各向异性对数凹测度的$L^\infty$最优输运与Fisher无穷小模型中的指数收敛性
$L^\infty$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model
AI总结：
该研究证明各向异性强对数凹密度最优输运的$L^\infty$-Wasserstein距离上界，改进输运-信息不等式，并用Langevin动力学方法将Fisher无穷小模型的指数收敛结果从一维推广到任意维。
摘要：
我们证明了强对数凹概率密度与对数Lipschitz扰动之间最优输运的$L^\infty$-Wasserstein距离的上界。在最简单的情形下，这类上界等价于一个涉及$L^\infty$-Wasserstein度量和相对$L^\infty$-Fisher信息的输运-信息不等式。我们表明，当所涉及的密度具有各向异性时，该不等式可被显著改进。我们的证明基于使用Langevin动力学的概率技术。作为这些结果的一项应用，我们得到了数量遗传学中Fisher's infinitesimal model的精确指数收敛速率，将Calvez、Poyato和Santambrogio近期的一维结果推广到了任意维度。
英文摘要：
We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the $L^\infty$-Wasserstein metric and the relative $L^\infty$-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher's infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.
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