Stochastic Loewner evolution driven by Levy processes

Abstract

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.

Published version, minor typos corrected, added references

References in corpus (6)

• SLE for theoretical physicists
• A Guide to Stochastic Loewner Evolution and its Applications
• Conformally Invariant Fractals and Potential Theory
• Exact Solutions for Loewner Evolutions
• Exact Multifractal Spectra for Arbitrary Laplacian Random Walks
• Trace for the Loewner Equation with Singular Forcing

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• Stochastic Loewner Evolution Relates Anomalous Diffusion and Anisotropic Percolation
• Theta-point polymers in the plane and Schramm-Loewner evolution
• q-deformed Loewner evolution
• On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth
• Stochastic Loewner Evolutions, Fuchsian Systems and Orthogonal Polynomials