[Des modèles stochastiques d’évolution aux équations de Hamilton-Jacobi]
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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Keywords: Stochastic birth death models, large population approximation, selection, mutation, viscosity solution, maximum principle, Hamilton-Jacobi equation
Mots-clés : Processus de naissances et morts, processus aléatoire, approximation grande population, sélection, mutation, solution de viscosité, principe du maximum, équation de Hamilton-Jacobi
Champagnat, Nicolas 1 ; Méléard, Sylvie 2 ; Mirrahimi, Sepideh 3 ; Tran, Viet Chi 4
CC-BY 4.0
@article{JEP_2023__10__1247_0,
author = {Champagnat, Nicolas and M\'el\'eard, Sylvie and Mirrahimi, Sepideh and Tran, Viet Chi},
title = {Filling the gap between individual-based evolutionary models and {Hamilton-Jacobi} equations},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {1247--1275},
year = {2023},
publisher = {Ecole polytechnique},
volume = {10},
doi = {10.5802/jep.244},
language = {en},
url = {https://www.numdam.org/articles/10.5802/jep.244/}
}
TY - JOUR AU - Champagnat, Nicolas AU - Méléard, Sylvie AU - Mirrahimi, Sepideh AU - Tran, Viet Chi TI - Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 1247 EP - 1275 VL - 10 PB - Ecole polytechnique UR - https://www.numdam.org/articles/10.5802/jep.244/ DO - 10.5802/jep.244 LA - en ID - JEP_2023__10__1247_0 ER -
%0 Journal Article %A Champagnat, Nicolas %A Méléard, Sylvie %A Mirrahimi, Sepideh %A Tran, Viet Chi %T Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 1247-1275 %V 10 %I Ecole polytechnique %U https://www.numdam.org/articles/10.5802/jep.244/ %R 10.5802/jep.244 %G en %F JEP_2023__10__1247_0
Champagnat, Nicolas; Méléard, Sylvie; Mirrahimi, Sepideh; Tran, Viet Chi. Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1247-1275. doi: 10.5802/jep.244
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