Reverse Loewner flow

For a continuous driver $U$ and terminal time $T$, the reverse Loewner flow solvesIt satisfies $h_T(z)=g_T^{-1}(U_T+z)-U_T$.

For the reverse ${\mathrm{SLE}}_{\kappa }$ flow, write $X_s+i{Y}_s=h_s(z)-(U_{T-s}-U_T)$. Thenis a nonnegative local martingale and hence a supermartingale.

If $\kappa >8$, then for every $0<\alpha <(1-8/\kappa )/2$ and fixed $T$, there is an almost surely finite random $C$ such thatfor $|x|\le 1$ and $0<y\le 1$. The proof combines the derivative martingale, Markov inequality, a dyadic lattice, the Borel-Cantelli lemmas, and the Koebe distortion theorem.