Brownian motion conditioned to exit at a boundary point ($h(z)=P_D(z,a)/P_D(z_0,a)$)

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.

For a Brownian path in the unit disc from $0$ conditioned to exit at $1$, let ${\Phi }_U:U\to D$ fix $0$ and $1$, and suppose $U$ agrees with the disc near $1$. The probability of avoiding $D\setminus U$ is the ratio of the boundary Poisson-kernel densities, hence $|{\Phi }_U^{\prime }(1)|$. Given avoidance, the conformally mapped path has the original law up to the conformal Brownian clock. There is no conformal-radius factor at the interior starting point.