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

TL;DR

This paper extends the chordal Komatu-Loewner equation to families of growing hulls in slit domains, providing a conformally invariant framework and proving locality of a stochastic version, addressing an open problem in the field.

Contribution

It introduces a conformally invariant characterization of the Komatu-Loewner evolution for growing hulls and proves locality of the stochastic Komatu-Loewner evolution in general settings.

Findings

Established a conformally invariant formulation of the Komatu-Loewner equation.

Proved the locality property of the stochastic Komatu-Loewner evolution.

Solved an open problem regarding the invariance of SKLE in general domains.

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].