On the reversal of radial SLE, I: Commutation Relations in Annuli

We aim at finding the reversal of radial SLE and proving the reversibility of wholeplane SLE. For this purpose, we define annulus SLE(κ,Λ) processes in doubly connected domains with one marked boundary point. We derive some partial differential equation for Λ, which is sufficient for the annulus SLE(κ,Λ) process to satisfy commutation relation. If Λ satisfies this PDE, then using a coupling technique, we are able to construct a global commutation coupling of two annulus SLE(κ,Λ) processes. If more conditions are satisfied, the coupling exists in the degenerate case, which becomes a coupling of two whole-plane SLEκ processes. The reversibility of whole-plane SLEκ follows from this coupling together with the assumption that such annulus SLE(κ,Λ) trace ends at the marked point. We then conclude that the limit of such annulus SLE(κ,Λ) trace is the reversal of radial SLEκ trace. In the end, we derive some particular solutions to the PDE for Λ.