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

Zhen-Qing Chen University of Washington Masatoshi Fukushima Osaka University Steffen Rohde University of Washington

Probability mathscidoc:1904.28001

Trans. Amer. Math. Soc., 368, 4065-4114, 2016
Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ 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 $\HH$. 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 righthand 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.
Komatu-Loewner equation, Brownian motion with darning, harmonic function with zero period, Green function, complex Poisson kernel, multiply connected domain
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@inproceedings{zhen-qing2016chordal,
  title={Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains},
  author={Zhen-Qing Chen, Masatoshi Fukushima, and Steffen Rohde},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430131947887285296},
  booktitle={Trans. Amer. Math. Soc.},
  volume={368},
  pages={4065-4114},
  year={2016},
}
Zhen-Qing Chen, Masatoshi Fukushima, and Steffen Rohde. Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains. 2016. Vol. 368. In Trans. Amer. Math. Soc.. pp.4065-4114. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430131947887285296.
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