Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 36 / 4 / b

Let $g_t$ map out the original filled hull and let $g_t^{\ast }$ map out the filled hull generated by $\Phi (\gamma [0,t])$. Both have hydrodynamic normalization at infinity. WriteAlthough the original $\Phi$ is specified only on $N$, this conjugacy is defined in the surviving neighborhood of the tip. Boundary preservation gives a real analytic extension there by the Schwarz reflection principle, with $a_t>0$. The identities below are first applied on compactly localized intervals before exit and then continued up to $T$.

Set $u(t)=\mathrm{hcap}(K_t^{\ast })/2$, the capacity clock of the image hull. The Chordal Loewner equation for each flow and the chain rule giveAnalyticity at the driving point cancels the coefficient of $(z-{\xi }_t{)}^{-1}$, forcing $\dot{u}(t)=a_t^2$. The constant term, computed as in Question 3, is then ${\partial }_t{h}_t({\xi }_t)=-3b_t$. For a general SLE driver $d{\xi }_t=\sqrt{\kappa }d{B}_t$, the Itô formula consequently gives the transformed SLE driving functionAt $\kappa =6$, this becomesThus $W$ is a continuous local martingale up to the exit time. If the source notation ${\xi }_t^{\ast }$ indexes the image driver by original time, it denotes $W_t$. If it uses the image's own half-plane-capacity time, it denotes $W_{t(u)}$, with $t(u)$ the inverse clock; that time-changed process is also a local martingale. This distinguishes the two clocks without changing the requested conclusion.