A Bakry-\'Emery approach to Lipschitz transportation on manifolds

TL;DR

This paper establishes the existence of Lipschitz transport maps on weighted Riemannian manifolds under curvature conditions, with applications to measure perturbations and growth estimates for specific distributions.

Contribution

It introduces a Bakry-Émery approach to Lipschitz transportation on manifolds, extending Kim and Milman's diffusion transport map under new curvature-dimension conditions.

Findings

Preservation of Poincaré inequality under measure perturbations

New growth estimate for the Monge map on gamma distribution

Existence of Lipschitz transport maps under curvature conditions

Abstract

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 $\mathrm{CD}(\rho_{1}, \infty)$ holds, as well as a second order version of it, namely $\Gamma_{3} \geq \rho_{2} \Gamma_{2}$. We get new results as corollaries to this result, as the preservation of Poincar\'e'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.