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Strong clumping of critical space-time branching models in subcritical dimensions

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  • Dawson, Donald A.
  • Fleischmann, Klaus

Abstract

For critical spatially homogeneous branching processes of finite intensity the following dichotomy is well-known: convergence to non-trivial steady states, or local extinction. In the latter case the underlying phenomenon is the growth of large clumps at spatially rare sites. For this situation a precise description is given in terms of a scaling limit theorem provided that the dimension of the ambient space is small enough. In fact, a space-time-mass scaling limit exists and is a critical measure-valued branching process without a motion component. The clumps are located at Poissonian points and their sizes evolve according to critical continuous-state Galton-Watson processes. The spatial irregularities (intermittency) will grow in the sense that clumps will disappear as time increases in spite of the fact that the overall density remains constant in time.

Suggested Citation

  • Dawson, Donald A. & Fleischmann, Klaus, 1988. "Strong clumping of critical space-time branching models in subcritical dimensions," Stochastic Processes and their Applications, Elsevier, vol. 30(2), pages 193-208, December.
    • Handle: RePEc:eee:spapps:v:30:y:1988:i:2:p:193-208
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    Citations

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    Cited by:

    1. Donald A. Dawson & Klaus Fleischmann, 1997. "A Continuous Super-Brownian Motion in a Super-Brownian Medium," Journal of Theoretical Probability, Springer, vol. 10(1), pages 213-276, January.
    2. I. Kaj & S. Sagitov, 1998. "Limit Processes for Age-Dependent Branching Particle Systems," Journal of Theoretical Probability, Springer, vol. 11(1), pages 225-257, January.
    3. Fatheddin, Parisa & Xiong, Jie, 2015. "Large deviation principle for some measure-valued processes," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 970-993.
    4. Klaus Fleischmann & Achim Klenke & Jie Xiong, 2006. "Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process," Journal of Theoretical Probability, Springer, vol. 19(3), pages 557-588, December.
    5. D. A. Dawson & Z. Li & X. Zhou, 2004. "Superprocesses with Coalescing Brownian Spatial Motion as Large-Scale Limits," Journal of Theoretical Probability, Springer, vol. 17(3), pages 673-692, July.
    6. Zhou, Xiaowen, 2008. "A zero-one law of almost sure local extinction for (1+[beta])-super-Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 1982-1996, November.
    7. Stanislav Molchanov & Joseph Whitmeyer, 2017. "Stationary distributions in Kolmogorov-Petrovski- Piskunov-type models with an infinite number of particles," Mathematical Population Studies, Taylor & Francis Journals, vol. 24(3), pages 147-160, July.
    8. Dawson, Donald A. & Fleischmann, Klaus, 1997. "Longtime behavior of a branching process controlled by branching catalysts," Stochastic Processes and their Applications, Elsevier, vol. 71(2), pages 241-257, November.

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