Stochastic Komatu-Loewner evolutions and SLEs

A stochastic Komatu-Loewner evolution SKLEα,b has been introduced in Chen and Fukushima (2016) on a standard slit domain determined by certain continuous homogeneous functions α and b. We show that after a suitable reparametrization, SKLEα,b has the same distribution as the chordal Loewner evolution on H driven by a continuous semimartingale. When α is constant, we show that the distribution of SKLEα,b, after a suitable reparametrization and up to some random hitting time, is equivalent to that of SLEα2. Moreover, a reparametrized SKLE6,−bBMD has the same distribution as SLE6 for BMD domain constant bBMD.