Conditioned Brownian motion

Summary

A conditioned Brownian motion in a domain is the Doob h-transform of Brownian motion killed on leaving the domain, for a positive harmonic function $h$. Take for $h$ the Poisson kernel or the Martin kernel with pole at a boundary point. Then the process is Brownian motion conditioned to leave the domain at that point. Conditioned Brownian motion gives probabilistic proofs of boundary limit theorems, such as Fatou-type theorems and the relative Fatou theorem.