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A scale-invariant law of unparameterized chords from zero to infinity in the complex upper half-plane is local when a boundary-preserving conformal map between neighborhoods of zero, fixing zero, transports its curve stopped on leaving the first neighborhood to the same-law curve stopped on leaving the second. Equality concerns the stopped curves up to increasing reparameterization. For SLE with parameter six, the transformed SLE driving function is a continuous local martingale with bracket six times the image capacity clock. The Lévy characterization of Brownian motion makes that driver Brownian in the new clock, proving the local stopped-curve invariance.

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.

By the tip correspondence in the conjugacy,The time-dependent Itô formula is valid after localization away from the pole and the terminal time $S$. The map's time variation is of finite variation, so there is no extra stochastic cross-variation term. Using the supplied conjugacy derivative,ConsequentlyThe cancellation is special to $\kappa =6$: for a general SLE parameter the drift would be $(\kappa /2-3){\Phi }_t^{\prime \prime }({\xi }_t)dt$. This is the transformed SLE driving function, and explains the special Locality property of SLE at six.

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:

Changing the target boundary point by a conformal automorphism of the upper half-plane preserves the local chordal SLE law at parameter six, up to a capacity time change and until the relevant boundary point is swallowed. The transformed SLE driving function has zero drift at six; the Dambis-Dubins-Schwarz theorem makes it Brownian in image capacity time.