= Divergence of the conformal Brownian clock for an entire function
{title2=$f\text{ nonconstant entire}\implies\int_0^t|f'(Z_s)|^2ds\to\infty\text{ a.s.}$}
For a nonconstant <entire function>, its derivative is nonzero somewhere and therefore bounded away from zero on a small disc. <Recurrence of planar Brownian motion> and the <Strong Markov property> imply infinite total occupation time in that disc: repeat entrances into a smaller concentric disc after exits from the larger one. On each entrance there is a uniformly positive chance of remaining inside the larger disc for a fixed positive duration. Iterating the conditional failure bound gives infinitely many such sojourns almost surely.
The displayed integral is bounded below by a positive constant times this occupation time, so it diverges. The same conclusion follows by comparing with the <divergence of a positive planar Brownian occupation integral>. At every finite time the clock is finite, because the Brownian path has compact range and $f'$ is continuous. Thus the <conformal Brownian clock> covers every nonnegative image time; <conformal invariance of planar Brownian motion> then yields a Brownian image with no missing finite or infinite lifetime.
Back to article page
