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

Abstract

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