Past exam of the Brownian motion in complex analysis course of the University of Oxford / 2011 / 1 / b

Apply part (a) to the entire function $f(z)=e^{i\theta }z$. Its derivative has modulus one, so its conformal Brownian clock is $A_t=t$. Thuswhich is a planar Brownian motion starting from the rotated initial point. In particular, if the initial point is zero, its law is unchanged. This is rotational invariance of planar Brownian motion.

Start a planar Brownian motion $B$ at zero and apply part (a) to the entire function $f(z)=e^z$. ThenThe left side never vanishes. The clock is continuous, starts at zero, and tends to infinity almost surely, so its range is all nonnegative times. Consequently the complex Brownian motion $1+{\tilde{Z}}_s$ never reaches zero at any time $s$. Every planar Brownian motion started at one has that same law, henceThis is the complex exponential construction of planar Brownian motion and proves that a fixed point is a polar point for planar Brownian motion. The divergence of the clock is essential: it makes this a conclusion about all Brownian times.