SLE curves and natural parametrization

TL;DR

This paper establishes the well-defined nature of the natural parametrization for SLE curves with parameter <8, using a two-interior-point local martingale approach, advancing the mathematical understanding of SLE dynamics.

Contribution

It introduces a proof that the natural parametrization of <8 SLE curves is well defined, utilizing a novel two-interior-point local martingale method.

Findings

Natural parametrization is well defined for all <8 SLE curves.

The proof employs a two-interior-point local martingale.

Advances understanding of SLE curve parametrization.

Abstract

Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_{\kappa}$ curves is well defined for all $\kappa<8$. Our proof uses a two-interior-point local martingale.