Fractional Stochastic Loewner Evolution and Scaling Curves

Abstract: The Stochastic Loewner equation, introduced by Schramm, gives us a powerful way to study and classify critical random curves and interfaces in two-dimensional statistical mechanics. New kind of stochastic Loewner equation, called fractional stochastic Loewner evolution (FSLE), has been proposed for the first time. Using the fractional time series as the driving function of the Loewner equation and local fractional integrodifferential operators, we introduce a large class of fractal curves. We argue that the FSLE curves, besides the fractal dimension calculations, have crucial differences which caused by the Hurst index of the driving function. This extension opens a new way to classify different types of scaling curves based on the Hurst index of the corresponding driving function. Such formalism appear to be suitable to deal with the study of a wide range of two-dimensional curves appearing in statistical mechanics or natural phenomena.