= Solution
Let $D$ be the unit disc and normalize its positive <Poisson kernel> at the boundary point $1$ by
$$
h(z)=\frac{1-|z|^2}{|1-z|^2},\qquad h(0)=1.
$$
It is harmonic in $D$. Let $p_D(t,z,w)$ be the killed <Brownian transition density>, for <planar Brownian motion> with generator $\tfrac12\Delta$. The <Doob h-transform> has sub-Markov transition density
$$
\boxed{p_D^h(t,z,w)=p_D(t,z,w)\frac{h(w)}{h(z)}.}
$$
Its missing mass represents paths already absorbed at the distinguished boundary point. Equivalently, for a stopping time $\sigma$ before exit from a compact subdomain, its law is weighted relative to ordinary killed <Brownian motion> by $h(B_\sigma)/h(B_0)$. These stopped laws are consistent and define a diffusion up to its lifetime. Its generator is
$$
\mathcal L^hf=\frac1h\frac12\Delta(hf)=\frac12\Delta f+\nabla\log h\cdot\nabla f.
$$
This is a rigorous <Brownian motion conditioned to exit at a boundary point>, not conditioning on an event of positive probability. Exhaustion of $D$ shows that its terminal boundary limit is $1$; its lifetime is finite. For example its expected lifetime from $0$ is $\int_DG_D^{\mathrm{BM}}(0,w)h(w)\,dw$, which is finite: the Green function vanishes linearly near the smooth boundary and cancels the Poisson-kernel singularity there, while its logarithmic singularity at $0$ is integrable.
One can also condition ordinary <Brownian motion> on exiting through an arc $I_\epsilon$ about $1$, and then let its length decrease to zero. The conditional harmonic functions $\omega_D(z,I_\epsilon)/\omega_D(0,I_\epsilon)$ converge locally uniformly to $h(z)$. Thus their stopped conditional laws converge to this same diffusion. Both constructions specify the conditioning unambiguously.
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