Stochastic Loewner evolution driven by L\'evy 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 L\'evy process. The situation is defined by the usual SLE parameter, $\kappa$, as well as $\alpha$ which defines the shape of the stable L\'evy 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. These descriptors are shown to change qualitatively and singularly at critical values of $\kappa$ and $\alpha$. 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.