Rescaled Lotka–Volterra Models Converge to Super-Stable Processes

Hui He ()
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Hui He: Beijing Normal University

Journal of Theoretical Probability, 2011, vol. 24, issue 3, 688-728

Abstract: Abstract 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).

Keywords: Super-stable process; Lotka–Volterra; Voter model; Domain of attraction; Stable law; Stable random walk; 60K35; 60G57; 60F17; 60J80 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-009-0273-z

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