An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure

Details An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure An integral test on time dependent local extinction for super-coalescing Brownian motion with Lebesgue initial measure

2009-11-04

arxiv.org/abs/0911.0774v2

Mathematics

Probability

14 pages

Scientific paper

This paper concerns the almost sure time dependent local extinction behavior for super-coalescing Brownian motion $X$ with $(1+\beta)$-stable branching and Lebesgue initial measure on $\bR$. 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 $g$, put \tau:=\sup \{t\geq 0: X_t([-g(t),g(t)])>0 \}. We prove that $\bP\{\tau=\infty\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/\beta} dt$ is finite or infinite.

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