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.