Ergodicity of the Fisher infinitesimal model with quadratic selection Calvez, Vincent Lepoutre, Thomas Poyato Sánchez, Jesús David Asymptotic behavior Nonlinear spectral theory Quantitative genetics VC and DP have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 865711). DP has also received founding from the European Union’s Horizon Europe research and innovation program under the Marie Sklodowska-Curie grant agreement No 101064402, and partially from the State Research Agency (SRA) of the Spanish Ministry of Science and Innovation and European Regional Development Fund (ERDF), project PID2022-137228OB-I00, and by Modeling Nature Research Unit, project QUAL21-011. We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher’s infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogs, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein–Rutman theorem in a genuinely non-linear, and non-monotone setting. 2023-12-07T08:47:59Z 2023-12-07T08:47:59Z 2024 journal article V. Calvez, T. Lepoutre and D. Poyato. Ergodicity of the Fisher infinitesimal model with quadratic selection. Nonlinear Analysis 238 (2024) 113392 [https://doi.org/10.1016/j.na.2023.113392] https://hdl.handle.net/10481/86060 10.1016/j.na.2023.113392 eng info:eu-repo/grantAgreement/ERC/H2020/865711 info:eu-repo/grantAgreement/EC/H2020/MSC 101064402 http://creativecommons.org/licenses/by/4.0/ open access Atribución 4.0 Internacional Elsevier