发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了Fathi等人的猜想，证明双边缘测度经任意全局利普希茨扰动时，布伦尼尔映射仍满足卡法雷利估计，推导了高斯测度下的相关界并扩展到一般势函数情形。
AI 中文摘要
我们研究在两个边缘测度的一阶扰动下，布伦尼尔（Brenier）映射的无量纲微分估计。当源势和目标势满足逐点黑塞（Hessian）比较条件时，卡法雷利收缩定理可给出此类估计。我们证明，在两个边缘测度受到任意全局利普希茨（Lipschitz）扰动时，无需额外假设即可得到相同结论，这尤其解答了法蒂（Fathi）、米库林瑟（Mikulincer）和申费尔德（Shenfeld）提出的猜想。更准确地说，设 $d\ge1$，$L\ge0$，且 $B:\mathbb{R}^d\to\mathbb{R}$ 是全局 $L$-利普希茨函数，我们证明从标准高斯测度 $\gamma_d$ 到与 $\mathrm{e}^{-B}\gamma_d$ 成比例的概率测度的布伦尼尔映射具有全局利普希茨代表，其界仅依赖于 $L$。该布伦尼尔势也属于 $C^{1,1}(\mathbb{R}^d)$，并满足无量纲双侧黑塞界。设 $\mathfrak{C}(0,L)$ 表示上界常数，则 $\log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \\ (L\to\infty)$，且具有布伦尼尔映射 $T_L$ 的一维例子满足 $\log\operatorname{Lip}(T_L)\ge L^2/2$。我们从各向异性双边缘测度结果推导该高斯估计：设 $V,W:\mathbb{R}^d\to\mathbb{R}$ 为势函数，$Q,P$ 为正定矩阵，利用分布曲率界 $D^2V\preceq Q$ 和 $D^2W\succeq P$，我们建立参考边缘测度的任意全局利普希茨扰动之间布伦尼尔映射的矩阵黑塞界，最重要的是，常数保留了基本梯度范围的方向几何；对于仿射扰动，我们的估计恢复了尖锐的非交换卡法雷利张量，还进一步得到高斯扰动的逐点位移界和紧凑高斯混合的谱黑塞估计。
英文摘要
We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let $d\ge1$, $L\ge0$, and let $B:\mathbb{R}^d\to\mathbb{R}$ be globally $L$-Lipschitz. We prove that the Brenier map from the standard Gaussian measure $γ_d$ to the probability measure proportional to $\mathrm{e}^{-B}γ_d$ has a globally Lipschitz representative whose bound depends only on $L$. The Brenier potential also belongs to $C^{1,1}(\mathbb{R}^d)$ and satisfies dimension-free two-sided Hessian bounds. Let $\mathfrak{C}(0,L)$ denote the upper-bound constant, then \[ \log\mathfrak{C}(0,L)=4L^2+\log L+\mathcal{O}(1) \qquad(L\to\infty), \] and a one-dimensional example with Brenier map $T_L$ satisfies $\log\operatorname{Lip}(T_L)\ge L^2/2$. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let $V,W:\mathbb{R}^d\to\mathbb{R}$ be potentials, and let $Q,P$ be positive-definite matrices. Using the distributional curvature bounds $D^2V\preceq Q$ and $D^2W\succeq P$, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.
