Past exam of the stochastic differential equations and Brownian motion in complex analysis course of the University of Oxford / 2010 / B (Brownian Motion in Complex Analysis)

There is a printed domain error: $\mathrm{Re}z>0$ defines the right half-plane. Taken literally, continuity forces $\mathrm{Re}{Z}_{T_H}=0$ almost surely, so its law is a point mass at zero. The displayed density belongs to the intended upper half-plane $\mathrm{Im}z>0$, with $b>0$. We prove that intended assertion explicitly.

Write $Z_t=a+X_t+i(b+Y_t)$ for independent standard Brownian motions. The exit time $T$ is the first time $Y$ hits $-b$, and depends only on $Y$. Its density, by the supplied Brownian first-passage time law, is $b(2\pi t^3{)}^{-1/2}{e}^{-b^2/(2t)}$. Conditional on $T=t$, the real exit coordinate is normally distributed with mean $a$ and variance $t$. Independence therefore givesWith $u=[b^2+(x-a{)}^2]/(2t)$, the integral is $2/[b^2+(x-a{)}^2]$. Thus the Cauchy exit law from a Brownian half-plane isFor the literal right half-plane, the corresponding nondegenerate coordinate is $\mathrm{Im}{Z}_{T_H}$, whose density is $a/\{\pi [a^2+(y-b{)}^2]\}$ for $a>0$.

If $f:D\to D^{\prime }$ is a conformal map, the image of planar Brownian motion in $D$ is Brownian motion in $D^{\prime }$ after the increasing clock changeMore precisely, $f(Z_{A^{-1}(u)})$ is planar Brownian motion, starting at $f(Z_0)$ and run up to the relevant exit time. The Itô formula and the Cauchy-Riemann equations give image-coordinate local martingales with common bracket $A$ and zero cross bracket; the Lévy characterization of multidimensional Brownian motion identifies the time-changed process. In particular, conformal invariance of planar Brownian motion transports exit locations and harmonic measure, while changing traversal speed.

For $z=x+iy$ in the strip,so $\mathrm{Im}f(z)>0$. This is a bijective conformal map onto the upper half-plane. One way to verify bijectivity is to factor it as $w=e^{\pi z/a}$, followed by $(w+w^{-1})/2$. The first map gives $\mathrm{Im}w>0$, $|w|>1$. The two inverse roots of the second map have product one; for any point in the upper half-plane exactly one root has modulus greater than one, and its imaginary part is positive because $\mathrm{Im}[(w+w^{-1})/2]=(1-|w{|}^{-2})\mathrm{Im}w/2$.

The three boundary pieces map as follows: the lower edge goes to $(1,\infty )$, the upper edge to $(-\infty ,-1)$, and the vertical edge to $(-1,1)$. Thus the desired exit event becomes exit from the upper half-plane to the left of $-1$. WriteBy conformal invariance of harmonic measure and the Cauchy exit law from a Brownian half-plane,The strip exit is finite almost surely, since its imaginary coordinate exits $(0,a)$ in finite time; corners have probability zero because individual points are polar points for planar Brownian motion. This is the Top-edge exit probability of a Brownian half-strip.

Let $\alpha =\pi q/a\in (0,\pi )$. With $q$ and $a$ fixed,The principal arctangent satisfies $\arctan (\cot \alpha )=\pi /2-\alpha$ throughout this interval. Substitution givesFar from the vertical boundary, the limit agrees with the one-dimensional Brownian motion probability of reaching height $a$ before zero.

The disk exit time is finite almost surely; for example, either coordinate of the planar Brownian motion eventually leaves $(-1,1)$. By path continuity, the exit lies on the unit circle. Starting at the origin, rotational invariance of planar Brownian motion makes the exit law invariant under every circle rotation. The unique rotation-invariant probability measure is normalized arc length. Consequently $Z_{T_D}$ is uniform on the unit circle, or equivalently its angle has density $1/(2\pi )$ on $[0,2\pi )$.

By conformal invariance of planar Brownian motion, $w(Z_{T_D})$ has the uniform boundary law from part (a), because the image process starts at zero. A disk automorphism extends to an orientation-preserving boundary diffeomorphism. Choose a continuous angular lift $\varphi :\mathbb{R}\to \mathbb{R}$ satisfyingThe principal-angle version may have a branch jump, which is harmless for the density. The change of variables for probability densities yields the exit-angle density ${\varphi }^{\prime }(\theta )/(2\pi )$. Differentiating the boundary relation givesTaking absolute values proves ${\varphi }^{\prime }(\theta )=|w^{\prime }(e^{i\theta })|$. ThereforeThe angular lift explains rigorously the derivative appearing in the printed hint despite the principal-angle branch cut.

By Brownian scaling, $r^{-1}{Z}_{r^{2}t}$ is planar Brownian motion started at $w_0/r$. Its unit-disk exit angle is exactly the original radius-$r$ exit angle. Substitute $z_0=w_0/r$ into the supplied Poisson kernel for the unit disk:This density is with respect to angle $d\theta$. Its density with respect to arc length would be smaller by a factor $r$.

Put $\rho =|z|$. The triangle inequality gives $1-\rho \le |z-e^{i\theta }|\le 1+\rho$, so the Poisson kernel for the unit disk obeysIf $\rho \le r$, replace the lower bound by $(1-r)/(1+r)$ and the upper bound by $(1+r)/(1-r)$. Integrating against the positive measure $h(e^{i\theta })d\theta /(2\pi )$ givesHere $u(0)$ is exactly the uniform boundary average from part (a). Positivity permits multiplication of the pointwise kernel bounds; continuity on the compact circle ensures all integrals are finite. This is the Harnack inequality on the unit disk.

Use first that planar Brownian motion avoids a fixed point, so the path starting at $1$ never reaches zero. Its continuous logarithmic lift, initialized at zero, has real part $U_t=\log |Z_t|$ and imaginary part the continuous unwrapped angle ${\Theta }_t$. Local branches of the complex logarithm, patched along the path, and the Itô formula show that both coordinates are continuous local martingales withThe clock is finite on every compact time interval, since a continuous nonvanishing path stays a positive distance from zero there. It is continuous and strictly increasing. Moreover $\tau (\infty )=\infty$: by recurrence, planar Brownian motion spends infinite total time in a fixed disk, and inside the unit disk the clock integrand is at least one. This is the divergence of a positive planar Brownian occupation integral.

The common-clock time change of orthogonal local martingales, using the Lévy characterization of multidimensional Brownian motion, supplies independent Brownian coordinates ${\zeta }^1,{\zeta }^2$ such that $U_t+i{\Theta }_t={\zeta }_{\tau (t)}$. Exponentiating givesTo identify the inverse clock, write $A={\tau }^{-1}$. The derivative of an inverse function and the representation just obtained giveConsequentlyThe strict inequality in this generalized inverse has the same infimum as the unique inverse value, since $A$ is continuous and strictly increasing onto the half-line.

Assume $0<r<1<R$, as required for the annulus and logarithms. The logarithmic radius of planar Brownian motion is a local martingale. Stopping at $T=T_{r,R}$ keeps it between the constants $\log r$ and $\log R$, so the stopped process is a bounded martingale. The exit time is finite almost surely because the annulus is bounded and Brownian motion eventually exits its containing disk.

The optional stopping theorem followed by dominated convergence gives $\mathbb{E}\log |Z_T|=\log |Z_0|=0$. Continuity puts the exit radius at either $r$ or $R$. If $p$ is the inner-exit probability, then $p\log r+(1-p)\log R=0$. HenceStopping before taking expectations matters: the unstopped logarithmic-radius process is a strict local martingale, so one cannot assume its unrestricted expectation is constant.

The second coordinate ${\zeta }^2$ is one-dimensional Brownian motion, whose zero set is unbounded almost surely by recurrence. The clock $\tau$ and its inverse are continuous, increasing, and unbounded, so the unwrapped angle ${\Theta }_t={\zeta }_{\tau (t)}^2$ has zeros at arbitrarily large original times.

To ensure the times are stopping times for the original Brownian filtration, define directlyThe logarithmic lift makes $\Theta$ a continuous adapted process. Hitting a closed set after a stopping time is a stopping time under the usual filtration conditions, and the unbounded zero set proves inductively that each $T_n$ is finite almost surely. Since $T_n\ge T_{n-1}+1$, the sequence tends to infinity, and continuity givesGeometrically, the Brownian path returns to the positive real ray at arbitrarily late times with zero net unwrapped winding angle about the origin. The assertion is stronger than principal argument zero, which also allows any integer number of completed turns.

The long-time Spitzer winding theorem states that, for the continuous unwrapped angle ${\Theta }_t$ of planar Brownian motion started at $1$,Here $C$ has the standard symmetric Cauchy distribution. The notation means convergence in distribution: for every real $x$,Equivalently, expectations converge for every bounded continuous test function. This is a statement about the laws at large deterministic times, without asserting almost-sure convergence of sample paths. The angle is unwrapped; a principal argument is bounded and would instead give zero after division by $\log t$.