Generalized Space–Time Fractional Dynamics in Networks and Lattices

Abstract

We analyze generalized space–time fractional motions on undirected networks and lattices. The continuous-time random walk (CTRW) approach of Montroll and Weiss is employed to subordinate a space fractional walk to a generalization of the time fractional Poisson renewal process. This process introduces a non-Markovian walk with long-time memory effects and fat-tailed characteristics in the waiting time density. We analyze ‘generalized space–time fractional diffusion’ in the infinite d-dimensional integer lattice \( \mathbb {Z}^d\). We obtain in the diffusion limit a ‘macroscopic’ space–time fractional diffusion equation. Classical CTRW models such as with Laskin’s fractional Poisson process and standard Poisson process which occur as special cases are also analyzed. The developed generalized space–time fractional CTRW model contains a four-dimensional parameter space and offers therefore a great flexibility to describe real-world situations in complex systems.