We establish sufficient conditions for the existence of globally Lipschitz transport maps between probability measures and their log-Lipschitz perturbations, with dimension-free bounds. Our results include Gaussian measures on Euclidean spaces and uniform measures on spheres as source measures. More generally, we prove results for source measures on manifolds satisfying strong curvature assumptions. These seem to be the first examples of dimension-free Lipschitz transport maps in non-Euclidean settings, which are moreover sharp on the sphere. We also present some applications to functional inequalities, including a new dimension-free Gaussian isoperimetric inequality for log-Lipschitz perturbations of the standard Gaussian measure. Our proofs are based on the Langevin flow construction of transport maps of Kim and Milman.

我们为概率测度及其对数 Lipschitz 扰动之间存在全局 Lipschitz 传输图（具有无量纲边界）建立了充分的条件。我们的结果包括欧几里得空间上的高斯测度和球体上的统一测度作为源测度。更一般地说，我们证明了满足强曲率假设的流形上的源测量结果。这些似乎是非欧几里得设置中无维利普希茨输运图的第一个例子，而且在球面上很清晰。我们还提出了一些在函数不等式上的应用，包括用于标准高斯测度的对数李普希茨扰动的新的无维高斯等周不等式。我们的证明基于 Kim 和 Milman 交通图的 Langevin 流构建。