Radial Schramm–Loewner evolution (${radial\; SLE}_{\kappa }$)

Radial SLE grows from a boundary point to an interior target. In the unit disc from $1$ to $0$, its conformal-radius parameterization satisfiesThe target remains in the remaining simply connected component; the time parameter records its decreasing conformal radius.

For radial SLE at parameter two, the normalized slit-domain Poisson kernel at the growing tip is a local martingale. This observable provides the coupling to loop-erasure of planar Brownian motion, oriented from the boundary point to the interior target. Its discrete counterpart is a loop-erased random walk.