Boundary derivative diffusion under conformal Loewner conjugacy ($d{a}_t=\sqrt{\kappa }{b}_{t}d{B}_t+[b_t^2/(2a_t)+(\kappa /2-4/3)c_t]dt$)

Let $h_t$ intertwine two Loewner chains, with driver ${\xi }_t=\sqrt{\kappa }{B}_t$, and set $a_t=h_t^{\prime }({\xi }_t)$, $b_t=h_t^{\prime \prime }({\xi }_t)$, $c_t=h_t^{\prime \prime \prime }({\xi }_t)$. Expanding the conjugacy equation at the driver gives ${\partial }_t{h}_t^{\prime }({\xi }_t)=b_t^2/(2a_t)-4c_t/3$; the Itô formula gives the displayed diffusion. For a real exponent $p$, the drift of $a_t^p$ is $p[1+\kappa (p-1)]a_t^{p-2}{b}_t^2/2+p(\kappa /2-4/3)a_t^{p-1}{c}_t$. At $\kappa =8/3$, $p=5/8$ cancels both terms and yields the SLE eight-thirds restriction martingale.