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Journal of Theoretical Probability

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

Rescaled Lotka–Volterra Models Converge to Super-Stable Processes

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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 

Mathematics Subject Classification (2000)

60K35 60G57 60F17 60J80 

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References

  1. 1.
    Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354(7), 2933–2953 (2002) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bramson, M., Cox, J.T., Le Gall, J.-F.: Super-Brownian limits of voter model clusters. Ann. Probab. 29, 1001–1032 (2001) MathSciNetMATHGoogle Scholar
  3. 3.
    Bramson, M., Griffeath, D.: Asymptotics for interacting particle systems on ℤd. Z. Wahrsch. Verw. Geb. 53, 183–196 (1980) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Cox, J.T., Durrett, R., Perkins, E.A.: Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28(1), 185–234 (2000) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cox, J.T., Klenke, A.: Rescaled interacting diffusions converge to super Brownian motion. Ann. Appl. Probab. 13(2), 501–514 (2003) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cox, J.T., Perkins, E.A.: An application of the voter model–super-Brownian motion invariance principle. Ann. Inst. H. Poincaré Probab. Stat. 40(1), 25–32 (2004) MathSciNetMATHGoogle Scholar
  7. 7.
    Cox, J.T., Perkins, E.A.: Rescaled Lotka–Volterra models converge to Super-Brownian motion. Ann. Probab. 33(3), 904–947 (2005) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cox, J.T., Perkins, E.A.: Survival and coexistence in stochastic spatial Lotka–Volterra models. Probab. Theory Relat. Fields 139, 89–142 (2007) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cox, J.T., Perkins, E.A.: Renormalization of the two-dimensional Lotka–Volterra model. Ann. Appl. Probab. 18(2), 747–812 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dawson, D.A.: Measure-valued Markov processes. In: Lecture Notes in Math., vol. 1541, pp. 1–260. Springer, Berlin (1993) Google Scholar
  11. 11.
    Durrett, R., Perkins, E.A.: Rescaled contact processes converge to super-Brownian motion in two or more dimensions. Probab. Theory Relat. Fields 114(3), 309–399 (1999) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, New York (1986) MATHGoogle Scholar
  13. 13.
    Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971) MATHGoogle Scholar
  14. 14.
    He, H.: Rescaled Lotka–Volterra models converge to super stable processes (2009). arXiv:0809.4520
  15. 15.
    Le Gall, J.-F., Rosen, J.: The range of stable random walks. Ann. Probab. 19(2), 650–705 (1991) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Liggett, T.M.: Interacting Particle Systems. Springer, New York (1985) MATHGoogle Scholar
  17. 17.
    Neuhauser, C., Pacala, S.: An explicitly spatial version of the Lotka–Volterra model with interspecific competition. Ann. Appl. Probab. 9, 1226–1259 (1999) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Perkins, E.A.: Dawson–Watanabe superprocesses and measure-valued diffusions. In: Lectures on Probability Theory and Statistics (Saint-Flour, 1999). Lecture Notes in Math., vol. 1781, pp. 125–324. Springer, Berlin (2002) Google Scholar
  19. 19.
    Pruitt, W.E.: The growth of random walks and Levy processes. Ann. Probab. 9(2), 948–956 (1981) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999). English edition MATHGoogle Scholar
  21. 21.
    Sawyer, S.: A limit theorem for patch sizes in a selectively-neutral migration model. J. Appl. Probab. 16, 482–495 (1979) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Slade, G.: Scaling limits and super-Brownian motion. Not. Am. Math. Soc. 49(9), 1056–1067 (2002) MathSciNetMATHGoogle Scholar
  23. 23.
    Spitzer, F.L.: Principles of Random Walk, 2nd edn. Springer, New York (1976) MATHGoogle Scholar

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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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