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Uniform contractivity of the Fisher infinitesimal model with strongly convex selection

Vincent Calvez, David Poyato and Filippo Santambrogio

Vol. 18 (2025), No. 8, 1835–1874
Abstract

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 L 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 W 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
Mathematical Subject Classification
Primary: 35B40, 35P30
Secondary: 35Q92, 47G20, 92D15
Milestones
Received: 28 April 2023
Revised: 15 May 2024
Accepted: 20 September 2024
Published: 25 July 2025
Authors
Vincent Calvez
CNRS, Université de Bretagne Occidentale, UMR 6205
Laboratoire de Mathématiques de Bretagne Atlantique
Brest
France
David Poyato
Departamento de Matemática Aplicada and Research Unit “Modeling Nature” (MNat)
Facultad de Ciencias
Universidad de Granada
Granada
Spain
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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