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

Write $Z_t=z+B_t^1+i{B}_t^2$, where the coordinates are independent standard Brownian motions, and write $f=u+iv$. The Cauchy-Riemann equations imply $\Delta u=\Delta v=0$. Itô's formula therefore makescontinuous local martingales. Their quadratic variations and quadratic covariation areIndeed the two gradient norms are equal to $|f^{\prime }|$, and their dot product vanishes by the Cauchy-Riemann equations.

The clock is finite at every finite time, since a continuous Brownian path has compact range there and $f^{\prime }$ is continuous. To show its lifetime is infinite, choose a small disc on which $|f^{\prime }{|}^2\ge c>0$; such a disc exists because the entire function is nonconstant. A supporting recurrence result is that planar Brownian motion spends infinite total time in every nonempty open disc almost surely. One proof combines recurrent returns to a smaller disc with the strong Markov property: each visit has a uniformly positive chance of lasting a fixed positive time in the larger disc. Equivalently, use the divergence of a positive planar Brownian occupation integral. ThereforeFinally the common-clock time change of orthogonal local martingales states that two continuous local martingales starting at zero with common bracket $A$, zero cross bracket, and $A_{\infty }=\infty$ are the coordinates of a planar Brownian motion evaluated at $A_t$. Applying it gives a planar Brownian motion $\tilde{Z}$ starting at zero withThis proves conformal invariance of planar Brownian motion in the entire-function setting, including the divergence of the conformal Brownian clock for an entire function.

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.