Strong path convergence from Loewner driving function convergence

Details Strong path convergence from Loewner driving function convergence Strong path convergence from Loewner driving function convergence

2010-03-24

arxiv.org/abs/1003.4675v2

Annals of Probability 2012, Vol. 40, No. 2, 578-610

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP627 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP627

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.

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