SLE two-boundary-point ratio diffusion ($Z_t=\frac{g_t(y)-W_t}{g_t(y)-g_t(x)}$)

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 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.

For $\kappa \ge 8$, almost surely the positive-boundary swallowing times satisfy $T(x)<T(y)$ simultaneously for every $0<x<y$. The SLE two-boundary-point ratio diffusion has unbounded scale, ruling out escape before hitting $1$. For $4<\kappa <8$, its total scale $S_{\kappa }$ is finite, and the simultaneous-swallowing probability is $s_{\kappa }(y/(y-x))/S_{\kappa }$. At $\kappa =4$, fixed positive points have infinite swallowing times.

A continuous capacity-parameterized Loewner trace with finite, strictly ordered positive swallowing times visits every positive real point. An unvisited swallowed point has an interval disjoint from the compact trace up to its swallowing time; that interval is swallowed simultaneously, contradicting strict order. Reflection yields the negative-axis conclusion for chordal SLE at $\kappa \ge 8$.