Well-posedness of Hamilton-Jacobi equations in population dynamics and applications to large deviations

TL;DR

This paper establishes large deviation principles for population models with immigration and harvesting by analyzing Hamilton-Jacobi equations, including cases with discontinuous Hamiltonians at boundaries, advancing the mathematical understanding of these stochastic processes.

Contribution

It introduces a novel analytic approach to prove well-posedness of Hamilton-Jacobi equations with discontinuous Hamiltonians, applicable to various population dynamics models.

Findings

Proved large deviation principles for multiple population models.

Established well-posedness of Hamilton-Jacobi equations with boundary discontinuities.

Provided partial results for multi-dimensional models.

Abstract

We prove Freidlin-Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and prey-predator models. The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.