Non-Lévy stable random walk propagators for a non-Markovian walk with both superdiffusive and subdiffusive regimes

Abstract

One of the most important quantities of interest in the theory of diffusion and transport is the random walk propagator. For Markovian processes, such as the standard Brownian random walk and Lévy flights, the functional form of the random walk propagator is well understood. Similarly, for certain kinds of simple non-Markovian processes, such as Lévy walks, the problem can be mapped to a solvable Markovian model. However, more complicated non-Markovian walks pose a challenge. Here we study a non-Markovian model that is rich enough to exhibit superdiffusion, normal diffusion and subdiffusion regimes (Kumar, Harbola, and Lindenberg (2010)). We numerically estimate propagators for this model and obtain good fits with a family of non-Lévy propagators based on the Tsallis q-exponential function. We conclude that stops and restrictions play similar roles in the long time limit of the propagator.