Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 203 / 3 / c / Solution

Under the Loewner correspondence, mapping out the past replaces the driver byThe conformal Markov property therefore makes $U$ a continuous process with stationary independent increments. Every such process has the formScale invariance of SLE says that $r^{-1}{U}_{r^{2}t}$ has the same law as $U_t$. Comparing means forces $ar=a$ for every $r>0$, hence $a=0$; comparison of variances leaves the constant $\kappa$. Nondegeneracy gives $\kappa >0$, so