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Transactions of the American Mathematical Society

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Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains


Authors: Zhen-Qing Chen, Masatoshi Fukushima and Steffen Rohde
Journal: Trans. Amer. Math. Soc. 368 (2016), 4065-4114
MSC (2010): Primary 60H30; Secondary 30C20
Published electronically: October 2, 2015
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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
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

DOI: https://doi.org/10.1090/tran/6441
Keywords: Komatu-Loewner equation, Brownian motion with darning, harmonic function with zero period, Green function, complex Poisson kernel, multiply connected domain
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
Article copyright: © Copyright 2015 Zhen-Qing Chen, Masatoshi Fukushima and Steffen Rohde