Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 203 / 4

A conformal automorphism of the upper half-plane is a real Möbius transformationwith coefficients determined up to a common nonzero factor. The boundary interval $(\infty ,1)$ is the oriented real boundary arc from infinity to $1$, namely the finite real interval $(-\infty ,1)$. The map must take the left endpoint $-1$ to infinity and the right endpoint infinity to $1$, while fixing zero. These three boundary values determine the Möbius transformation uniquely:Its determinant is positive, $\mathrm{Im}\varphi (z)=\mathrm{Im}z/|1+z{|}^2>0$, and on $(-1,\infty )$ its derivative is $(1+x{)}^{-2}>0$. Its endpoint limits are $-\infty$ at $-1+$ and $1$ at $+\infty$, so it has exactly the required boundary extension.

The scale is precisely the inverse conformal map:By definition, a chordal SLE in a specified scale in $(\mathbb{H},0,1)$ is a curve whose image under $\sigma$ is the half-plane-capacity-parameterized SLE from zero to infinity. Hereand the right-hand side is the given ${\mathrm{SLE}}_6$. Equivalently, this is Conformal invariance of SLE, with the target point transported from infinity to $1$. Thus $\tilde{\gamma }$ is ${\mathrm{SLE}}_6$ in $(\mathbb{H},0,1)$ in scale $\sigma$, using the original parameter $t$. This scaled parameter is distinguished from the ordinary half-plane-capacity parameterization of its image hulls used next.

Put ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$ and $p_t={\varphi }_t^{\prime }({\xi }_t)$. The given conformal change of half-plane capacity makes the image mapping-out equationIndeed its capacity speed is $2p_t^2$, and the stated Loewner local growth property identifies the indicated single driver. For $t<T$, differentiate the conjugacy identityat a fixed original point. The Chordal Loewner equation then gives the conformal conjugacy derivative for the chordal Loewner equationThe change of variable $z=g_t(w)$ covers the whole upper half-plane. The conformal automorphism of the upper half-plane ${\varphi }_t$ is a real Möbius transformation. Its pole is the image of the still-unswallowed boundary point $-1$, so it is holomorphic in a neighbourhood of ${\xi }_t$ before $T$.

To evaluate the apparent singularity, write $h=z-{\xi }_t$, $p={\varphi }_t^{\prime }({\xi }_t)>0$ and $q={\varphi }_t^{\prime \prime }({\xi }_t)$. The Taylor series isThus the first term in the derivative is $2p/h-q+O(h)$, while the second is $2p/h+2q+O(h)$. Their poles cancel and their constant terms differ by $-3q$. Therefore

For the SLE driver $d{\xi }_t=\sqrt{6}d{W}_t$, apply the Itô formula to ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$. At a fixed spatial point, ${\varphi }_t$ has finite variation locally in time by the derivative formula in part (c); there is no extra spatial martingale term. ConsequentlyThis exact cancellation is the target-change locality of SLE6. For a general parameter the transformed SLE driving function would have drift $(\kappa /2-3){\varphi }_t^{\prime \prime }({\xi }_t)$.

Define the increasing clock and its inverse byThe derivative $p_t$ is real and strictly positive before $T$, so this is a valid continuous time change of a continuous process. The driver is a continuous local martingale with quadratic variation $[\tilde{\xi }{]}_t=6a(t)$ and initial value zero. The Dambis-Dubins-Schwarz theorem gives a standard Brownian motion $\hat{W}$ such thatIf the terminal clock is finite, the Brownian motion may be extended beyond that stopping time; only its stopped part is used here. Reparameterizing the image maps givesThese are exactly the defining Chordal Loewner equation and half-plane-capacity parameterization for ${\mathrm{SLE}}_6$. The trace is ${\tilde{\gamma }}_{t(s)}$. Hence the image curve is also ordinary chordal ${\mathrm{SLE}}_6$ after this time change, up to the specified stopping time: