Reverse SLE derivative martingale ($M_s$)

Fix $\kappa >0$ and a deterministic terminal time $T$. The reversed driver $U_{T-s}-U_T$ has the law of $\sqrt{\kappa }{B}_s$ in its reversed-increment filtration. For the Reverse Loewner flow, put $Z_s=X_s+i{Y}_s=h_s(z)-\sqrt{\kappa }{B}_s$, $q_s=X_s/Y_s$ and $J_s=|h_s^{\prime }(z)|$. Direct differentiation and the Itô formula giveConsequentlyThe drift cancellation includes the quadratic variation correction. Thus $M$ is a positive local martingale, hence a supermartingale; a true martingale assertion needs additional integrability. This yields useful expected value bounds for the inverse-map derivative without assuming the estimate itself is integrable in advance. The exponent $4/\kappa$ excludes the deterministic case $\kappa =0$.

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.