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Uniform contractivity
of the Fisher infinitesimal model with strongly convex
selection
Vincent Calvez, David Poyato and Filippo Santambrogio |
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Vol. 18 (2025), No. 8, 1835–1874
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Abstract |
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The Fisher infinitesimal model is a classical model of phenotypic trait inheritance in quantitative genetics. Here, we prove that it encompasses a remarkable convexity structure which is compatible with a selection function having a convex shape. It yields uniform contractivity along the flow, as measured by an version of the Fisher information. It induces in turn asynchronous exponential growth of solutions, associated with a well-defined, log-concave, equilibrium distribution. Although the equation is nonlinear and nonconservative, our result shares some similarities with the Bakry–Emery approach to the exponential convergence of solutions to the Fokker–Planck equation with a convex potential. Indeed, the contraction takes place at the level of the Fisher information. Moreover, the key lemma for proving contraction involves the Wasserstein distance between two probability distributions of a (dual) backward-in-time process, and it is inspired by a maximum principle by Caffarelli for the Monge–Ampère equation. |
Keywords
integrodifferential equations, asymptotic behavior,
nonlinear spectral theory, quantitative genetics,
Monge–Ampère equation, maximum principle
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Mathematical Subject Classification
Primary: 35B40, 35P30
Secondary: 35Q92, 47G20, 92D15
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Milestones
Received: 28 April 2023
Revised: 15 May 2024
Accepted: 20 September 2024
Published: 25 July 2025
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Authors |
| Vincent Calvez | |
| CNRS, Université de Bretagne
Occidentale, UMR 6205 Laboratoire de Mathématiques de Bretagne Atlantique Brest France |
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| David Poyato | |
| Departamento de Matemática Aplicada
and Research Unit “Modeling Nature” (MNat) Facultad de Ciencias Universidad de Granada Granada Spain |
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| Filippo Santambrogio | |
| Université Claude Bernard Lyon 1,
CNRS Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet Institut Camille Jordan UMR5208 Villeurbanne France |
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| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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