# Reformulation of Laplacian-$b$ motion in terms of stochastic Komatu-Loewner evolution in the chordal case

## TL;DR

This paper establishes a connection between Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) in multiply connected domains, showing they are equivalent under certain conditions and analyzing their finite-time explosion behavior.

## Contribution

It reformulates Laplacian-$b$ motion in terms of SKLE, providing new insights into their relationship and demonstrating finite-time blow-up for the case corresponding to SLE_6.

## Key findings

- SKLE with a specific stochastic differential equation matches Laplacian-$b$ motion after a time change
- Finite time explosion occurs for SKLE related to Laplacian-$0$ motion (SLE_6)
- Solution to the Komatu-Loewner equation for SLE_6 blows up in finite time

## Abstract

We investigate the relation between the Laplacian-$b$ motion and stochastic Komatu-Loewner evolution (SKLE) on multiply connected subdomains of the upper half-plane, both of which are analogues to SLE. In particular, we show that, if the driving function of an SKLE is given by a certain stochastic differential equation, then this SKLE is the same as a time-changed Laplacian-$b$ motion. As an application, we prove the finite time explosion of SKLE corresponding to Laplacian-$0$ motion, or $\mathrm{SLE_6}$, in the sense that the solution to the Komatu-Loewner equation for the slits blows up.