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Occupation Time Fluctuations in Branching Systems

Author

Listed:
  • D. A. Dawson

    (The Fields Institute)

  • L. G. Gorostiza

    (Centro de Investigación y de Estudios Avanzados)

  • A. Wakolbinger

    (Johann Wolfgang Goethe–Universität)

Abstract

We consider particle systems in locally compact Abelian groups with particles moving according to a process with symmetric stationary independent increments and undergoing one and two levels of critical branching. We obtain long time fluctuation limits for the occupation time process of the one- and two-level systems. We give complete results for the case of finite variance branching, where the fluctuation limits are Gaussian random fields, and partial results for an example of infinite variance branching, where the fluctuation limits are stable random fields. The asymptotics of the occupation time fluctuations are determined by the Green potential operator G of the individual particle motion and its powers G 2,G 3, and by the growth as t→∞ of the operator $$G_t = \int_0^t {T_s } ds$$ and its powers, where T t is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric α-stable Lévy process in $$\mathbb{R}^d (0

Suggested Citation

  • D. A. Dawson & L. G. Gorostiza & A. Wakolbinger, 2001. "Occupation Time Fluctuations in Branching Systems," Journal of Theoretical Probability, Springer, vol. 14(3), pages 729-796, July.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:3:d:10.1023_a:1017597107544
    DOI: 10.1023/A:1017597107544
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    References listed on IDEAS

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    1. Deuschel, Jean-Dominique & Wang, Kongming, 1994. "Large deviations for the occupation time functional of a Poisson system of independent Brownian particles," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 183-209, August.
    2. Gorostiza, L. G. & Lopez-Mimbela, J. A., 1994. "An occupation time approach for convergence of measure-valued processes, and the death process of a branching system," Statistics & Probability Letters, Elsevier, vol. 21(1), pages 59-67, September.
    3. Grabner, Peter J. & Woess, Wolfgang, 1997. "Functional iterations and periodic oscillations for simple random walk on the Sierpinski graph," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 127-138, July.
    4. Dawson, Donald A. & Hochberg, Kenneth J. & Vinogradov, Vladimir, 1996. "High-density limits of hierarchically structured branching-diffusing populations," Stochastic Processes and their Applications, Elsevier, vol. 62(2), pages 191-222, July.
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    Cited by:

    1. Cheng, Ziling, 2024. "Occupation times for age-structured branching processes," Statistics & Probability Letters, Elsevier, vol. 211(C).
    2. D. A. Dawson & L. G. Gorostiza, 2018. "Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space," Journal of Theoretical Probability, Springer, vol. 31(1), pages 494-526, March.

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