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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.

Use the PDF's starred image hull $K_t^{\ast }$ and write $h_t={\Phi }_t=g_{K_t^{\ast }}\circ \Phi \circ g_{K_t}^{-1}$. For $t<T$, this map is analytic across a real neighborhood of the driver. SetLet $u(t)=\mathrm{hcap}(K_t^{\ast })/2$. The classical conformal change of half-plane capacity gives $\dot{u}(t)=a_t^2$, and the image driver, indexed by original time, is $h_t({\xi }_t)$. Differentiating the conjugacy relation givesThese are precisely the standard classical identities allowed in this question. The singularities cancel at $z={\xi }_t$. To calculate the remaining derivative, put $\delta =z-{\xi }_t$ and expand at fixed $t$:Thereforewhere each ${\partial }_t$ keeps the spatial variable fixed before evaluation. Applying the Itô formula to $h_t^{\prime }({\xi }_t)$ gives the boundary derivative diffusion under conformal Loewner conjugacyAt $\kappa =8/3$, the third-derivative drift cancels. A second application of the Itô formula to $a_t^p$ givesChoosing $p=1-1/\kappa =5/8$ cancels the remaining drift. Hence the SLE eight-thirds restriction martingale isLocalizing away from $T$ and from vanishing derivatives makes the displayed stochastic integral a martingale on each localized interval, so it defines a continuous local martingale up to $T$.

For the Locality property of SLE, take a compact H-hull $A$ away from $0$, and map $\mathbb{H}\setminus A$ conformally to $\mathbb{H}$ by ${\psi }_A$, fixing $0$ and infinity and with derivative one at infinity. Up to the first contact with $A$, the image under ${\psi }_A$ of an ${\mathrm{SLE}}_6$ grown in $\mathbb{H}$ has the same law as ${\mathrm{SLE}}_6$ grown in $\mathbb{H}$, after the change to its own half-plane-capacity parameterization. Equivalently, ${\mathrm{SLE}}_6$ in the smaller domain has the same initial law as the curve in the larger domain until the latter encounters the removed hull. There is no conditioning on eventual avoidance.

The chordal restriction property for ${\mathrm{SLE}}_{8/3}$ concerns the whole curve. Conditional on avoiding $A$, applying ${\psi }_A$ gives the original ${\mathrm{SLE}}_{8/3}$ law, modulo increasing reparameterization:Its associated avoidance probability, from the SLE eight-thirds restriction martingale, isThus locality identifies a stopped, unconditioned law, whereas restriction identifies a conditioned law of the entire curve.

Let $J_t={\psi }_t^{\prime }(U_t)$, $q_t={\psi }_t^{\prime \prime }(U_t)/{\psi }_t^{\prime }(U_t)$, and $r_t=({\partial }_z^3{\psi }_t)(U_t)/{\psi }_t^{\prime }(U_t)$. Since $d{U}_t=\sqrt{8/3}d{B}_t$, the supplied identity and Itô formula giveFor $M_t=J_t^{\alpha }$, its drift coefficient isThus the nonzero choice isand $M_{t\wedge \tau }$ is a continuous local martingale. The boundary Schwarz lemma for mapping-out maps gives $0\le J_{t\wedge \tau }\le 1$, so $0\le M_{t\wedge \tau }\le 1$. A bounded local martingale is a true martingale. This is the SLE eight-thirds restriction martingale.