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

2004 ◽  
Vol 41 (03) ◽  
pp. 623-638 ◽  
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.

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

2004 ◽  
Vol 41 (3) ◽  
pp. 623-638 ◽  
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.

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

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

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.

scholarly journals Large Deviations for Continuous Time Random Walks

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 697 ◽  
Author(s):  
Wanli Wang ◽  
Eli Barkai ◽  
Stanislav Burov
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Exponential Decay ◽  
Continuous Time ◽  
Large Deviation ◽  
Waiting Times ◽  
Continuous Time Random Walk ◽  
Random Walk Model ◽  
Logarithmic Correction ◽  
Continuous Time Random Walks

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
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.

scholarly journals Limit theorems for additive functionals of continuous time random walks

2020 ◽  
pp. 1-22
Author(s):  
Yuri Kondratiev ◽  
Yuliya Mishura ◽  
Georgiy Shevchenko
Keyword(s):  
Continuous Time ◽  
Limit Theorems ◽  
Domain Of Attraction ◽  
Weak Limit ◽  
Additive Functional ◽  
Stable Law ◽  
Additive Functionals ◽  
Lévy Motion ◽  
Continuous Time Random Walks ◽  
Poisson Shot Noise

Abstract For a continuous-time random walk X = {X t , t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$ , t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

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

1998 ◽  
Vol 7 (4) ◽  
pp. 397-401 ◽  
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.

Simple random walk statistics. Part II: Continuous time results

1996 ◽  
Vol 33 (02) ◽  
pp. 331-339 ◽  
Author(s):  
W. Böhm ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Simple Random Walk ◽  
Laplace Transforms ◽  
Time Process ◽  
Basic Approach

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.

scholarly journals On the csáki-vincze transformation

2013 ◽  
Vol 50 (2) ◽  
pp. 266-279
Author(s):  
Hatem Hajri
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Asymptotic Properties ◽  
Scaling Limit ◽  
Simple Random Walk ◽  
Precise Description ◽  
Discrete Transformation

Csáki and Vincze have defined in 1961 a discrete transformation T which applies to simple random walks and is measure preserving. In this paper, we are interested in ergodic and asymptotic properties of T. We prove that T is exact: ∩k≧1σ(Tk(S)) is trivial for each simple random walk S and give a precise description of the lost information at each step k. We then show that, in a suitable scaling limit, all iterations of T “converge” to the corresponding iterations of the continuous Lévy transform of Brownian motion.

Simple random walk statistics. Part II: Continuous time results

1996 ◽  
Vol 33 (2) ◽  
pp. 331-339 ◽  
Author(s):  
W. Böhm ◽  
W. Panny
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Discrete Time ◽  
Continuous Time ◽  
Simple Random Walk ◽  
Laplace Transforms ◽  
Time Process ◽  
Basic Approach

In this paper various statistics for randomized random walks and their distributions are presented. The distributional results are derived by means of a limiting procedure applied to the pertaining discrete time process, which has been considered in part I of this work (Katzenbeisser and Panny 1996). This basic approach, originally due to Meisling (1958), seems to offer certain technical advantages, since it avoids the use of Laplace transforms and is even simpler than Feller's randomization technique.

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