Conformal conjugacy derivative for the chordal Loewner equation (${\dot{\varphi }}_t({\xi }_t)=-3{\varphi }_t^{\prime \prime }({\xi }_t)$)

For conjugated Loewner chains, ${\varphi }_t={\tilde{g}}_t\circ \varphi \circ g_t^{-1}$ and ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$. The image capacity speed is $2{\varphi }_t^{\prime }({\xi }_t{)}^2$. Differentiating this identity gives ${\dot{\varphi }}_t(z)=2{\varphi }_t^{\prime }({\xi }_t{)}^2/({\varphi }_t(z)-{\varphi }_t({\xi }_t))-2{\varphi }_t^{\prime }(z)/(z-{\xi }_t)$. The removable singularity at the driver has value $-3{\varphi }_t^{\prime \prime }({\xi }_t)$.

Apply the Itô formula to the transformed Loewner driving function ${\tilde{\xi }}_t={\varphi }_t(\sqrt{\kappa }{W}_t)$. The conformal conjugacy derivative for the chordal Loewner equation cancels the Itô drift exactly when $\kappa =6$. Its quadratic variation is $\kappa {\int }_0^t{\varphi }_s^{\prime }({\xi }_s{)}^{2}ds$.