Scaling invariance of SLE

For every $a>0$, if $(K_t)$ is ${\mathrm{SLE}}_{\kappa }$, then $(a^{-1}{K}_{a^{2}t})$ has the same law. Its driving function is $a^{-1}{U}_{a^{2}t}$, which has the same law as $U_t$ by Brownian scaling.

For a chordal SLE trace $\gamma$, the Scaling invariance of SLE makes $\mathbb{P}({\sup }_t\mathrm{Im}\gamma (t)>r)$ independent of $r>0$. The trace must enter the upper half-plane because its positive-time hull has positive half-plane capacity. Letting $r\downarrow 0$ therefore makes this probability one, and continuity implies that every positive height is reached. This argument does not assume Transience of chordal SLE.