SLE and α-SLE driven by Levy processes

• 成果类型: Article
• 署名作者: Guan, Qing-Yang; Winkel, Matthias
• 署名单位: Chinese Academy of Sciences; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/07-AOP355
• 发表日期: 2008
• 页码: 1221-1266
• 关键词: erased random-walks; conformal-invariance; FORMULA; plane

摘要

Stochastic Loewner evolutions (SLE) with a multiple root kappa B of Brownian motion B as driving process are random planar curves (if kappa < 4) or growing compact sets generated by a curve (if kappa > 4). We consider here more general Levy 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 root kappa B + theta(1/alpha) S for a symmetric a-stable Levy process S, the cluster has zero or positive Lebesgue measure according to whether kappa <= 4 or kappa > 4. We also give mathematical evidence that a further phase transition at alpha = 1 is attributable to the recurrence/transience dichotomy of the driving Levy process. We introduce a new class of evolutions that we call alpha-SLE. They have alpha-self-similarity properties for alpha-stable Levy driving processes. We show the phase transition at a critical coefficient theta = theta(0)(alpha) analogous to the kappa = 4 phase transition.