 Solution

The Conformal Markov property of SLE says that, conditionally on the curve through time $t$, mapping out the initial hull by $g_t-U_t$ turns the future into an independent Schramm–Loewner evolution in $(\mathbb{H},0,\infty )$. More precisely,has driving function ${\tilde{U}}_s=U_{t+s}-U_t$. Since $U_t=\sqrt{\kappa }{B}_t$, the stationary increments and independent increments of Brownian motion give the asserted independence and equality in law.