Conditioned random walks — paths pinned to endpoints or constrained to stay positive
Bridge: B(t|B(1)=0)
Covariance: K(s,t) = min(s,t) − st
Excursion: bridge conditioned on B(t) ≥ 0 ∀t (Itô-McKean)
Max of bridge ~ Kolmogorov-Smirnov distribution
A Brownian bridge is a standard Brownian motion B(t) conditioned on returning to 0 at time T=1. Its mean is E[B(t)] = 0 and covariance K(s,t) = min(s,t) − st. It can be constructed explicitly as B_bridge(t) = B(t) − t·B(1), or via the SDE dX = −X/(1−t) dt + dW.
The Brownian excursion adds the constraint B(t) ≥ 0 for all t ∈ [0,1]. It arises in combinatorics (ballot problems, Catalan numbers), queueing theory (busy periods), and the Vervaat transformation. Its distribution at the midpoint t=1/2 is a half-normal. The area under an excursion has the Airy distribution — famously appearing in the analysis of algorithms (Quicksort, BST).
The arc-sine law states that the fraction of time a Brownian motion spends positive follows the arc-sine distribution p(x) = 1/(π√(x(1−x))) — peaks at 0 and 1, not uniform! Lévy proved this in 1939.