Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet
Volume 570 / 2021
Dissertationes Mathematicae 570 (2021), 1-130
MSC: Primary 60G60; Secondary 60G17, 60G15.
DOI: 10.4064/dm811-9-2021
Published online: 4 November 2021
Abstract
We first consider the additive Brownian motion process $(X(s_1,s_2),\ (s_1,s_2) \in \mathbb R^2)$ defined by $X(s_1,s_2) = Z_1(s_1) - Z_2 (s_2)$, where $Z_1$ and $Z_2 $ are two independent (two-sided) Brownian motions. We show that with probability $1$, the Hausdorff dimension of the boundary of any connected component of the random set $\{(s_1,s_2)\in \mathbb R^2: X(s_1,s_2) >0\}$ is equal to $$ \textstyle\tfrac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421. $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the non-negative quadrant.
