Loop-Erasure of Plane Brownian Motion

TL;DR

This paper proves the existence of a loop-erasure for plane Brownian motion stopped at domain exit, showing it corresponds to the reversal of a radial SLE$_2$ curve, using coupling techniques.

Contribution

It introduces a coupling method to establish the loop-erasure of plane Brownian motion as the reversal of a radial SLE$_2$ curve, a novel connection in stochastic processes.

Findings

Loop-erasure of plane Brownian motion exists.

The loop-erased curve is the reversal of a radial SLE$_2$.

Coupling technique is used for the proof.

Abstract

We use the coupling technique to prove that there exists a loop-erasure of a plane Brownian motion stopped on exiting a simply connected domain, and the loop-erased curve is the reversal of a radial SLE$_2$ curve.