Welding of the Backward SLE and Tip of the Forward SLE
Update: 2015-06-30
Description
Co-author: Steffen Rohde (University of Washington)
Let κ∈(0,4]. A backward chordal SLEκ process generates a conformal welding ϕ, which is a random auto-homeomorphism of R that satisfies ϕ−1=ϕ and has a single fixed point: 0. Using a stochastic coupling technique, we proved that the welding ϕ satisfies the following symmetry: Let h(z)=−1/z. Then h∘ϕ∘h has the same law as ϕ. Combining this symmetry result with the forward/backward SLE symmetry and the conformal removability of forward SLE curve, we then derived some ergodic property of the tip of a forward SLEκ curve for κ∈(0,4).
Let κ∈(0,4]. A backward chordal SLEκ process generates a conformal welding ϕ, which is a random auto-homeomorphism of R that satisfies ϕ−1=ϕ and has a single fixed point: 0. Using a stochastic coupling technique, we proved that the welding ϕ satisfies the following symmetry: Let h(z)=−1/z. Then h∘ϕ∘h has the same law as ϕ. Combining this symmetry result with the forward/backward SLE symmetry and the conformal removability of forward SLE curve, we then derived some ergodic property of the tip of a forward SLEκ curve for κ∈(0,4).
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