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

TL;DR

This paper derives free-boundary Hamilton-Jacobi equations from a stochastic population model considering extinction, mutations, and large population limits, integrating probabilistic and PDE methods.

Contribution

It introduces a novel derivation of free-boundary Hamilton-Jacobi equations incorporating extinction phenomena from stochastic individual-based models.

Findings

Derived Hamilton-Jacobi equations with state constraints from stochastic models.

Extended classical equations by accounting for possible extinction regions.

Combined PDE analysis with probabilistic large deviations techniques.

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.