Trace of a Loewner chain ($\gamma (t)={\lim }_{y\downarrow 0}{g}_t^{-1}(U_t+iy)$)

When the indicated limit exists continuously and the hulls are generated by its past, it defines the trace of the Loewner chain. The hull $K_t$ includes the curve up to time $t$ and any regions it disconnects from infinity. Therefore swallowing a point by $K_t$ is different from visiting that point by the Loewner trace. For SLE, the continuous trace is part of its basic existence theory.

For a continuous Loewner trace generating its hulls, the inverse mapping-out function of a compact H-hull $g_t^{-1}$ extends continuously to the closed complex upper half-plane. The finite trace is a continuous map image of an interval, and the boundary of its unbounded complementary component has local connectedness; the Caratheodory boundary extension theorem gives the extension. In particular $g_t^{-1}(U_t)=\gamma (t)$, while $g_t^{-1}$ maps the open complex upper half-plane into $D_t$. This is the deterministic boundary-extension input of Rohde and Schramm, Theorem 4.1. Under a Conformal Markov property of SLE restart, this relates boundary contacts of the mapped future to contacts with the old hull, without assuming the old trace is simple.