Chordal Komatu-Loewner equation for a family of continuously growing hulls

Abstract

In this paper, we discuss the chordal Komatu-Loewner equation on standard slit domains in a manner applicable not just to a simple curve but also a family of continuously growing hulls. Especially a conformally invariant characterization of the Komatu-Loewner evolution is obtained. As an application, we prove a sort of conformal invariance, or locality, of the stochastic Komatu-Loewner evolution ${\mathrm{SKLE}}_{\sqrt{6},-b_{\mathrm{BMD}}}$ in a fully general setting, which solves an open problem posed by Chen, Fukushima and Suzuki [Stochastic Komatu-Loewner evolutions and SLEs, Stoch. Proc. Appl. 127 (2017), 2068-2087].

The results on the explosion of Komatu-Loewner evolutions contained in the first version will appear in another paper