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

@article{Meerschaert2004LimitTF,
  title={Limit theorems for continuous-time random walks with infinite mean waiting times},
  author={Mark M. Meerschaert and Hans-Peter Scheffler},
  journal={Journal of Applied Probability},
  year={2004},
  volume={41},
  pages={623 - 638}
}
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… 
Triangular array limits for continuous time random walks
Semi-Markov approach to continuous time random walk limit processes
Continuous time random walks (CTRWs) are versatile models for anomalous diffusion processes that have found widespread application in the quantitative sciences. Their scaling limits are typically
Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks
TLDR
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 are established.
Limit theorems for continuous time random walks with continuous paths
The continuous time random walks (CTRWs) are typically defned in the way that their trajectories are discontinuous step fuctions. This may be a unwellcome feature from the point of view of
Chung-type law of the iterated logarithm for continuous time random walk
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 a Chung-type law of the iterated logarithm
Continuous time random walks and space-time fractional differential equations
The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random
ORACLE CONTINUOUS TIME RANDOM WALKS
In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW model is useful in physics, to model dif- fusing particles. Its scaling limit is a time-changed
...
...

References

SHOWING 1-10 OF 63 REFERENCES
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
Random walks with infinite spatial and temporal moments
The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding
From continuous time random walks to the fractional fokker-planck equation
  • Barkai, Metzler, Klafter
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
TLDR
The domain of validity of the fractional kinetic equation is discussed, and the CTRW solution and that of the FFPE are compared for the force free case.
Random Walk Approach to Relaxation in Disordered Systems
A detailed study of the limiting probability distributions of R(t) — the location at time t — for one-dimensional random walks with waiting-time distributions having long tails, is presented. In the
Fractional kinetic equations: solutions and applications.
TLDR
Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed, presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process.
The fractional‐order governing equation of Lévy Motion
A governing equation of stable random walks is developed in one dimension. This Fokker‐Planck equation is similar to, and contains as a subset, the second‐order advection dispersion equation (ADE)
Stable non-Gaussian random processes
The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the
Lévy flights and related topics in physics : proceedings of the international workshop held at Nice, France, 27-30 June 1994
Variability of anomalous transport exponents versus different physical situations in geophysical and laboratory turbulence.- Conditionally-averaged dynamics of turbulence, new scaling and stochastic
Some Useful Functions for Functional Limit Theorems
TLDR
This paper facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence.
...
...