Strong path convergence from Loewner driving function convergence

TL;DR

This paper proves that convergence of Loewner driving functions implies uniform convergence of simple planar curves, extending to random curves like SLE, and clarifies conditions needed for such convergence proofs.

Contribution

It establishes that Loewner driving function convergence guarantees uniform convergence of curves, including in the random case, under mild assumptions and specific conditions.

Findings

Driving function convergence implies curve convergence.

Extension to random curves such as SLE.

Conditions for driving function convergence in both directions.

Abstract

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version applies in particular to random lattice paths that have chordal $\mathrm {SLE}_{\kappa}$ as a scaling limit, with $\kappa <8$ (nonspace-filling). Existing $\mathrm {SLE}_{\kappa}$ convergence proofs often begin by showing that the Loewner driving functions of these paths (viewed from $\infty$) converge to Brownian motion. Unfortunately, this is not sufficient, and additional arguments are required to complete the proofs. We show that driving function convergence is sufficient if it can be established for both parametrization directions and a generic observation point.