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Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets — arXiv:2608.15906

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.15906
License (as archived): CC0-1.0
Archived: 2026-10-18T20:25:38.956Z
SHA-256 of the archived text: ee0b602b9eb293a335cd96c2169e8458c48ac6769a596b6cc8e3ba2c88a61b0d
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How to cite

Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets — arXiv:2608.15906 — arXiv (en), https://arxiv.org/abs/2608.15906 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_ea59cb49fba0 (Metraipolis archive dc_ea59cb49fba0, sha256 ee0b602b9eb293a335cd96c2169e8458c48ac6769a596b6cc8e3ba2c88a61b0d)

BibTeX

@misc{dcea59cb49fba0,
  title = {Dimension-Free Lipschitz Bounds for Brenier Maps to Compactly Supported Log-Concave Targets — arXiv:2608.15906},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.15906},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_ea59cb49fba0; sha256 ee0b602b9eb293a335cd96c2169e8458c48ac6769a596b6cc8e3ba2c88a61b0d},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.15906. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla\Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-18 in math.PR).
Categorie arXiv: Analysis of PDEs (math.AP) ; Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.15906 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T19:45:36.397Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).