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

Hui He (), Zenghu Li () and Xiaowen Zhou ()
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Hui He: Beijing Normal University
Zenghu Li: Beijing Normal University
Xiaowen Zhou: Concordia University

Journal of Theoretical Probability, 2013, vol. 26, issue 1, 31-45

Abstract: 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.

Keywords: Super-coalescing Brownian motion; Almost sure local extinction; Excursion representation; Integral test; 60G57; 60J80; 60F20 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-011-0372-5

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