Realization of Lévy walks as Markovian stochastic processes.

Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.