Logarithmic martingale for SLE4 ($d\log Z_t=-2Z_t^{-1}d{W}_t$)

For Schramm–Loewner evolution with $\kappa =4$ and $Z_t=g_t(z)-2W_t$, the Itô formula for the complex logarithm gives $d\log Z_t=-2Z_t^{-1}d{W}_t$ before swallowing. The drift cancels exactly at this parameter. Its real part is $\log |Z_t|$ and its imaginary part is the SLE4 angle martingale. The real-part quadratic variation is $4\int X_t^2/|Z_t{|}^{4}dt$, which also gives a direct non-swallowing argument.

Every fixed $z\in \mathbb{H}$ has infinite interior-point swallowing time for a Loewner chain under Schramm–Loewner evolution at $\kappa =4$. The centered flow is bounded on any finite horizon, making $\log |Z_t|$ bounded above. The one-sided bound criterion for a martingale clock gives a finite logarithmic limit at any putative finite swallowing time. Thus $|Z_t|$ stays away from zero and the imaginary coordinate remains positive, so the differential equation extends, a contradiction. In particular the trace misses each fixed interior point almost surely.