The dimension of the boundary of super-Brownian motion

TL;DR

This paper determines the Hausdorff dimension of the boundary of super-Brownian motion in dimensions 1, 2, and 3, revealing specific fractal dimensions for each case.

Contribution

It provides exact Hausdorff dimensions of the boundary of super-Brownian motion in low dimensions, advancing understanding of its geometric properties.

Findings

Dimension is 0 for d=1

Dimension is 4-2√2 for d=2

Dimension is (9-√17)/2 for d=3

Abstract

We show that the Hausdorff dimension of the boundary of $d$-dimensional super-Brownian motion is $0$, if $d=1$, $4-2\sqrt2$, if $d=2$, and $(9-\sqrt{17})/2$, if $d=3$.