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

Khudiakova K, Maas J, Pedrotti F. 2025. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. 35(3), 1913–1940.

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Abstract
We prove upper bounds on the L∞-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∞-Wasserstein metric and the relative L∞-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.
Publishing Year
Date Published
2025-06-01
Journal Title
The Annals of Applied Probability
Publisher
Institute of Mathematical Statistics
Acknowledgement
This research was funded in part by the Austrian Science Fund (FWF) project 10.55776/F65 and the Austrian Academy of Science, DOC fellowship nr. 26293.
Volume
35
Issue
3
Page
1913-1940
ISSN
IST-REx-ID

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Khudiakova K, Maas J, Pedrotti F. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. 2025;35(3):1913-1940. doi:10.1214/25-aap2162
Khudiakova, K., Maas, J., & Pedrotti, F. (2025). L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/25-aap2162
Khudiakova, Kseniia, Jan Maas, and Francesco Pedrotti. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” The Annals of Applied Probability. Institute of Mathematical Statistics, 2025. https://doi.org/10.1214/25-aap2162.
K. Khudiakova, J. Maas, and F. Pedrotti, “L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model,” The Annals of Applied Probability, vol. 35, no. 3. Institute of Mathematical Statistics, pp. 1913–1940, 2025.
Khudiakova K, Maas J, Pedrotti F. 2025. L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model. The Annals of Applied Probability. 35(3), 1913–1940.
Khudiakova, Kseniia, et al. “L∞-Optimal Transport of Anisotropic Log-Concave Measures and Exponential Convergence in Fisher’s Infinitesimal Model.” The Annals of Applied Probability, vol. 35, no. 3, Institute of Mathematical Statistics, 2025, pp. 1913–40, doi:10.1214/25-aap2162.
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