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An Integral Test on Time-Dependent Local Extinction for Super-coalescing Brownian Motion with Lebesgue Initial Measure

Author

Listed:
  • Hui He

    (Beijing Normal University)

  • Zenghu Li

    (Beijing Normal University)

  • Xiaowen Zhou

    (Concordia University)

Abstract

This paper concerns the almost sure time-dependent local extinction behavior for super-coalescing Brownian motion X with (1+β)-stable branching and Lebesgue initial measure on ℝ. We first give a representation of X using excursions of a continuous-state branching process and Arratia’s coalescing Brownian flow. For any nonnegative, nondecreasing, and right-continuous function g, let $$\tau:=\sup\bigl\{t\geq0: X_t\bigl(\bigl[-g(t),g(t)\bigr]\bigr )>0 \bigr \}.$$ We prove that ℙ{τ=∞}=0 or 1 according as the integral $\int_{1}^{\infty}\! g(t)t^{-1-1/\beta} dt$ is finite or infinite.

Suggested Citation

  • Hui He & Zenghu Li & Xiaowen Zhou, 2013. "An Integral Test on Time-Dependent Local Extinction for Super-coalescing Brownian Motion with Lebesgue Initial Measure," Journal of Theoretical Probability, Springer, vol. 26(1), pages 31-45, March.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:1:d:10.1007_s10959-011-0372-5
    DOI: 10.1007/s10959-011-0372-5
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    References listed on IDEAS

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    1. 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.
    2. 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.
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