Department of Mathematics

Analysis and PDE

  •  Rami Fakhry, MSU
  •  Canceled due to health issue of the speaker
  •  05/05/2021
  •  4:10 PM - 5:00 PM
  •  Online (virtual meeting) (Virtual Meeting Link)
  •  Dapeng Zhan (zhan@msu.edu)

For a Chordal SLE$_\kappa$ ($\kappa \in (0,8)$) curve in a simply connected domain $D$ with smooth boundary, the $n$-point boundary Green's function valued at distinct points $z_1, ..., z_n\in \partial{D}$ is defined by} \[ G(z_1,...,z_n)= \lim_{r_1,...,r_n \to 0+} \prod_{j=1}^{n} {r_j}^ {- \alpha} \mathbb{P} \left[ dist(\gamma, z_k) \leqq r_k, 1 \leqq k \leqq n \right] ,\]where $ \alpha = \frac{8}{\kappa} - 1 $ is the boundary exponent of SLE$_\kappa$, provided that the limit converges. In this talk, we will show that such Green's function exists for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them. We use the same zoom link and passcode as before.

 

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Michigan State University
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