The total mass of super-Brownian motion upon exiting balls and Sheu's compact support condition

TL;DR

This paper analyzes the total mass of super-Brownian motion exiting expanding balls, characterizing its branching mechanism, and linking it to Sheu's compact support condition through Grey's classical criterion.

Contribution

It provides a detailed characterization of the time-dependent branching mechanism of super-Brownian motion's total mass and connects Sheu's support condition to Grey's criterion.

Findings

Converges to the branching mechanism of 1D super-Brownian motion crossing levels

Identifies Sheu's compact support condition as Grey's classical criterion

Characterizes the process as a time-inhomogeneous continuous-state branching process

Abstract

We study the total mass of a d-dimensional super-Brownian motion as it first exits an increasing sequence of balls. The process of the total mass is a time-inhomogeneous continuous-state branching process, where the increasing radii of the balls are taken as the time parameter. We are able to characterise its time-dependent branching mechanism and show that it converges, as time goes to infinity, towards the branching mechanism of the total mass of a one-dimensional super-Brownian motion as it first crosses above an increasing sequence of levels. Our results allow us to identify the compact support criterion given in Sheu (1994) as a classical Grey condition (1974) for the aforementioned limiting branching mechanism.