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

Journal article (2020)

Authors

R.C. Kraaij Applied Probability - EEMCS
Louis Mahé University Paris-Saclay
Research Group
Applied Probability (EEMCS) (TU Delft)
Copyright: © 2020 R.C. Kraaij, Louis Mahé
DOI
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https://doi.org/10.1016/j.spa.2020.03.013
Boundary conditions Large deviations Hamilton–Jacobi equations Population dynamics
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Published Date

2020

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Applied Probability

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.