On the generation of anomalous and ultraslow diffusion

Abstract

We consider an ensemble of particles propagating randomly in a general space. The particles’ trajectories are independent and identically distributed copies of a general random spatial curve. The motions of the particles along their trajectories are subordinated to nonlinear clocks with random exponents. We characterize the class of Poissonian randomizations of the clocks’ exponents that render the statistical behavior of the ensemble invariant with respect to the particles’ trajectories. In particular, we show how anomalous diffusion and ultraslow diffusion statistical behaviors can be universally generated—regardless of the specific details of the particles’ trajectories.