Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 203 / 2 / c / i / Solution

For $x>0$, the Chordal Loewner equation and $U_t=\sqrt{\kappa }{B}_t$ givePut $D_t=V_t^r-V_t^1$. The common Brownian term cancels, soThe Itô formula applied to $Z_t=\log D_t-\log V_t^1$ givesSincethe clock $q(u)={\int }_0^u(V_s^1{)}^{-2}ds$ and its inverse $\sigma$ turn the local-martingale term into Brownian motion by the Dambis-Dubins-Schwarz theorem. ThereforeThis is the SLE boundary-point logarithmic separation diffusion.