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

TL;DR

This paper investigates the almost sure time-dependent local extinction behavior of super-coalescing Brownian motion with stable branching, establishing a zero-one law based on an integral criterion involving the initial measure and coalescence dynamics.

Contribution

It provides a new integral test for local extinction in super-coalescing Brownian motion with Lebesgue initial measure, linking extinction probability to an explicit integral condition.

Findings

Extinction probability is zero or one depending on the finiteness of a specific integral.

Representation of the process via excursions of a continuous state branching process and coalescing flow.

Established a criterion for almost sure local extinction based on initial measure and process parameters.

Abstract

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.