scholarly journals Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube

2008 ◽  
Vol 45 (03) ◽  
pp. 703-713 ◽  
Author(s):  
Stephen Connor ◽  
Saul Jacka
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Continuous Time Random Walks

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z 2 n . We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

scholarly journals Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube

2008 ◽  
Vol 45 (3) ◽  
pp. 703-713 ◽  
Author(s):  
Stephen Connor ◽  
Saul Jacka
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Continuous Time Random Walks

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

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 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.

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.

scholarly journals Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

2013 ◽  
Vol 50 (4) ◽  
pp. 1117-1130
Author(s):  
Stephen Connor
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Complete Graph ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Coupling Inequality ◽  
Maximal Coupling

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

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.

scholarly journals Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

2013 ◽  
Vol 50 (04) ◽  
pp. 1117-1130
Author(s):  
Stephen Connor
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Complete Graph ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Coupling Inequality ◽  
Maximal Coupling

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks onZ2dwas considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk onKnd, whereKnis the complete graph withnvertices. Moreover, we show that although this coupling is not maximal for anyn(i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling asn→ ∞.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (03) ◽  
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 {X n , n ≥ 0} and two observables, τ(∙) and V(∙), 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 {X n , n ≥ 0} is a sequence of independent and identically distributed random variables.

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.

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.

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