Mathematics – Probability
Scientific paper
2010-04-12
Mathematics
Probability
The last version of this paper is shortened. Some small errors and typos are corrected
Scientific paper
The main result of this paper is that for $\kappa\in(0,4]$, whole-plane SLE$_\kappa$ satisfies reversibility, which means that the time-reversal of a whole-plane SLE$_\kappa$ trace is still a whole-plane SLE$_\kappa$ trace. In addition, we find that the time-reversal of a radial SLE$_\kappa$ trace for $\kappa\in(0,4]$ is a disc SLE$_\kappa$ trace with a marked boundary point. The main tool used in this paper is the stochastic coupling technique, which was introduced to prove the reversibility of chordal SLE$_\kappa$ for $\kappa\in(0,4]$. The reversibility of whole-plane SLE is closely related to the reversibility of certain SLE traces in doubly connected domains connecting two boundary points. Such SLE is defined using the annulus Loewner equation and a drift function $\Lambda$. For the reversibility to hold, the drift function $\Lambda$ must satisfy certain PDE. We use the Feynman-Kac representation to find a solution to this PDE, and finally use the solution to prove the reversibility of whole-plane SLE$_\kappa$.
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