Domain Markov property of a chordal Loewner chain

Conditionally on the entire past up to $t$, the future domains mapped and centered by $g_t-U_t$ have the original chain law and are independent of that past. For Schramm–Loewner evolution this follows from independent stationary Brownian motion increments and the composition rule for chordal Loewner driving functions. The Strong Markov property gives the version at almost surely finite stopping times.

Suppose a chordal SLE trace stays in the open complex upper half-plane at every positive time almost surely. For every deterministic rational time $s$, the domain Markov property of a chordal Loewner chain makes its mapped and centered future a fresh SLE, so its positive-time future stays in the open half-plane as well. The inverse mapping-out function then places that future outside the entire old hull. Intersecting these probability-one events over rational $s$ rules out every self-intersection: if two times $u<v$ shared a trace point, choose rational $s$ strictly between them. This proves simplicity using the allowed boundary-avoidance assertion for $0<\kappa \le 4$.