Continuous LERW Started from Interior Points

TL;DR

This paper introduces a new family of continuous loop-erased random walks (LERW) starting from interior points in finitely connected domains, demonstrating their conformal invariance and convergence as scaling limits of discrete LERW.

Contribution

It defines continuous LERW from interior points using the whole-plane Loewner equation and establishes their conformal invariance and relation to discrete LERW.

Findings

Continuous LERW are conformally invariant.

They preserve certain continuous local martingales.

They are the scaling limits of discrete LERW.

Abstract

We use the whole-plane Loewner equation to define a family of continuous LERW in finitely connected domains that are started from interior points. These continuous LERW satisfy conformal invariance, preserve some continuous local martingales, and are the scaling limits of the corresponding discrete LERW on the discrete approximation of the domains.