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 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 $d{M}_t/M_t=-\rho X_t/(\sqrt{\kappa }|Z_t{|}^2)d{B}_t$.