June 2023 The spatial Λ-Fleming–Viot process in a random environment
Aleksander Klimek, Tommaso Cornelis Rosati
Author Affiliations +
Ann. Appl. Probab. 33(3): 2426-2492 (June 2023). DOI: 10.1214/22-AAP1871

Abstract

We study the large scale behaviour of a population consisting of two types which evolve in dimension d=1,2 according to a spatial Lambda-Fleming–Viot process subject to random time-independent selection. If one of the two types is rare compared to the other, we prove that its evolution can be approximated by a super-Brownian motion in a random (and singular) environment. Without the sparsity assumption, a diffusion approximation leads to a Fisher–KPP equation in a random potential. The proofs build on two-scale Schauder estimates and semidiscrete approximations of the Anderson Hamiltonian.

Funding Statement

AK acknowledges that the funding for this work has been provided by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research.
TR gratefully acknowledges support by the IRTG 1740: this paper was developed within the scope of the IRTG 1740/TRP2015/50122-0, funded by the DFG/FAPESP.

Acknowledgments

We would like to thank Nicolas Perkowski for many helpful discussions and comments, Guglielmo Feltrin for an enlightening conversation and the anonymous referees for pointing out certain mistakes and their numerous suggestions to improve the article.

Citation

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Aleksander Klimek. Tommaso Cornelis Rosati. "The spatial Λ-Fleming–Viot process in a random environment." Ann. Appl. Probab. 33 (3) 2426 - 2492, June 2023. https://doi.org/10.1214/22-AAP1871

Information

Received: 1 April 2020; Revised: 1 November 2021; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583675
zbMATH: 1516.35576
Digital Object Identifier: 10.1214/22-AAP1871

Subjects:
Primary: 35R60 , 60F05 , 60J68
Secondary: 60G51 , 60J70 , 92D15

Keywords: Anderson Hamiltonian , scaling limits , Schauder estimates , Spatial Lambda-Fleming–Viot model , Super-Brownian motion , super-processes

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 3 • June 2023
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