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

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 $U$, $\partial \mathcal{R}\cap U\ne \varnothing$ implies $$\text{dim}(U\cap \partial \mathcal{R})=\{\begin{array}{ll}4-2\sqrt{2}\approx 1.17& \text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44& \text{ if }d=3.\end{array}$$ 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.