SLE and \(\alpha \)-SLE driven by Lévy processes. (English) Zbl 1151.60025

From the authors’ abstract: Stochastic Loewner evolutions (SLE) with a multiple \(\sqrt {\kappa}B\) of Brownian motion \(B\) as driving process are random planar curves (if \(\kappa \leq 4\)) or growing compact sets generated by a curve (if \(\kappa >4\)). Now, consider more general Lévy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. The authors show that when the driving force is of the form \(\sqrt {\kappa} B + \theta^{1/\alpha}S\) for a symmetric \(\alpha \)-stable Lévy process \(S\), the cluster has zero or positive Lebesgue measure according to whether \(\kappa \leq 4\) or \(\kappa >4\). They also give mathematical evidence that a further phase transition at \(\alpha =1\) is attributable to the recurrence/transience dichotomy of the driving Lévy process. The authors introduce a new class of evolutions that we call \(\alpha \)-SLE. They have \(\alpha \)-self-similarity properties for \(\alpha \)-stable Lévy driving processes. The authors show the phase transition at a critical coefficient \(\theta =\theta _{0}(\alpha )\) analogous to the \(\kappa =4\) phase transition.


60G52 Stable stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G51 Processes with independent increments; Lévy processes
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J45 Probabilistic potential theory
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