Levy flight processes solutions

The characteristic changes brought about by fractional dynamics in comparison to the Brownian case include the temporal nonlocality of the approach manifest in the convolution character of the fractional Riemann-Liouville operator. Initial conditions relax slowly, and thus they influence the evolution of the system even for long times [62, 116] furthermore, the Mittag-Leffler behavior replaces the exponential relaxation patterns of Brownian systems. Still, the associated fractional equations are linear and thus extensive, and the limit solution equilibrates toward the classical Gibbs-B oltzmann and Maxwell distributions, and thus the processes are close to equilibrium, in contrast to the Levy flight or generalised thermostatistics models under discussion. 

This exponent corresponds to a symmetric a-stable Levy process 8 (0, a Levy flight, which is self-similar with Hurst exponent H = Xja. It follows from (3.89) that the mesoscopic density of particles is the solution to the space-fractional diffusion equation [371] ... 

---