On the large scale behavior of super-Brownian motion in three dimensions with a single point source

TL;DR

This paper investigates the large-scale behavior and scaling properties of a super-Brownian motion in three dimensions with a point source, building on prior analytical constructions involving heat equations with point potentials.

Contribution

It analyzes the scaling and large-scale mean behavior of super-Brownian motion in 3D with a point source, extending previous existence results.

Findings

Characterization of the process's scaling behavior

Analysis of the large-scale mean behavior in 3D

Extension of analytical methods to understand super-Brownian motion with point sources

Abstract

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean.