Doob h-transform ($P_t^{h}f(x)=h(x{)}^{-1}{P}_t(hf)(x)$)

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, $h(z)=\mathrm{Im}z$ conditions killed planar Brownian motion to remain in that half-plane.

If $h>0$ is smooth and $Z_t=h(X_t,t)$ is the unit-mean density martingale of a locally absolutely continuous probability measure relative to Wiener measure, then $h_t+\frac{1}{2}{h}_{xx}=0$. The Itô formula gives $Z=\mathcal{E}(\int (h_x/h)(X_s,s)d{X}_s)$ when $Z_0=1$. The true-martingale hypothesis is already supplied by the density process. On each finite horizon the Girsanov theorem makes $X-\int (h_x/h)(X_s,s)ds$ Brownian under the new measure. Positivity provides local equivalence; global infinite-horizon equivalence and the Novikov condition are not additional requirements.

Killed planar Brownian motion conditioned to exit at a smooth boundary point $a$ is the Doob h-transform by its positive Poisson kernel, normalized at the starting point. Its generator is $\frac{1}{2}\Delta +\nabla \log h\cdot \nabla$. Conditioning on shrinking boundary arcs yields the same localized laws and specifies a canonical version of an otherwise probability-zero conditioning event.

Consider a path of planar Brownian motion in the unit disc $D$, started at $0$ and conditioned to exit at $1$. Let $U\subseteq D$ be a simply connected domain containing $0$ and agreeing with $D$ near $1$. Let ${\Phi }_U:U\to D$ be a conformal map normalized by ${\Phi }_U(0)=0$ and ${\Phi }_U(1)=1$. ThenThe conditional probability is the ratio of the boundary Poisson kernel densities for exit at $1$ in $U$ and in $D$. The conformal transformation rule for the Poisson kernel contributes the boundary derivative $|{\Phi }_U^{\prime }(1)|$; the normalized interior image is still $0$, so no additional conformal-radius factor appears. Given avoidance of $D\setminus U$, the mapped path has the original conditioned law, with the conformal Brownian clock accounting for its time parametrization.