Anomalous transport in the presence of truncated lévy flights

The role of truncated Lévy flights on anomalous diffusion and front propagation is studied. Starting from the Continuous Time Random Walk for general Lévy stochastic processes, an integro-differential equation describing the dynamics in the long-wavelength limit is obtained. In the case of exponentially tempered Lévy flights, the transport equation involves exponentially truncated fractional derivatives describing the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from super-diffusion to sub-diffusion. The decay of the tail of the Green’s function changes from algebraic at short times, to stretched exponential at long times. The role of truncation in the super-diffusive propagation of fronts in the tempered fractional Fisher–Kolmogorov equation is also studied. In the absence of truncation, the fronts have exponential acceleration and algebraic decaying tails. In the presence of truncation, this phenomenology prevails in an intermediate asymptotic regime. Outside this regime, the front’s tail exhibits tempered decay, the acceleration is transient, and the front velocity exhibits an algebraically slow convergence to a terminal velocity. In the over-truncated regime, fronts have exponential tails and move at a constant velocity.