The Green's function for the radial Schramm-Loewner evolution

TL;DR

This paper establishes the existence of the Green's function for radial SLE(k) processes for k<8, providing explicit formulas for k=4 and a conditional expectation representation for other k values.

Contribution

It proves the existence of the Green's function for radial SLE(k) and derives explicit formulas for specific cases, advancing understanding of SLE's potential theory.

Findings

Green's function exists for radial SLE(k) with k<8

Explicit formula provided for k=4

Conditional expectation formula for other k values

Abstract

We prove the existence of the Green's function for radial SLE(k) for k<8. Unlike the chordal case where an explicit formula for the Green's function is known for all values of k<8, we give an explicit formula only for k=4. For other values of k, we give a formula in terms of an expectation with respect to SLE conditioned to go through a point.