 SLE two-boundary-point ratio diffusion

For $0<x<y$, let $D_t=g_t(y)-g_t(x)$. Before the first boundary-point swallowing time for a Loewner chain, the ratio obeysThe clock $u=\int D_t^{-2}dt$ removes the denominator. An increasing scale function of a one-dimensional diffusion is $s_{\kappa }(z)={\int }_1^z[v(v-1){]}^{-4/\kappa }dv$. For $\kappa >4$ the endpoint $1$ represents strict swallowing; escape at infinite clock represents simultaneous swallowing.