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

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.