Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2007 / B / 7

First, planar Brownian motion started away from zero almost surely never hits zero. To see this without confusing it with neighborhood recurrence, stop in an annulus with inner radius $ϵ$ and fixed outer radius $R>|Z_0|$. The preceding logarithmic calculation gives probability $\log (R/|Z_0|)/\log (R/ϵ)$ of reaching the inner circle first. A hit of zero before the outer exit would require all these inner hits, so its probability is zero as $ϵ\downarrow 0$. On every finite path interval the continuous path is bounded; a countable union over integer outer radii and integer times proves the all-time assertion.

Consequently the integrand $1/Z$ is locally square integrable: on every compact time interval the nonvanishing continuous path has a positive minimum modulus. Write $Z=X+iY$ with independent standard real coordinates. Complex quadratic covariation satisfies $d[Z,Z]=dt-dt+2id[X,Y]=0$, so analytic Itô's lemma has no second-derivative drift. For $L=\int Z^{-1}dZ$ one likewise has $d[L,L]=0$, and $d[e^{-L},Z]=0$. The product rule therefore givesIts initial value is $Z_0$, proving, simultaneously for all times by continuity,Thus $\Re L_t=\log (|Z_t|/|Z_0|)$ and $\Im L_t$ is the continuous accumulated change of complex argument of the path about zero, measured in radians. It is a winding angle, without reduction modulo $2\pi$. A single-valued logarithm on the whole punctured plane is unnecessary; the path has a continuous stochastic logarithm of planar Brownian motion.

Write $Z_0=r_0{e}^{i{\alpha }_0}$ with $0<r_0<1$ and choose its initial argument once. The exponential map in the source is a covering of the punctured disc, rather than a bijective map onto the full disc. Its pathwise inverse is the continuous liftFor $t<T_{\mathbb{D}}$ this lies in the upper half-plane, and at the disc exit it lies on the real line. Local conformal invariance of planar Brownian motion, or the quadratic variation of its analytic derivative $-i/Z$, shows that $W_t$ is planar Brownian motion run at the clockThe disc exit time is finite almost surely. The path avoids zero on the compact interval through its exit, so its minimum modulus there is positive and $A_{T_{\mathbb{D}}}<\infty$. Before this terminal clock time the lifted imaginary coordinate is positive, and at it the coordinate is zero. Hence it is exactly the upper-half-plane exit time of the time-changed Brownian motion, not an additional arbitrary stopping time.

The starting point of the lift is ${\alpha }_0+ib$ with $b=-\log r_0>0$. Its real exit coordinate has Cauchy distribution $b/[\pi ((s-{\alpha }_0{)}^2+b^2)]$. Since that coordinate is ${\alpha }_0+{\theta }_{T_{\mathbb{D}}}$, we obtain the stronger exact lawIts first absolute moment diverges:There is no contradiction with the terminal point lying on the compact unit circle: reducing its argument modulo $2\pi$ loses the number and direction of turns accumulated before exit.