Transience of chordal SLE

The chordal SLE Loewner trace eventually leaves every bounded subset of the complex upper half-plane. For $0<\kappa <4$, its time-one compact H-hull is a simple slit and does not contain a half-disc. Instead, the boundary approach forces a small SLE Bessel gap estimate shows that every fixed nonzero real point is outside the closure of the full Loewner trace. After mapping out the initial simple slit, the starting point has two distinct real prime end images. A fresh independent SLE avoids approaching either image, so the original future stays a positive distance from its starting point. Scaling invariance of SLE gives ${\inf }_{t\ge T}|\gamma (t)|\stackrel{d}{=}\sqrt{T}{\inf }_{t\ge 1}|\gamma (t)|$, and decreasing-event continuity proves escape from every fixed disc. The separate argument based on a swallowed half-disc at positive time applies in the nonsimple regime $\kappa >4$; it must not be applied to a simple slit. The critical parameters require their own limiting or trace-existence arguments.