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 Brownianmotion and a stable Lévy process. The situation is defined by the usual SLE parameter,κ, aswell as α, which defines the shape of the stable Lévy distribution. The resulting behaviour is characterized by twodescriptors: p, the probability that the trace self-intersects, and , 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 ofκ and α. It is reasonable to call such changes ‘phase transitions’. These transitions occur asκ passes through four (a well-known result) and asα passes through one (a new result). Numerical simulations are then used to explore theassociated touching and near-touching events.