SLE boundary-point logarithmic separation diffusion

For two boundary points $1<r$, let $V_t^x=g_t(x)-U_t$ and set $Z_t=\log (V_t^r-V_t^1)-\log V_t^1$. After the clock $q(u)={\int }_0^u(V_s^1{)}^{-2}ds$, the Dambis-Dubins-Schwarz theorem givesThis one-dimensional diffusion process compares the swallowing times of nearby boundary points.

For every $x\in \mathbb{R}\setminus \{0\}$, the probability that a chordal ${\mathrm{SLE}}_{\kappa }$ trace in $(\mathbb{H},0,\infty )$ passes through $x$ is zero. The SLE boundary-point logarithmic separation diffusion shows that the chance for $x$ to be swallowed strictly before a nearby point tends to zero as that point approaches $x$; Scaling invariance of SLE and reflection then handle every nonzero $x$.