Langevin Picture of L,vy Walks and Their Extensions

In this paper we derive Langevin picture of L,vy walks. Applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled L,vy walk. Next, we introduce extensions of Levy walks, in which jump sizes are some functions of waiting times. We prove that under proper scaling conditions, such generalized L,vy walks converge in distribution to the appropriate limiting processes. We also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes, showing that different coupling functions lead to various types of anomalous diffusion.