SLE boundary-point logarithmic separation diffusion
= SLE boundary-point logarithmic separation diffusion
{c}
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> gives
$$
d\widetilde Z_t=\sqrt\kappa\,dW_t+\left(\frac{\kappa-4}{2}-\frac2{1+e^{\widetilde Z_t}}\right)dt.
$$
This one-dimensional <diffusion process> compares the swallowing times of nearby boundary points.