SLE angle process (${\theta }_t=\arg (g_t(z)-U_t)$)

With the upper-half-plane branch, ${\theta }_t\in (0,\pi )$ before the Loewner swallowing time. Write $Z_t=X_t+i{Y}_t$. The Itô formula and the Chordal Loewner equation giveAt parameter four its drift vanishes. At other parameters the drift describes the competition between the deterministic conformal evolution and the Brownian driver.

For the SLE angle process, the bounded Dirichlet problem on the unbounded surviving domain has data $\pi$ on the intrinsic boundary mapped to the left of the Loewner driving function and zero on the boundary mapped to its right. For a simple Loewner trace these are its left and right banks, with the negative and positive real rays respectively. The tip is a discontinuity point of the boundary data.

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.

For a fixed interior point and parameter four, the SLE angle process has zero drift and satisfies ${\theta }_t=\arg z+2{\int }_0^t{Y}_s|Z_s{|}^{-2}d{B}_s$. Its values in $(0,\pi )$ make this continuous local martingale a true bounded martingale. The identity $\log ({{\rm Y}}_t/{{\rm Y}}_0)=-[\theta {]}_t$, together with the Dambis-Dubins-Schwarz theorem, prevents its Loewner conformal radius from vanishing at a finite lifetime. Since the parameter-four trace is simple and does not meet the real boundary at positive times, finite swallowing would require a visit to the point and vanishing radius. Thus its Loewner swallowing time is infinite almost surely. This gives a global continuous bounded martingale, not just one defined before swallowing.

For $\kappa >0$, $\rho >0$, let $J_t=|g_t^{\prime }(z)|$, $S_t=\sin {\theta }_t$ and ${{\rm Y}}_t=Y_t/J_t$. Thenis a positive continuous local martingale before the Loewner swallowing time. Indeed the Itô formula gives $d\log J_t=-2(X_t^2-Y_t^2)|Z_t{|}^{-4}dt$, $d\log {{\rm Y}}_t=-4Y_t^2|Z_t{|}^{-4}dt$, andCombining the exponents cancels the drift of $M_t$, including the quadratic-variation correction, leaving $d{M}_t/M_t=-\rho X_t/(\sqrt{\kappa }|Z_t{|}^2)d{B}_t$.