On the topological boundary of the range of super-Brownian motion-extended version

TL;DR

This paper characterizes the Hausdorff dimension of the topological boundary of the range of super-Brownian motion in dimensions 2 and 3, providing precise local dimension results and improving previous work.

Contribution

It establishes the Hausdorff dimension of the topological boundary of super-Brownian motion's range in dimensions 2 and 3, focusing on local boundary properties.

Findings

Dimension of boundary in 2D is approximately 1.17.

Dimension of boundary in 3D is approximately 2.44.

Results improve previous bounds by analyzing the actual topological boundary.

Abstract

We show that if $\partial\mathcal{R}$ is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$ implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases} 4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{ if }d=3. \end{cases}$$ This improves recent results of the last two authors (arxiv:1711.03486) by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.