Stochastic Komatu-Loewner evolutions and SLEs

Let $D={mathbb H}setminus bigcup_{j=1}^N C_j$ be a standard slit domain, where ${mathbb H}$ is the upper half plane and $C_j,1le jle N,$ are mutually disjoint horizontal line segments in ${mathbb H}$. A stochastic Komatu-Loewner evolution denoted by ${rm SKLE}_{alpha,b}$ has been introduced in cite{CF} as a family ${F_t}$ of random growing hulls with $F_tsubset D$ driven by a diffusion process $xi(t)$ on $partial {mathbb H}$ that is determined by certain continuous homogeneous functions $alpha$ and $b$ defined on the space ${cal S}$ of all labelled standard slit domains. We aim at identifying the distribution of a suitably reparametrized ${rm SKLE}_{alpha,b}$ with that of the Loewner evolution on ${mathbb H}$ driven by the path of a certain continuous semimartingale and thereby relating the former to the distribution of ${rm SLE}_{alpha^2}$ when $alpha$ is a constant. We then prove that, when $alpha$ is a constant, ${rm SKLE}_{alpha,b}$ up to some random hitting time and modulo a time change has the same distribution as ${rm SLE}_{alpha^2}$ under a suitable Girsanov transformation. We further show that a reparametrized ${rm SKLE}_{sqrt{6},-b_{rm BMD}}$ has the same distribution as ${rm SLE}_6$, where $b_{rm BMD}$ is the BMD-domain constant indicating the discrepancy of $D$ from ${mathbb H}$ relative to Brownian motion with darning (BMD in abbreviation). A key ingredient of the proof is a hitting time analysis for the absorbing Brownian motion on ${mathbb H}.$ We also revisit and examine the locality property of ${rm SLE}_6$ in several canonical domains. Finally K-L equations and SKLEs for other canonical multiply connected planar domains than the standard slit one are recalled and examined.