# Hausdorff dimension of the boundary of bubbles of additive Brownian motion and of the Brownian sheet

## TL;DR

This paper determines the Hausdorff dimension of the boundary of bubbles in additive Brownian motion and Brownian sheet, revealing a precise fractal dimension for these boundary sets with probability one.

## Contribution

It establishes the exact Hausdorff dimension of the boundary of bubbles in additive Brownian motion and Brownian sheet, a previously unknown fractal property.

## Key findings

- Hausdorff dimension of boundary is approximately 1.421
- Dimension result holds for both additive Brownian motion and Brownian sheet
- Boundary sets are fractal with a specific, calculable dimension

## 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 one, 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 $$ \frac{1}{4}\left(1 + \sqrt{13 + 4 \sqrt{5}}\right) \simeq 1.421\, . $$ Then the same result is shown to hold when $X$ is replaced by a standard Brownian sheet indexed by the nonnegative quadrant.