Abstract
In two-dimensional statistical mechanics, the Stochastic Loewner equation, introduced by Oded Schramm, provides a powerful technique to study and categorize critical random curves and interfaces. For the first time, a new type of stochastic Loewner equation entitled fractional stochastic Loewner evolution (FSLE) has been proposed. We introduce a wide class of fractal curves using the fractional time series as the driving function of the Loewner equation and local fractional integrodifferential operators. We suggest that, in addition to fractal dimension estimations, the Hurst index of the driving function leads significant differences in the FSLE curves. This modification introduces a new approach to categorize different types of scaling curves based on the Hurst index of the driving function. Such formalizations appear to be appropriate for studying a wide range of two-dimensional curves seen in statistical mechanics and natural phenomena.
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