Conformal Markov property of SLE

Conditionally on an ${\mathrm{SLE}}_{\kappa }$ initial segment through time $t$, mapping out that segment by $g_t-U_t$ turns the future into an independent ${\mathrm{SLE}}_{\kappa }$ in $(\mathbb{H},0,\infty )$. This follows because its driving function is $U_{t+s}-U_t$, and Brownian motion has stationary increments and independent increments.

The conformal Markov property gives the continuous Loewner driver stationary independent increments. Scale invariance removes its deterministic drift. Thereforefor a standard Brownian motion and a constant $\kappa \ge 0$.