SLE and α-SLE driven by Lévy processes

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.