$L^{\infty }$-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model

Abstract

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.

arXiv commentPublished in AAP. 28 pages, 1 figure

References in corpus (4)

• Regularization under diffusion and anti-concentration of the information content
• Ergodicity of the Fisher infinitesimal model with quadratic selection
• On the Lipschitz properties of transportation along heat flows
• Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map