## 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.## References

- Lars V. Ahlfors,
*Complex analysis*, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR**510197** - Robert O. Bauer and Roland M. Friedrich,
*On radial stochastic Loewner evolution in multiply connected domains*, J. Funct. Anal.**237**(2006), no. 2, 565–588. MR**2230350**, DOI 10.1016/j.jfa.2005.12.023 - Robert O. Bauer and Roland M. Friedrich,
*On chordal and bilateral SLE in multiply connected domains*, Math. Z.**258**(2008), no. 2, 241–265. MR**2357634**, DOI 10.1007/s00209-006-0041-z - Z.-Q. Chen,
*Brownian Motion with Darning*. Lecture notes for talks given at RIMS, Kyoto University. 2012. - Zhen-Qing Chen and Masatoshi Fukushima,
*Symmetric Markov processes, time change, and boundary theory*, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR**2849840** - John B. Conway,
*Functions of one complex variable. II*, Graduate Texts in Mathematics, vol. 159, Springer-Verlag, New York, 1995. MR**1344449**, DOI 10.1007/978-1-4612-0817-4 - Shawn Drenning,
*Excursion reflected Brownian motion and loewner equations in multiply connected domains*, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–The University of Chicago. MR**2992562** - Peter L. Duren,
*Univalent functions*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR**708494** - Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda,
*Dirichlet forms and symmetric Markov processes*, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR**1303354**, DOI 10.1515/9783110889741 - P. R. Garabedian,
*Partial Differential Equations*, AMS Chelsia, 2007, republication of 1964 edition - John B. Garnett and Donald E. Marshall,
*Harmonic measure*, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR**2150803**, DOI 10.1017/CBO9780511546617 - Yûsaku Komatu,
*On conformal slit mapping of multiply-connected domains*, Proc. Japan Acad.**26**(1950), no. 7, 26–31. MR**46437** - Gregory F. Lawler,
*Conformally invariant processes in the plane*, Mathematical Surveys and Monographs, vol. 114, American Mathematical Society, Providence, RI, 2005. MR**2129588**, DOI 10.1090/surv/114 - Gregory F. Lawler,
*The Laplacian-$b$ random walk and the Schramm-Loewner evolution*, Illinois J. Math.**50**(2006), no. 1-4, 701–746. MR**2247843** - John W. Milnor,
*Topology from the differentiable viewpoint*, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR**0226651** - Zeev Nehari,
*Conformal mapping*, Dover Publications, Inc., New York, 1975. Reprinting of the 1952 edition. MR**0377031** - Sidney C. Port and Charles J. Stone,
*Brownian motion and classical potential theory*, Probability and Mathematical Statistics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**0492329** - M. Tsuji,
*Potential theory in modern function theory*, Maruzen Co. Ltd., Tokyo, 1959. MR**0114894**

## 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