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

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 $\text{bR}$. 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 $\text{bP}\{\tau =\mathrm{\infty }\}=0$ or 1 according as the integral $\int_1^\infty g(t)t^{-1-1/\beta} dt$ is finite or infinite.