Brownian bricklayer: a random space-filling curve

Let $(B(t),t\ge 0)$ denote the standard, one-dimensional Wiener process and $(\mathrm{\ell }(y,t);y\in \mathbb{R},t\ge 0)$ its local time at level $y$ up to time $t$. Then $((B(t),\mathrm{\ell }(B(t),t)),t\ge 0)$ is a random path that fills the upper half-plane, covering one unit of area per unit time.