Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1247-1275.

We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh is much smaller than the size of mutation steps. When considering the evolution of the population in long time scales, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of subpopulation sizes on a logarithmic scale. We establish that under our rescaling, the stochastic discrete process converges to the viscosity solution of a Hamilton-Jacobi equation. The proof makes use of almost sure maximum principles and careful control of the martingale parts.

Nous considérons un modèle stochastique pour l’évolution d’une population discrète structurée en trait à valeurs dans une grille finie du tore, avec mutation et sélection. On se place dans une limite d’échelle de grande population, de petites mutations (mais pas rares), et où le maillage tend vers zéro. En temps long, la contribution de petites sous-populations peut fortement influencer la dynamique. Nous montrons que dans ce cadre, le processus stochastique discret converge sur une échelle logarithmique vers la solution de viscosité d’une équation de Hamilton-Jacobi. La preuve fait appel à un principe du maximum presque-sûr et à des estimées fines des parties martingales.

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DOI: 10.5802/jep.244
Classification: 92D25, 92D15, 60J80, 60F99, 35F21
Keywords: Stochastic birth death models, large population approximation, selection, mutation, viscosity solution, maximum principle, Hamilton-Jacobi equation
Mot clés : Processus de naissances et morts, processus aléatoire, approximation grande population, sélection, mutation, solution de viscosité, principe du maximum, équation de Hamilton-Jacobi

Nicolas Champagnat 1; Sylvie Méléard 2; Sepideh Mirrahimi 3; Viet Chi Tran 4

1 Université de Lorraine, CNRS, Inria, IECL F-54000 Nancy, France
2 École Polytechnique & Institut Universitaire de France, CNRS, Institut polytechnique de Paris route de Saclay, 91128 Palaiseau Cedex, France
3 Institut Montpelliérain Alexander Grothendieck, Univ. Montpellier, CNRS Place Eugène Bataillon, 34090 Montpellier, France
4 LAMA, Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS F-77454 Marne-la-Vallée, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Nicolas Champagnat; Sylvie Méléard; Sepideh Mirrahimi; Viet Chi Tran. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1247-1275. doi : 10.5802/jep.244. https://jep.centre-mersenne.org/articles/10.5802/jep.244/

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