Brownian bricklayer: A random space-filling curve

Let (B(t),t≥0) denote the standard, one-dimensional Wiener process and (ℓ(y,t);y∈R,t≥0) its local time at level y up to time t. Then ((B(t),ℓ(B(t),t)),t≥0) is a random, continuous path that fills the upper half-plane, covering one unit of area per unit time.