Random Walks in a One-Dimensional Lévy Random Environment (original) (raw)
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Journal of Physics A, 2017
We consider super-diffusive Lévy walks in d 2 dimensions when the duration of a single step, i.e., a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite variance and finite mean. We demonstrate that the probability density function (PDF) of the coordinate of the random walker has two different scaling limits at large times. One limit describes the bulk of the PDF. It is the d−dimensional generalization of the one-dimensional Lévy distribution and is the counterpart of central limit theorem (CLT) for random walks with finite dispersion. In contrast with the one-dimensional Lévy distribution and the CLT this distribution does not have universal shape. The PDF reflects anisotropy of the single-step statistics however large the time is. The other scaling limit, the so-called 'infinite density', describes the tail of the PDF which determines second (dispersion) and higher moments of the PDF. This limit repeats the angular structure of PDF of velocity in one step. Typical realization of the walk consists of anomalous diffusive motion (described by anisotropic d−dimensional Lévy distribution) intermitted by long ballistic flights (described by infinite density). The long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion. We illustrate the concept by considering two types of Lévy walks, with isotropic and anisotropic distributions of velocities. Furthermore, we show that for isotropic but otherwise arbitrary velocity distribution the d−dimensional process can be reduced to one-dimensional Lévy walk.
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We investigate a Lévy-Walk alternating between velocities ±v0 with opposite sign. The sojourn time probability distribution at large times is a power law lacking its mean or second moment. The first case corresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at large times is x 2 ∝ t 2 , the latter to enhanced diffusion with x 2 ∝ t ν , 1 < ν < 2. The correlation function and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSD from a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function. In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yet very slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averaged MSDs.
Asymptotic densities of ballistic Lévy walks
Physical review. E, Statistical, nonlinear, and soft matter physics, 2015
We propose an analytical method to determine the shape of density profiles in the asymptotic long-time limit for a broad class of coupled continuous-time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random-walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard nonballistic random walks.
Transport properties of Lévy walks: An analysis in terms of multistate processes
EPL (Europhysics Letters), 2014
Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of Lévy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker, we are led to a description of such Lévy walks in terms of multistate processes whose time-evolution is shown to obey a set of coupled delay differential equations. Using simple arguments, we obtain asymptotic solutions to these equations and rederive the scaling laws for the mean squared displacement of such processes. Our calculation includes the computation of all relevant transport coefficients in terms of the parameters of the models.
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Probability Theory and Related Fields, 2004
We consider a general model of discrete-time random walk X t on the lattice ν, ν = 1,..., in a random environment ξ={ξ(t,x):(t,x)∈ ν+1} with i.i.d. components ξ(t,x). Previous results on the a.s. validity of the Central Limit Theorem for the quenched model required a small stochasticity condition. In this paper we show that the result holds provided only that an obvious non-degeneracy condition is met. The proof is based on the analysis of a suitable generating function, which allows to estimate L 2 norms by contour integrals.
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Let X be a Lévy process with regularly varying Lévy measure ν. We obtain sample-path large deviations for scaled processes X̄n(t) , X(nt)/n and obtain a similar result for random walks with regularly varying increments. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but a full large deviation principle does not hold.
Convergence of UUU-statistics indexed by a random walk to stochastic integrals of a Lévy sheet
Bernoulli, 2017
A U -statistic indexed by a Z d 0 -random walk (Sn)n is a process Un := n i,j=1 h(ξ S i , ξ S j ) where (ξ k ) k is a sequence of iid random variables (independent of the walk) and h some real-valued function. We assume that either (Sn)n is transient or the distribution of S 1 − S 0 belongs to the normal domain of attraction of a strictly stable distribution of exponent α ∈ (0, 2]. We assume also that the distribution of h(ξ 1 , ξ 2 ) belongs to the normal domain of attraction of a strictly stable distribution of exponent β ∈ (0, 2). We establish the convergence in distribution of ((U ⌊nt⌋ /an)t)n to some observable of a Lévy sheet Z (for a suitable sequence (an)n). The limit process is (Zt,t)t when the walk is transient or null recurrent and some stochastic integral with respect to Z when the walk is positive recurrent. This behaviour is in some sense analogous to that of random walks in random scenery.
Random Walks with Bivariate Lévy-Stable Jumps in Comparison with Lévy Flights
Acta Physica Polonica B - ACTA PHYS POL B, 2009
In this paper we compare the Lévy flight model on a plane with the random walk resulting from bivariate Lévy-stable random jumps with the uniform spectral measure. We show that, in general, both processes exhibit similar properties, i.e. they are characterized by the presence of the jumps with extremely large lenghts and uniformly distributed directions (reflecting the same heavy-tail behavior and the spherical symmetry of the jump distributions), connecting characteristic clusters of short steps. The bivariate Lévy-stable random walks, belonging to the class of the well investigated stable processes, can enlarge the class of random-walk models for transport phenomena if other than uniform spectral measures are considered.