= Recurrent disk occupation makes an entire conformal clock infinite
{title2=$\int_0^\infty|f'(Z_s)|^2ds=\infty$}
For a nonconstant <entire function> choose a point with nonzero derivative and a small closed disk around it where $|f'|^2$ has a positive lower bound. <recurrence of planar Brownian motion> gives infinitely many successive crossings from a concentric inner circle to the disk boundary. Their durations are positive iid variables by strong Markov and <rotational invariance of Brownian motion>. The strong law applied to their bounded positive truncations makes their sum infinite. Hence the conformal clock is infinite and the analytic Brownian image, under <time change of a continuous process>, has full infinite Brownian lifetime. Centering the disk at a noncritical point is essential.
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