Excursion Reflected Brownian Motion

TL;DR

This paper introduces Excursion Reflected Brownian Motion (ERBM), a Markov process in multiply connected domains, and develops its harmonic function theory to aid in studying SLE in complex geometries.

Contribution

It constructs ERBM using conformal invariance and develops its harmonic functions, Poisson kernel, and Green's function, providing tools for conformal mapping in multiply connected domains.

Findings

Constructed ERBM via conformal invariance.

Developed harmonic function theory for ERBM.

Showed applications to conformal mapping in multiply connected domains.

Abstract

Excursion reflected Brownian motion (ERBM) is a strong Markov process defined in a finitely connected domain $D \subset \mathbb{C}$ 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 construction of ERBM using its conformal invariance and develop the basic theory of its harmonic functions. One important reason for studying ERBM is the hope that it will be a useful tool in the study of SLE in multiply connected domains. To this end, we develop the basic theory of the Poisson kernel and Green's function for ERBM and show how it can be used to construct conformal maps into certain classes of multiply connected domains.