From stochastic individual-based models to free-boundary Hamilton-Jacobi equations

Nicolas Champagnat; Sylvie Méléard; Sepideh Mirrahimi; Viet Chi Tran

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arXiv · 2604.06516

From stochastic individual-based models to free-boundary Hamilton-Jacobi equations

Nicolas Champagnat·Sylvie Méléard·Sepideh Mirrahimi·Viet Chi Tran

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Abstract

We study a stochastic branching model for a population structured by a quantitative phenotypic trait and subject to births, deaths, and mutations. In a regime of large population and small mutations, and in logarithmic scales of size and time, we derive a certain class of free boundary Hamilton-Jacobi equations with state constraints from the stochastic individual-based system. This goes beyond the classical Hamilton-Jacobi equations obtained from deterministic models by taking into account the possible extinction of the system in certain regions of the trait space. The proof is obtained by combining methods for the analysis of Hamilton-Jacobi equations with probabilistic tools from the theory of large deviations and branching processes.

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Nicolas Champagnat, Sylvie Méléard, Sepideh Mirrahimi, Viet Chi Tran. 2026-04-07. From stochastic individual-based models to free-boundary Hamilton-Jacobi equations. https://arxiv.org/abs/2604.06516

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