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

For a continuous real driving function $U$, solve the Chordal Loewner equationup to the first time $T_z$ at which $g_t(z)-U_t$ tends to zero. The generated hulls areThey form an increasing Loewner chain of compact H-hulls, satisfy $\mathrm{hcap}(K_t)=2t$, and $g_t$ is the mapping-out function of $K_t$.

The Conformal Markov property of SLE says that, conditionally on $(K_s{)}_{s\le t}$, the future hullshave the law of a fresh SLE in $(\mathbb{H},0,\infty )$ and are independent of the past, with the usual interpretation of mapping out the old hull.

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

For $t<T_x$, differentiate the Loewner equation with respect to $x$:ThereforeThe exponent is nonpositive and decreases with $t$, so $0<g_t^{\prime }(x)\le 1$ and $g_t^{\prime }(x)$ is decreasing before $x$ is swallowed.