Abstract
We study the asymptotics of a two-dimensional stochastic differential system with a degenerate diffusion matrix. This system describes the dynamics of a population where individuals contribute to the degradation of their environment through two different behaviors, responding more or less intensively to their environmental perception. We exploit the almost one-dimensional form of the dynamical system to compute explicitly the Freidlin-Wentzell action functional. This allows us to give conditions under which the small noise regime of the invariant measure is concentrated around the equilibria of the dynamical system having the smallest diffusion coefficient.
Funding Statement
This work has been supported by the Chair Modélisation Mathématique et Biodiversité of Veolia - Ecole polytechnique - Museum national d’Histoire naturelle - Fondation X. It is also funded by the European Union (ERC AdG SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
Acknowledgments
We warmly thank the two anonymous referees for their helpul suggestions and comments.
Citation
Pierre Collet. Claire Ecotière. Sylvie Méléard. "Long time behavior of a degenerate stochastic system modeling the response of a population to its environmental perception." Electron. Commun. Probab. 30 1 - 12, 2025. https://doi.org/10.1214/24-ECP650
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