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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains
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by Zhen-Qing Chen, Masatoshi Fukushima and Steffen Rohde PDF
Trans. Amer. Math. Soc. 368 (2016), 4065-4114

Abstract:

Let $D={\mathbb H} \setminus \bigcup _{k=1}^N C_k$ be a standard slit domain where $\mathbb H$ is the upper half-plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $\mathbb {H}$. Given a Jordan arc $\gamma \subset D$ starting at $\partial {\mathbb H},$ let $g_t$ be the unique conformal map from $D\setminus \gamma [0,t]$ onto a standard slit domain $D_t$ satisfying the hydrodynamic normalization. We prove that $g_t$ satisfies an ODE with the kernel on its right-hand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$, generalizing the chordal Loewner equation for the simply connected domain $D={\mathbb H}.$ Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in $t$. We establish the differentiability of $g_t$ in $t$ to make the equation a genuine ODE. To this end, we first derive the continuity of $g_t(z)$ in $t$ with a certain uniformity in $z$ from a probabilistic expression of $\Im g_t(z)$ in terms of the BMD for $D$, which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.
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Additional Information
  • Zhen-Qing Chen
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • MR Author ID: 242576
  • ORCID: 0000-0001-7037-4030
  • Email: zqchen@uw.edu
  • Masatoshi Fukushima
  • Affiliation: Branch of Mathematical Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
  • Email: fuku2@mx5.canvas.ne.jp
  • Steffen Rohde
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
  • Email: rohde@math.washington.edu
  • Received by editor(s): March 3, 2013
  • Received by editor(s) in revised form: April 7, 2014
  • Published electronically: October 2, 2015
  • Additional Notes: The first author’s research was partially supported by NSF Grant DMS-1206276 and NNSFC Grant 11128101
    The second author’s research was supported by Grant-in-Aid for Scientific Research of MEXT No. 22540125
    The third author’s research was partially supported by NSF Grant DMS-1068105
  • © Copyright 2015 Zhen-Qing Chen, Masatoshi Fukushima and Steffen Rohde
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 4065-4114
  • MSC (2010): Primary 60H30; Secondary 30C20
  • DOI: https://doi.org/10.1090/tran/6441
  • MathSciNet review: 3453365