 Doob h-transform

For a killed Markov process and a positive harmonic function $h$, its Doob h-transform weights the path law through a stopping time $\tau$ by $h(X_{\tau })/h(X_0)$, with localization when necessary. It describes conditioning on a boundary behaviour of probability zero. In the complex upper half-plane, $h(z)=\mathrm{Im}z$ conditions killed planar Brownian motion to remain in that half-plane.