## On the topological boundary of the range of super-Brownian motion.(English)Zbl 1471.60072

Summary: 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 $\text{dim}(U\cap\partial\mathcal{R})=\begin{cases}4-2\sqrt{2}\approx1.17 &\text{if }d=2,\\ \frac{9-\sqrt{17}}{2}\approx2.44 &\text{if }d=3.\end{cases}$ This improves recent results of the last two authors by working with the actual topological boundary [Probab. Theory Relat. Fields 174, No. 3–4, 821–885 (2019; Zbl 1448.60145)], rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.

### MSC:

 60G57 Random measures 60J68 Superprocesses 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J65 Brownian motion 60J80 Branching processes (Galton-Watson, birth-and-death, etc.)

### Keywords:

super-Brownian motion; Hausdorff dimension

Zbl 1448.60145
Full Text:

### References:

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