Cadlag curves of SLE driven by Levy processes

TL;DR

This paper investigates the nature of hulls generated by Schramm Loewner Evolutions driven by Levy processes, establishing conditions under which these hulls are generated by Cadlag curves, extending known results beyond Brownian motion drivers.

Contribution

It proves that SLE driven by a combination of Brownian motion and symmetric alpha-stable processes produces hulls generated by Cadlag curves, broadening the understanding of SLE path regularity.

Findings

Hulls are generated by Cadlag curves for certain Levy-driven SLE.

Extends SLE theory to non-Brownian driving processes.

Provides conditions on parameters for Cadlag curve generation.

Abstract

Schramm Loewner Evolutions (SLE) are random increasing hulls defined through the Loewner equation driven by Brownian motion. It is known that the increasing hulls are generated by continuous curves. When the driving process is of the form \sqrt{\kappa} B+\theta^{1/\alpha}S for a Brownian motion B and a symmetric \alpha-stable process S with \kappa not equal to 4 and 8, we prove that the corresponding increasing hulls are generated by Cadlag curves.