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

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 $U$, $U\cap\partial\mathcal{R}\neq\varnothing$ implies \[\operatorname{dim}(U\cap\partial\mathcal{R})=\begin{cases}4-2\sqrt{2}\approx1.17\quad\text{if }d=2,\\\frac{9-\sqrt{17}}{2}\approx2.44\quad\text{if }d=3.\end{cases}\] This improves recent results of the last two authors 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.