A Bakry-Émery Approach to Lipschitz Transportation on Manifolds

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-04-11 DOI:10.1007/s11118-024-10138-4

Pablo López-Rivera

引用次数: 0

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 \(\varvec{\textrm{CD}(\rho _{1}, \infty )}\) 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.

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积分榜上的 Lipschitz Transportation 的 Bakry-Émery 方法

在加权黎曼流形上，我们通过 Kim 和 Milman 的扩散传输映射证明了权重（概率）度量和它的对数-利普希兹扰动之间存在全局利普希兹传输映射、假设曲率维度条件 \(\varvec{\textrm{CD}(\rho _{1}, \infty )}\) 成立，以及它的二阶版本，即 \(\varvec{\Gamma _{3} \ge \rho _{2} \Gamma _{2}}\) 成立。作为这一结果的推论，我们得到了新的结果，如当受到对数-利普斯奇兹势能的扰动时，\(\varvec{(0,+\infty )}\) 上指数量的波恩卡莱不等式的保留，以及通过拉盖尔生成器，在\(\varvec{(0,+\infty )}\) 上推前伽马分布的蒙日映射的新的增长估计（然后作为一种特殊情况得到指数分布）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.