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Fluctuation limits of a locally regulated population and generalized Langevin equations. (English) Zbl 1326.60141

Summary: We consider a locally regulated spatial population model introduced by B. Bolker and S. W. Pacala [Theor. Popul. Biol. 52, No. 3, 179–197, Art. No. TP971331 (1997; Zbl 0890.92020)]. Based on the deterministic approximation studied by N. Fournier and S. Méléard [Ann. Appl. Probab. 14, No. 4, 1880–1919 (2004; Zbl 1060.92055)], 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 the one-dimensional case, which is characterized as a time-inhomogeneous Ornstein-Uhlenbeck process.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
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