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Put $w=g_T^{-1}(U_T+z)$ and defineThen $f_0(z)=z$. Differentiating with the Chordal Loewner equation givesUniqueness for this ordinary differential equation shows that $f_s=h_s$. At $s=T$,which is the endpoint identity for the Reverse Loewner flow.

For $0<s<T$, the pathwise identity $h_s(z)=g_s^{-1}(U_s+z)-U_s$ is generally false. The left side is built from the reversed final driver segment $(U_{T-r}-U_T{)}_{0\le r\le s}$, whereas the right side is built from the initial segment $(U_r{)}_{0\le r\le s}$. By time reversal and symmetry of Brownian motion they have the same probability distribution, but they are not equal for the given Brownian path.

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 $\kappa =0$.