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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.)

Citations:

Zbl 1448.60145
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References:

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