Stochastic Loewner evolution in doubly connected domains

Abstract

This paper introduces the annulus SLE$_κ$ processes in doubly connected domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a time-change. For $κ\not=6$, some weak equivalence relation exists between annulus SLE$_κ$ and radial SLE$_κ$. Annulus SLE$_2$ is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE$_2$ satisfies the reversibility property. We also consider the disc SLE$_κ$ process defined as a limiting case of the annulus SLE's. Disc SLE$_6$ has the same law as stopped full plane SLE$_6$, up to a time-change. Disc SLE$_2$ is the scaling limit of loop-erased random walk, and is the reversal of radial SLE$_2$.