= Finite exit from a conformal image of a bounded planar domain
{title2=$\mathbb P(T'<\infty)=1$}
A <planar Brownian motion> exits a conformal image of a bounded planar domain in finite time <almost surely>. If the image lifetime were infinite, the inverse derivative is bounded away from zero on a small interior disc. <Recurrence of planar Brownian motion> and the <Strong Markov property> give infinite <Brownian occupation time> there, forcing the inverse <conformal Brownian clock> to diverge. It cannot exceed the finite original exit time. The expected image exit time can nevertheless be infinite.
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