An elementary approach to subdiffusion

We consider a basic one-dimensional model which allows to obtain a diversity of diffusive regimes whose speed depends on the moments of a per-site trapping time. This models a discrete subordinated random walk, closely related to the continuous time random walks widely studied in the literature. The model we consider lends itself to a detailed elementary treatment, based on the study of recurrence relation for the time-[Formula: see text] dispersion of the process, making it possible to study deviations from normality due to finite time effects.

Download Full-text

Limit theorems for some continuous-time random walks

Advances in Applied Probability ◽

10.1239/aap/1316792670 ◽

2011 ◽

Vol 43 (3) ◽

pp. 782-813 ◽

Cited By ~ 1

Author(s):

M. Jara ◽

T. Komorowski

Keyword(s):

Markov Chain ◽

Random Walk ◽

Random Walks ◽

Quantum Transport ◽

Continuous Time ◽

Limit Theorems ◽

Transport Theory ◽

Sufficient Conditions ◽

Continuous Time Random Walks ◽

Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

Download Full-text

From Dispersion to Laminarity in Dynamical Systems

Zeitschrift für Naturforschung A ◽

10.1515/zna-1994-1225 ◽

1994 ◽

Vol 49 (12) ◽

pp. 1241-1247 ◽

Cited By ~ 1

Author(s):

G. Zumofen ◽

J. Klafter

Keyword(s):

Random Walk ◽

Dynamical Systems ◽

Continuous Time ◽

Chaotic Behavior ◽

Continuous Time Random Walk ◽

Standard Map ◽

One Dimensional ◽

Enhanced Diffusion

Abstract We study transport in dynamical systems characterized by intermittent chaotic behavior with coexistence of dispersive motion due to periods of localization, and of enhanced diffusion due to periods of laminar motion. This transport is discussed within the continuous-time random walk approach which applies to both dispersive and enhanced motions. We analyze the coexistence for the standard map and for a one-dimensional map.

Download Full-text

Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks

Journal of Applied Mathematics ◽

10.1155/2012/906373 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13

Author(s):

Kyo-Shin Hwang ◽

Wensheng Wang

Keyword(s):

Random Walk ◽

Random Walks ◽

Anomalous Diffusion ◽

Renewal Process ◽

Continuous Time ◽

Waiting Times ◽

Continuous Time Random Walk ◽

Domains Of Attraction ◽

Iterated Logarithm ◽

Continuous Time Random Walks

A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.

Download Full-text

Continuous Time Random Walk Based Theory for a One-Dimensional Coarsening Model

Journal of Elliptic and Parabolic Equations ◽

10.1007/bf03377401 ◽

2016 ◽

Vol 2 (1-2) ◽

pp. 189-206 ◽

Cited By ~ 1

Author(s):

Diego Torrejon ◽

Maria Emelianenko ◽

Dmitry Golovaty

Keyword(s):

Random Walk ◽

Continuous Time ◽

Continuous Time Random Walk ◽

One Dimensional

Download Full-text

On a Conjecture of Bollobás and Brightwell Concerning Random Walks on Product Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548398003605 ◽

1998 ◽

Vol 7 (4) ◽

pp. 397-401 ◽

Cited By ~ 2

Author(s):

OLLE HÄGGSTRÖM

Keyword(s):

Random Walk ◽

Random Walks ◽

Discrete Time ◽

Complete Graph ◽

Continuous Time ◽

Product Graph ◽

Product Graphs ◽

Continuous Time Random Walks ◽

Discrete Time Case

We consider continuous time random walks on a product graph G×H, where G is arbitrary and H consists of two vertices x and y linked by an edge. For any t>0 and any a, b∈V(G), we show that the random walk starting at (a, x) is more likely to have hit (b, x) than (b, y) by time t. This contrasts with the discrete time case and proves a conjecture of Bollobás and Brightwell. We also generalize the result to cases where H is either a complete graph on n vertices or a cycle on n vertices.

Download Full-text

Limiting stochastic processes of shift-periodic dynamical systems

Royal Society Open Science ◽

10.1098/rsos.191423 ◽

2019 ◽

Vol 6 (11) ◽

pp. 191423

Author(s):

Julia Stadlmann ◽

Radek Erban

Keyword(s):

Brownian Motion ◽

Random Walk ◽

Dynamical Systems ◽

Stochastic Processes ◽

Discrete Time ◽

Real Line ◽

Continuous Time ◽

Initial Distribution ◽

Dynamical Behaviour ◽

One Dimensional

A shift-periodic map is a one-dimensional map from the real line to itself which is periodic up to a linear translation and allowed to have singularities. It is shown that iterative sequences x n +1 = F ( x n ) generated by such maps display rich dynamical behaviour. The integer parts ⌊ x n ⌋ give a discrete-time random walk for a suitable initial distribution of x 0 and converge in certain limits to Brownian motion or more general Lévy processes. Furthermore, for certain shift-periodic maps with small holes on [0,1], convergence of trajectories to a continuous-time random walk is shown in a limit.

Download Full-text

On certain unrestricted, linear, unit step, continuous time random walks

Journal of Applied Probability ◽

10.1017/s0021900200035385 ◽

1971 ◽

Vol 8 (02) ◽

pp. 374-380

Author(s):

A. E. Gibson ◽

B. W. Conolly

Keyword(s):

Random Walk ◽

Continuous Time ◽

Queueing Theory ◽

Basic Assumption ◽

The Other ◽

Time Intervals ◽

Queueing Process ◽

Unit Step ◽

Continuous Time Random Walks ◽

The One

A basic process of simple queueing theory is S(t), an integer valued stochastic process which represents the “number of clients present, including the one in service” at epoch t. The queueing context physically limits S(t) to non-negative values, but if this impenetrable barrier is removed thereby permitting S(t) to assume negative values as well, we have a “randomized random walk” in which S(t) represents the position at time t of a mythical particle which moves on the x-axis according to the rules of the walk. The nomenclature “randomized random walk” is due to Feller (1966a). S(t) changes by unit positive or negative amounts, and our basic assumption is that the time intervals τ/σ separating successive positive/negative steps are i.i.d. with continuous d.f.'s. A(t)/B(t) (A(0)/B(0) = 0) possessing p.d.f.'s. a(t)/b(t): τ and σ are further assumed to be statistically independent. When A(t) = 1 – e–λt and B(t) = 1 – e–μt , we have a generalization of the M/M/1 queueing process which we shall denote by M/M. The walk generated by A(t) = 1 – e–λt and B(t) arbitrary may similarly be denoted by M/G. For G/M we clearly mean A(t) arbitrary and B(t) = 1 – e–μt . M/M has received extensive treatment elsewhere (cf. Feller (1966a), (1966b); Takács (1967); Gibson (1968); Conolly (1971)), and will not be considered here. Our interest centers on certain aspects of M/G and G/M, but since each is the dual of the other (loosely, G/M is M/G “upside down”) it is sufficient to restrict attention to one of them, as convenient; pointing out, where necessary, the interpretation in terms of the other.

Download Full-text

Limit theorems for continuous-time random walks with infinite mean waiting times

Journal of Applied Probability ◽

10.1017/s002190020002043x ◽

2004 ◽

Vol 41 (03) ◽

pp. 623-638 ◽

Cited By ~ 31

Author(s):

Mark M. Meerschaert ◽

Hans-Peter Scheffler

Keyword(s):

Random Walk ◽

Random Walks ◽

Continuous Time ◽

Waiting Times ◽

Scaling Limit ◽

Domain Of Attraction ◽

Simple Random Walk ◽

Lévy Motion ◽

Hamiltonian Chaos ◽

Continuous Time Random Walks

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Download Full-text