We also show that the best dimensional constant in map stability is at least of order d1/3.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026onethird,
  title = {{One-third Hölder stability of optimal transport maps from log-concave sources}},
  author = {Songbo Wang and Xiaozhen Wang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05534},
  eprint = {2611.05534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05534},
}
RIS
TY  - UNPB
TI  - One-third Hölder stability of optimal transport maps from log-concave sources
AU  - Songbo Wang
AU  - Xiaozhen Wang
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.05534
UR  - https://arxiv.org/abs/2611.05534
AB  - We prove quantitative stability of quadratic optimal transport maps from arbitrary log-concave sources to bounded targets. If the source ρ is isotropic on ℝ^d and the targets μ, ν are supported in a common ball of radius R, their Brenier maps satisfy lVert T_μ−T_νrVert_L^2(ρ) leqslant 17d^2/3R^2/3W_2(μ,ν)^1/3. The one-third exponent is optimal for dgeqslant2. The proof combines a linear stability estimate for Brenier potentials in a weighted norm with a matching weighted reverse Poincaré inequality. The weight is chosen in terms of the negative log-density of the source to account for its tails. We also show that the best dimensional constant in map stability is at least of order d^1/3.
ER  -