Power-law statistics from nonlinear stochastic differential equations driven by Lévy stable noise

Abstract

Anomalous diffusion occurring in complex dynamical systems can often be described by Langevin equations driven by Lévy stable noise. Nonlinear stochastic differential equations yielding power-law steady state distribution and generating signals with 1/f power spectral density can be generalized by replacing the Gaussian noise with a more general Lévy stable noise. These nonlinear equations can generate signals exhibiting anomalous diffusion: either sub-diffusion or super-diffusion. In a special case when stability index is α=2, we retain the equations with the Gaussian noise. We investigate numerically the frequency range where the spectrum has 1/f form and demonstrate that this frequency range depends on power-law exponent in steady state distribution as well as on the index of stability α. We expect that this generalization may be useful for describing 1/f fluctuations in the complex systems exhibiting anomalous diffusion.