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July 2008 SLE and α-SLE driven by Lévy processes
Qing-Yang Guan, Matthias Winkel
Ann. Probab. 36(4): 1221-1266 (July 2008). DOI: 10.1214/07-AOP355

Abstract

Stochastic Loewner evolutions (SLE) with a multiple $\sqrt{\kappa}B$ of Brownian motion B as driving process are random planar curves (if κ≤4) or growing compact sets generated by a curve (if κ>4). We consider here more general Lévy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. We show that when the driving force is of the form $\sqrt{\kappa}B+\theta^{1/\alpha}S$ for a symmetric α-stable Lévy process S, the cluster has zero or positive Lebesgue measure according to whether κ≤4 or κ>4. We also give mathematical evidence that a further phase transition at α=1 is attributable to the recurrence/transience dichotomy of the driving Lévy process. We introduce a new class of evolutions that we call α-SLE. They have α-self-similarity properties for α-stable Lévy driving processes. We show the phase transition at a critical coefficient θ=θ0(α) analogous to the κ=4 phase transition.

Citation

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Qing-Yang Guan. Matthias Winkel. "SLE and α-SLE driven by Lévy processes." Ann. Probab. 36 (4) 1221 - 1266, July 2008. https://doi.org/10.1214/07-AOP355

Information

Published: July 2008
First available in Project Euclid: 29 July 2008

zbMATH: 1151.60025
MathSciNet: MR2435848
Digital Object Identifier: 10.1214/07-AOP355

Subjects:
Primary: 60H10
Secondary: 60G51 , 60G52 , 60J45

Keywords: hitting times , Lévy process , self-similarity , stochastic Loewner evolution , α-stable process

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 4 • July 2008
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