Doob h-transform
= Doob h-transform
{c}
{title2=$P_t^hf(x)=h(x)^{-1}P_t(hf)(x)$}
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>, $h(z)=\operatorname{Im}z$ conditions killed <planar Brownian motion> to remain in that half-plane.