Chordal Komatu-Koewner equation and brownian motion with darning in multiply connected domains

Let (Formula Presented) be a standard slit domain where H is the upper half-plane and C_k, 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 D_t satisfying the hydrodynamic normalization. We prove that gt 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 = 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 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 (Formula Presented)gt(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.