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Sharp potential and Brenier map W2-stability for log-concave sources

  • Yuanlong Ruan
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Abstract

We study stability of Brenier potentials and maps under changes of the target measure. Let κ>0, for a full support κ-strongly log-concave source ρ, possibly with a non-smooth potential, the normalized transport potentials satisfy |uμ−uν|L2(ρ)leqslant√(2/κ) W2(μ,ν) for arbitrary targets with finite second moments. The proof controls normalized Laguerre cell barycentres by their mass derivatives and centers the potential velocity to remove unbounded source cell locations. Bounded variation interpolation then yields the sharp map exponent 1/3 for targets in a common compact set, ‖ Tμ−Tν‖ L2(ρ)leqslant Const · W2( μ,ν) 1/3. Sharpness of the exponents for both the potential and Brenier map stability are established.

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Cite

BibTeXRIS
BibTeX
@misc{ruan2026sharp,
  title = {{Sharp potential and Brenier map $W_2$-stability for log-concave sources}},
  author = {Yuanlong Ruan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39272},
  eprint = {2610.39272},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39272},
}
RIS
TY  - UNPB
TI  - Sharp potential and Brenier map W_2-stability for log-concave sources
AU  - Yuanlong Ruan
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39272
UR  - https://arxiv.org/abs/2610.39272
AB  - We study stability of Brenier potentials and maps under changes of the target measure. Let κ>0, for a full support κ-strongly log-concave source ρ, possibly with a non-smooth potential, the normalized transport potentials satisfy |u_μ−u_ν|_L^2(ρ)leqslant√(2/κ) W_2(μ,ν) for arbitrary targets with finite second moments. The proof controls normalized Laguerre cell barycentres by their mass derivatives and centers the potential velocity to remove unbounded source cell locations. Bounded variation interpolation then yields the sharp map exponent 1/3 for targets in a common compact set, ‖ T_μ−T_ν‖ _L^2(ρ)leqslant Const · W_2( μ,ν) ^1/3. Sharpness of the exponents for both the potential and Brenier map stability are established.
ER  -