SLE and alpha-SLE driven by Levy processes

TL;DR

This paper generalizes Stochastic Loewner Evolutions by incorporating Levy processes as driving forces, revealing phase transitions related to measure and recurrence properties, and introducing alpha-SLE with self-similarity features.

Contribution

It introduces alpha-SLE driven by Levy processes, extending classical SLE, and identifies phase transitions linked to Levy process characteristics.

Findings

Cluster measure depends on kappa, being zero or positive.

Phase transition at alpha=1 related to Levy process recurrence.

Critical coefficient theta_0(alpha) marks phase change in alpha-SLE.

Abstract

Stochastic Loewner Evolutions (SLE) with a multiple sqrt(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 sqrt(kappa)B+theta^(1/alpha)S for a symmetric alpha-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 dychotomy 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.