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Non-Markovian Lévy dynamics and the effect of the underlying time correlation

The combined effect of symmetric α-stable Lévy flights and their underlying time correlations, namely, non-Markovian Lévy dynamics, is studied. By the use of the propagation function approach, which allows in a natural way for the inclusion of ‘correlated’ Lévy noise, we derive a version of the apparent width for the Lévy distribution. The correlated Lévy flights in the absence of nonlinearity are described by integral forms of coupling Langevin equations. Using a Monte-Carlo simulation of the motion of a particle in a metastable potential, we show the interfering phenomenon of the escape process. Finally, we expand the model to a Lévy-type random environment, i.e. a particle is coupled to a set of finite independent oscillators obeying the symmetric α-stable Lévy distributions. Thus, a non-thermalization generalized Langevin equation is proposed, where the fluctuation-dissipation theorem does not hold, which exhibits ballistic diffusion for any α. A numerical solution for Hamilton’s equations of motion with the initial random data confirms the present theoretical finding.