  Standard stochastic Loewner evolution (SLE) is driven by a continuous
Brownian motion, which then produces a continuous fractal trace. If jumps are
added to the driving function, the trace branches. We consider a generalized
SLE driven by a superposition of a Brownian motion and a stable Levy process.
The situation is defined by the usual SLE parameter, $\kappa$, as well as
$\alpha$ which defines the shape of the stable Levy distribution. The resulting
behavior is characterized by two descriptors: $p$, the probability that the
trace self-intersects, and $\tilde{p}$, the probability that it will approach
arbitrarily close to doing so. Using Dynkin's formula, these descriptors are
shown to change qualitatively and singularly at critical values of $\kappa$ and
$\alpha$. It is reasonable to call such changes ``phase transitions''. These
transitions occur as $\kappa$ passes through four (a well-known result) and as
$\alpha$ passes through one (a new result). Numerical simulations are then used
to explore the associated touching and near-touching events.
