Let $(B(t),\,t\ge0)$ denote the standard, one-dimensional Wiener process and
$(\ell(y,t);\, y\in\mathbb{R},\, t\ge0)$ its local time at level $y$ up to time
$t$. Then $\big( (B(t),\, \ell(B(t),t)),\; t\ge0 \big)$ is a random path that
fills the upper half-plane, covering one unit of area per unit time.