= SLE two-boundary-point ratio diffusion
{c}
{title2=$Z_t=\dfrac{g_t(y)-W_t}{g_t(y)-g_t(x)}$}
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 obeys
$$
dZ_t=-\frac{\sqrt\kappa}{D_t}d\beta_t+\frac2{D_t^2}\left(\frac1{Z_t}+\frac1{Z_t-1}\right)dt.
$$
The 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.
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