Asymptotic distributions of continuous-time random walks: A probabilistic approach

We provide a systematic analysis of the possible asymptotic distributions o one-dimensional continuous-time random walks (CTRWs) by applying the limit theorems of probability theory. Biased and unbiased walks of coupled and decoupled memory are considered. In contrast to previous work concerning decoupled memory and Lévy walks, we deal also with arbitrary coupled memory and with jump densities asymmetric about its mean, obtaining asymmetric Lévy-stable limits. Suprisingly, it is found that in most cases coupled memory has no essential influence on the form of the limiting distribution. We discuss interesting properties of walks with an infinite mean waiting time between successive jumps.     

Asymptotic solutions of continuous-time random walks

The continuous-time random walk of Montroll and Weiss has a complete separation of time (how long a walker will remain at a site) and space (how far a walker will jump when it leaves a site). The time part is completely described by a pausing time distribution(t). This paper relates the asymptotic time behavior of the probability of being at sitel at timet to the asymptotic behavior of(t). Two classes of behavior are discussed in detail. The first is the familiar Gaussian diffusion packet which occurs, in general, when at least the first two moments of(t) exist; the other occurs when(t) falls off so slowly that all of its moments are infinite. Other types of possible behavior are mentioned. The relationship of this work to solutions of a generalized master equation and to transient photocurrents in certain amorphous semiconductors and organic materials is discussed.This work was partially supported by NSF Grant No. 28501.     

A geometric theory for Lévy distributions

Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT.     

The influence of the environment on Lévy random search efficiency: Fractality and memory effects

An open problem in the field of random searches relates to optimizing the search efficiency in fractal environments. Here we address this issue through a systematic study of Lévy searches in landscapes encompassing several degrees of target aggregation and fractality. For scarce resources, non-destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets. In contrast, no such universal behavior occurs in the destructive case with forbidden revisits, in which the optimal strategy strongly depends on the degree of target aggregation. We also investigate how the presence of memory and learning skills of the searcher affect the search efficiency. By considering a limiting model in which the searcher learns through recent experience to recognize food-rich areas, we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches. Instead, there is an optimal extent of memory, dependent on specific details of the search space and stochastic dynamics, which maximizes the search efficiency. This finding suggests a more general result, namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects.     

A probabilistic walk up power laws

We establish a path leading from Pareto’s law to anomalous diffusion, and present along the way a panoramic overview of power-law statistics. Pareto’s law is shown to universally emerge from “Central Limit Theorems” for rank distributions and exceedances, and is further shown to be a finite-dimensional projection of an infinite-dimensional underlying object — Pareto’s Poisson process  . The fundamental importance and centrality of Pareto’s Poisson process is described, and we demonstrate how this process universally generates an array of anomalous diffusion statistics characterized by intrinsic power-law structures: sub-diffusion and super-diffusion, Lévy laws and the “Noah effect”, long-range dependence and the “Joseph effect”, 1/f$1/f$ noises, and anomalous relaxation.     

Non-linear Shot Noise: Lévy, Noah, & Joseph

We introduce and study a generic non-linear Shot Noise system-model. Shots of random magnitudes arrive to the system stochastically, following an arbitrary time-homogeneous Poisson point process. After ‘hitting’ the system, the magnitude of an arriving shot decays to zero. The decay is governed by an arbitrary differential-equation dynamics. Shots are independent, and their overall effect on the system is additive: the system's noise level at time t equals the sum of the magnitudes, at time t, of all the shots arriving to the system prior to time t.The resulting Shot Noise is: (i) a Lévy process when the decay-dynamics are degenerate; (ii) a Lévy-driven Ornstein–Uhlenbeck process when the decay-dynamics are linear; and, (iii) a stationary non-Markov process when the decay-dynamics are non-linear.The resulting Shot Noise admits an underlying Lévy structure—which we explicitly compute, and can yield both the Noah effect and the Joseph effect. Closed-form analytic formulae for various statistics are derived, including: the log-Laplace transform and cumulants of the stationary noise level; the process’ auto-covariance function; and, the process’ range-of-dependence.