Continuous-time random walk

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In mathematics, a continuous-time random walk (CTRW) is a generalization of a stochastic jump process with arbitrary distributions of jump lengths and waiting times.[1][2][3]

Motivation[edit]

CTRW was introduced by Montroll and Weiss [4] as a generalization of physical diffusion process to effectively describe anomalous diffusion, i.e., the super- and sub-diffusive cases. An equivalent formulation of the CTRW is given by generalized master equations. [5] A connection between CTRWs and diffusion equations with fractional time derivatives has been established. [6] Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices. [7]

Formulation[edit]

A simple formulation of a CTRW is to consider the stochastic process X(t) defined by


X(t) = X_0 + \sum_{i=1}^{N(t)} \Delta X_i,

whose increments \Delta X_i are iid random variables taking values in a domain \Omega and N(t) is the number of jumps in the interval  (0,t). The probability for the process taking the value X at time t is then given by


P(X,t) = \sum_{n=0}^\infty P(n,t) P_n(X).

Here P_n(X) is the probability for the process taking the value X after n jumps, and P(n,t) is the probability of having n jumps after time t.

Montroll-Weiss formula[edit]

Denoting the waiting time distribution in between two jumps of N(t) by \psi(\tau), its Laplace transform is defined by


\tilde{\psi}(s)=\int_0^{\infty} d\tau e^{-\tau s} \psi(\tau).

Similarly, for the jump distribution  f(\Delta X) of the increments, the Fourier transform is given by


\hat{f}(k)=\int_\Omega d(\Delta X) e^{i k\Delta X} f(\Delta X).

One can show that the Laplace-Fourier transform of the probability P(X,t) is given by


\hat{\tilde{P}}(k,s) = \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\tilde{\psi}(s)\hat{f}(k)}.

The above is called Montroll-Weiss formula.

Examples[edit]

The Wiener process is the standard example of a continuous time random walk in which the waiting times are exponential and the jumps are continuous and normally distributed.

References[edit]

  1. ^ Klages, Rainer; Radons, Guenther; Sokolov, Igor M. Anomalous Transport: Foundations and Applications. 
  2. ^ Paul, Wolfgang; Baschnagel, Jörg (2013-07-11). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. pp. 72–. ISBN 9783319003276. Retrieved 25 July 2014. 
  3. ^ Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014. 
  4. ^ Elliott W. Montroll and George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6: 167. doi:10.1063/1.1704269. 
  5. ^ . M. Kenkre, E. W. Montroll, M. F. Shlesinger (1973). "Generalized master equations for continuous-time random walks". Journal of Statistical Physics 9 (1): 45–50. doi:10.1007/BF01016796. 
  6. ^ Hilfer, R. (2003). "On fractional diffusion and continuous time random walks". Physica A 329 (1): 35–40. doi:10.1103/PhysRevE.51.R848. 
  7. ^ Gorenflo, Rudolf and Mainardi, Francesco and Vivoli, Alessandro (2005). "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons \& Fractals (Elsevier) 34 (1): 87–103. doi:10.1016/j.chaos.2007.01.052.