Journal of Theoretical Probability

, Volume 24, Issue 3, pp 688–728 | Cite as

Rescaled Lotka–Volterra Models Converge to Super-Stable Processes



Recently, it has been shown that stochastic spatial Lotka–Volterra models, when suitably rescaled, can converge to a super-Brownian motion. We show that the limit process can be a super-stable process if the kernel of the underlying motion is in the domain of attraction of a stable law. The corresponding results in the Brownian setting were proved by Cox and Perkins (Ann. Probab. 33(3):904–947, 2005; Ann. Appl. Probab. 18(2):747–812, 2008). As applications of the convergence theorems, some new results on the asymptotics of the voter model started from single 1 at the origin are obtained, which improve the results by Bramson and Griffeath (Z. Wahrsch. Verw. Geb. 53:183–196, 1980).


Super-stable process Lotka–Volterra Voter model Domain of attraction Stable law Stable random walk 

Mathematics Subject Classification (2000)

60K35 60G57 60F17 60J80 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Laboratory of Mathematics and Complex Systems, School of Mathematical SciencesBeijing Normal UniversityBeijingPeople’s Republic of China

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