Fluctuation Limits of a locally regulated population and generalized Langevin equations

TL;DR

This paper establishes fluctuation limits for a spatially regulated population model, demonstrating that the fluctuations converge to a generalized Langevin equation, with detailed analysis in one dimension revealing an Ornstein-Uhlenbeck process.

Contribution

It proves a fluctuation theorem for the model and characterizes the limiting process as an infinite-dimensional Gaussian solving a generalized Langevin equation, extending previous deterministic approximations.

Findings

Fluctuation theorem holds under mild moment conditions.

Limiting process is an infinite-dimensional Gaussian.

In one dimension, the process is a time-inhomogeneous Ornstein-Uhlenbeck process.

Abstract

We consider a locally regulated spatial population model introduced by Bolker and Pacala. Based on the deterministic approximation studied by Fournier and M\'el\'eard, we prove that the fluctuation theorem holds under some mild moment conditions. The limiting process is shown to be an infinite-dimensional Gaussian process solving a generalized Langevin equation. In particular, we further consider its properties in one-dimension case, which is characterized as a time-inhomogeneous Ornstein-Uhlenbeck process.