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Welding of the Backward SLE and Tip of the Forward SLE

Thursday 18th June 2015 - 15:20 to 16:20
INI Seminar Room 1
Co-author: Steffen Rohde (University of Washington)

Let $\kappa\in(0,4]$. A backward chordal SLE$_\kappa$ process generates a conformal welding $\phi$, which is a random auto-homeomorphism of $\mathbb R$ that satisfies $\phi^{-1}=\phi$ and has a single fixed point: $0$. Using a stochastic coupling technique, we proved that the welding $\phi$ satisfies the following symmetry: Let $h(z)=-1/z$. Then $h\circ \phi\circ h$ has the same law as $\phi$. 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 SLE$_\kappa$ curve for $\kappa\in(0,4)$.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons