Stochastic Loewner evolution driven by Levy processes

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 a, which defines the shape of the stable Levy distribution. The resulting behaviour is characterized by two descriptors: p, the probability that the trace self-intersects, and similar to (p) over tilde, 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. and a. It is reasonable to call such changes 'phase transitions'. These transitions occur as kappa passes through four (a well-known result) and as a passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.