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

Let D = H \ ∪k=1Ck be a standard slit domain where H is the upper half plane and Ck, 1 ≤ k ≤ N , are mutually disjoint horizontal line segments in H. Given a Jordan arc γ ⊂ D starting at ∂H, let gt be the unique conformal map from D \ γ[0, t] onto a standard slit domain Dt satisfying the hydrodynamic normalization. We prove that gt satisfies an ODE with the kernel on its righthand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for Dt, generalizing the chordal Loewner equation for the simply connected domain D = H. Such a generalization has been obtained by Y. Komatu in a radial case 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 gt in t to make the equation a genuine ODE. To this end, we first derive the continuity of gt(z) in t with a certain uniformity in z from a probabilistic expression of Igt(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. AMS 2000 Mathematics Subject Classification: Primary 60H30; Secondary 30C20