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

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.