= Radial restriction for a conditioned Brownian path
{title2=$\mathbb P(B\subset U)=|\Phi_U'(1)|$}
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 $|\Phi_U'(1)|$. 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.
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