Excursion Reflected Brownian Motions And Loewner Equations In Multiply Connected Domains

TL;DR

This paper introduces a new construction of excursion reflected Brownian motion (ERBM) in finitely connected domains, explores its conformal invariance, and derives Loewner equations for curve growth, aiming to advance SLE studies in complex domains.

Contribution

It provides a novel conformally invariant construction of ERBM and links it to Loewner equations in finitely connected domains, facilitating SLE analysis.

Findings

ERBM can be constructed via conformal invariance.

The Poisson kernel for ERBM relates to conformal maps.

Loewner equations for curve growth are derived using ERBM.

Abstract

Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \C$ that behaves like a Brownian motion away from the boundary of $D$ and picks a point according to harmonic measure from infinity to reflect from every time it hits a boundary component. We give a new construction of ERBM using its conformal invariance and discuss the relationship between the Poisson kernel and Green's function for ERBM and conformal maps into certain classes of finitely connected domains. One important reason for studying ERBM is the hope that it will be a useful tool in the study of SLE in finitely connected domains. To this end, we show how the Poisson kernel for ERBM can be used to derive a Loewner equation for simple curves growing in a certain class of finitely connected domains.