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

DOI:
https://doi.org/10.1090/tran/6441

Published electronically:
October 2, 2015

MathSciNet review:
3453365

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Abstract | References | Similar Articles | Additional Information

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

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