Commutation relations for two-sided radial SLE

Understand

We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points.

• We follow the framework introduced by Dub\'{e}dat.
• Under an additional requirement of the interchangeability of the two curves, we classify all locally commuting 2-radial SLE$_\kappa$ for $\kappa\in (0,8)$: it is either a two-sided radial SLE$_\kappa$ with spiral of constant spiraling rate or a chordal SLE$_\kappa$ weighted by a power of the conformal radius of its complement.
• Namely, for fixed $\kappa$ and starting points, we have exactly two one-parameter continuous families of locally commuting 2-radial SLE.