L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model

preprint L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model draft Kseniia Khudiakova author 4E6DC800-AE37-11E9-AC72-31CAE56974250000-0002-6246-1465 Jan Maas author 4C5696CE-F248-11E8-B48F-1D18A9856A870000-0002-0845-1338 Francesco Pedrotti author d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c JaMa department Taming Complexity in Partial Differential Systems project The impact of deleterious mutations on small populations project 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. 2024 eng arXiv 2402.0415110.48550/arXiv.2402.04151 https://research-explorer.ista.ac.at/record/20050 https://research-explorer.ista.ac.at/record/17336 Khudiakova, K., Maas, J., & Pedrotti, F. (n.d.). L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv. <a href="https://doi.org/10.48550/arXiv.2402.04151">https://doi.org/10.48550/arXiv.2402.04151</a> Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv. doi:<a href="https://doi.org/10.48550/arXiv.2402.04151">10.48550/arXiv.2402.04151</a> K. Khudiakova, J. Maas, and F. Pedrotti, “L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model,” arXiv. . Khudiakova, Kseniia, Jan Maas, and Francesco Pedrotti. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” ArXiv, n.d. <a href="https://doi.org/10.48550/arXiv.2402.04151">https://doi.org/10.48550/arXiv.2402.04151</a>. Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. arXiv, 2402.04151. Khudiakova, Kseniia, et al. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” ArXiv, 2402.04151, doi:<a href="https://doi.org/10.48550/arXiv.2402.04151">10.48550/arXiv.2402.04151</a>. K. Khudiakova, J. Maas, F. Pedrotti, ArXiv (n.d.). 173522024-07-31T08:07:40Z2026-04-07T13:00:02Z