A Brownian bridge is a Wiener process conditioned to start and end at fixed points. Unlike free Brownian motion, it must return — creating a "pinched" process with unique statistical properties.
Formula: B_bridge(t) = W(t) − (t/T)·W(T) + (t/T)·b where W is standard Brownian motion. The variance is Var[B(t)] = σ²·t(T−t)/T — maximal at t=T/2, zero at endpoints.