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 $\Lambda$.

It is covered by my later paper "Reversibility of whole-plane SLE", arXiv:1004.1865

References in corpus (4)

• Duality of Chordal SLE
• Dipolar SLEs
• Duality of Chordal SLE, II
• Continuous LERW Started from Interior Points

Cited by in corpus (2)

• The Gaussian free field and SLE(4) on doubly connected domains
• Conformal restriction: the radial case