Generalized continuous time random walks, master equations, and fractional Fokker-Planck equations

Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space- and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker–Planck equations in the diffusion limit. Thus the fractional Fokker–Planck equation with space- and time-dependent forcing is robust to these changes in the underlying stochastic process