Stochastic Komatu-Loewner evolutions and SLEs

TL;DR

This paper investigates stochastic Komatu-Loewner evolutions (SKLEs) in slit domains, relating them to Schramm-Loewner evolutions (SLEs), and demonstrates their distributional equivalence under certain conditions, including the special case of SLE_6.

Contribution

It establishes the distributional equivalence of SKLEs and SLEs in slit domains, especially showing SKLE_6 matches SLE_6, and explores their properties and equations in multiply connected domains.

Findings

SKLE_{α,b} can be identified with SLE driven by a semimartingale.

When α is constant, SKLE_{α,b} up to a hitting time matches SLE_{α^2} after a Girsanov transform.

Reparametrized SKLE_{√6, -b_BMD} is distributionally equivalent to SLE_6.

Abstract

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 $C_j,1\le j\le N,$ 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 $F_t\subset D$ 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.