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 \(\varvec{\textrm{CD}(\rho _{1}, \infty )}\) holds, as well as a second order version of it, namely \(\varvec{\Gamma _{3} \ge \rho _{2} \Gamma _{2}}\). We get new results as corollaries to this result, as the preservation of Poincaré’s inequality for the exponential measure on \(\varvec{(0,+\infty )}\) when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on \(\varvec{(0,+\infty )}\) (then getting as a particular case the exponential one), via Laguerre’s generator.

在加权黎曼流形上，我们通过 Kim 和 Milman 的扩散传输图证明了权重（概率）测度与其对数 Lipschitz 扰动之间存在全局 Lipschitz 传输图，假设曲率维数条件\(\varvec{\textrm {CD}(\rho _{1}, \infty )}\)及其二阶版本成立，即\(\varvec{\Gamma _{3} \ge \rho _{2} \伽马 _{2}}\)。作为该结果的推论，我们得到了新的结果，即当受到对数 Lipschitz 势的扰动和新的增长估计时，对\(\varvec{(0,+\infty )}\)的指数测度庞加莱不等式得以保留蒙日图通过拉盖尔的生成器推动\(\varvec{(0,+\infty )}\)上的伽玛分布（然后作为一种特殊情况得到指数分布）。