  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.
