Spatial gliding, temporal trapping, and anomalous transport

We introduce and study a model of stochastic dynamics, governed by Markovian laws of motion in a phase space filled with random traps. The resulting (non-Markovian) stochastic trapped motion alternates intermittently between periods of spatial gliding, where the motion ‘glides’ the underlying Markovian dynamics, and periods of temporal trapping, where the motion is halted in random traps. We investigate the asymptotics and scaling limits of this model. We prove that when the random trappings are heavy-tailed then Mittag–Leffler functions and probability laws emerge and govern the functional structure and statistics of the system, and that the time flow has a random fractal structure whose fractal exponent (dimension) is determined by the ‘heaviness’ of the trappings. We study the effect of random trapping on general Lévy dynamics. We prove that subjecting Lévy dynamics to heavy-tailed trapping will always result in: (i) sub-diffusive behavior—when the underlying Lévy dynamics are of finite variance; and (ii) space–time fractal behavior—when the underlying Lévy dynamics are scale-invariant. Furthermore, we explore the issue of first exit times. To that end, a general Feynman–Kac framework for trapped processes is developed, and a method of transforming ‘trapped’ Feynman–Kac equations to ‘standard’ ones is established. The study of first exit times enables us to quantitatively connect macroscopic observations to microscopic behavior in general Markovian dynamics subjected to random trapping. In the case of Lévy dynamics, first exit times from balls are computed, and the relationships between their statistics and the statistics of the trapped motion are derived.