= Brownian motion conditioned to exit at a boundary point
{c}
{title2=$h(z)=P_D(z,a)/P_D(z_0,a)$}
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 $\tfrac12\Delta+\nabla\log h\cdot\nabla$. Conditioning on shrinking boundary arcs yields the same localized laws and specifies a canonical version of an otherwise probability-zero conditioning event.
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