The Annals of Applied Probability

Rescaled interacting diffusions converge to super Brownian motion

J. Theodore Cox and Achim Klenke

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Abstract

Super Brownian motion is known to occur as the limit of properly rescaled interacting particle systems such as branching random walk, the contact process and the voter model.

In this paper we show that certain linearly interacting diffusions converge to super Brownian motion if suitably rescaled in time and space. The results comprise nearest neighbor interaction as well as long range interaction.

Article information

Source
Ann. Appl. Probab., Volume 13, Number 2 (2003), 501-514.

Dates
First available in Project Euclid: 18 April 2003

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1050689591

Digital Object Identifier
doi:10.1214/aoap/1050689591

Mathematical Reviews number (MathSciNet)
MR1970274

Zentralblatt MATH identifier
1030.60090

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G57: Random measures
Secondary: 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Martingale problem spatially rescaled particle systems diffusion limit long range limit

Citation

Cox, J. Theodore; Klenke, Achim. Rescaled interacting diffusions converge to super Brownian motion. Ann. Appl. Probab. 13 (2003), no. 2, 501--514. doi:10.1214/aoap/1050689591. https://projecteuclid.org/euclid.aoap/1050689591


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References

  • [1] BAILLON, J.-B., CLÉMENT, PH., GREVEN, A. and DEN HOLLANDER, F. (1995). On the attracting orbit of a non-linear transformation arising from renormalization of hierarchically interacting diffusions. I. The compact case. Canad. J. Math. 47 3-27.
  • [2] COX, J. T., DURRETT, R. and PERKINS, E. A. (2000). Rescaled voter models converge to super-Brownian motion. Ann. Probab. 28 185-234.
  • [3] COX, J. T., FLEISCHMANN, K. and GREVEN, A. (1996). Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Related Fields 105 513-528.
  • [4] COX, J. T. and GREVEN, A. (1994). Ergodic theorems for infinite sy stems of locally interacting diffusions. Ann. Probab. 22 833-853.
  • [5] DAWSON, D. A. (1993). Measure-valued Markov processes. École d'Été de Probabilités de Saint-Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, Berlin.
  • [6] DURRETT, R. and PERKINS, E. A. (1999). Rescaled contact processes converge to superBrownian motion in two or more dimensions. Probab. Theory Related Fields 114 309- 399.
  • [7] LIGGETT, T. M. and SPITZER, F. (1981). Ergodic theorems for coupled random walks and other sy stems with locally interacting components. Z. Wahrsch. Verw. Gebiete 56 443- 468.
  • [8] MUELLER, C. and TRIBE, R. (1995). Stochastic P.D.E.'s arising from the long range contact and long range voter processes. Probab. Theory Related Fields 102 519-545.
  • [9] SHIGA, T. (1980). An interacting sy stem in population genetics. J. Math. Ky oto Univ. 20 213- 242.
  • [10] SHIGA, T. and SHIMIZU, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Ky oto Univ. 20 395-416.
  • Sy RACUSE, NEW YORK 13244 E-MAIL: jtcox@gumby.sy r.edu MATHEMATISCHES INSTITUT UNIVERSITÄT ZU KÖLN WEy ERTAL 86-90 50931 KÖLN GERMANY E-MAIL: math@aklenke.de