Randomly Trapped Random Walks

The model introduced in this work model, Randomly Trapped Random Walk (RTRW), is a general framework to study a one-dimensional motion in a random trapping environment. We develop tools to study its scaling limits. We discover that all possible limits of RTRW can be represented as a Brownian motion time-changed by a Levy subordinator. More precisely, we will show that if the trapping landscape is homogeneous, RTRW exhibits diffusive behavior. On the other hand, in inhomogeneous environments, RTRW is non-diffusive. The set of possible limits of RTRW (in inhomogeneous environments) is extremely rich. In particular, it includes Fractional Kinetics, which was initially introduced as a scaling limit of Continuous Time Random Walk, as well as a new broad class of processes, which we call Randomly Trapped Brownian Motions. This class contains a singular diffusion, which was recently introduced by Fontes, Isopi and Newman. We give various examples, some of the of a geometric nature, i.e. the Comb Model.