流形上Lipschitz输运的Bakry-Émery方法
A Bakry-Émery approach to Lipschitz transportation on manifolds
AI总结：
该研究在加权黎曼流形满足特定曲率-维数条件下，通过扩散输运映射证明了权重测度与其对数Lipschitz扰动间存在全局Lipschitz输运映射，并得到Poincaré不等式保持性、Monge映射增长估计等推论。
摘要：
在加权黎曼流形上，我们假设曲率-维数条件$\text{CD}(ρ_{1}, \text{∞})$及其二阶形式即$Γ_{3} \text{≥} ρ_{2} Γ_{2}$成立，通过Kim和Milman的扩散输运映射，证明了权重（概率）测度与其对数Lipschitz扰动之间存在全局Lipschitz输运映射。作为该结果的推论，我们得到了新结论：$(0,+∞)$上的指数测度在被对数Lipschitz势扰动时仍保持Poincaré不等式，以及通过Laguerre生成元推前$(0,+∞)$上伽马分布（指数分布为其特例）的Monge映射的新增长估计。
英文摘要：
On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman's diffusion transport map, assuming that the curvature-dimension condition $\mathrm{CD}(ρ_{1}, \infty)$ holds, as well as a second order version of it, namely $Γ_{3} \geq ρ_{2} Γ_{2}$. We get new results as corollaries to this result, as the preservation of Poincaré's inequality for the exponential measure on $(0,+\infty)$ when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on $(0,+\infty)$ (then getting as a particular case the exponential one), via Laguerre's generator.
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