A Bakry-Émery approach to Lipschitz transportation on manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913310941642752 |
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| author | López-Rivera, Pablo |
| author_facet | López-Rivera, Pablo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_02478 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A Bakry-Émery approach to Lipschitz transportation on manifolds López-Rivera, Pablo Probability Analysis of PDEs Functional Analysis 39B62, 47D07, 60J60, 58J65, 58J35 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. |
| title | A Bakry-Émery approach to Lipschitz transportation on manifolds |
| topic | Probability Analysis of PDEs Functional Analysis 39B62, 47D07, 60J60, 58J65, 58J35 |
| url | https://arxiv.org/abs/2310.02478 |