Continuous-time random walk

In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting times.^[1]^[2]^[3] More generally it can be seen to be a special case of a Markov renewal process.

Motivation

CTRW was introduced by Montroll and Weiss^[4] as a generalization of physical diffusion processes 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

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

We denote by $\tau$ the waiting time in between two jumps of $N(t)$ and by $\psi (\tau )$ its distribution. The Laplace transform of $\psi (\tau )$ is defined by

{\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).

Similarly, the characteristic function of the jump distribution $f(\Delta X)$ is given by its Fourier transform:

{\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\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 the Montroll–Weiss formula.

Examples

References

^ Klages, Rainer; Radons, Guenther; Sokolov, Igor M. (2008-09-08). Anomalous Transport: Foundations and Applications. ISBN 9783527622986.
^ 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.
^ Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014.
^ Elliott W. Montroll; George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6 (2): 167. Bibcode:1965JMP.....6..167M. doi:10.1063/1.1704269.
^ . 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. Bibcode:1973JSP.....9...45K. doi:10.1007/BF01016796.
^ Hilfer, R.; Anton, L. (1995). "Fractional master equations and fractal time random walks". Phys. Rev. E. 51 (2): R848–R851. Bibcode:1995PhRvE..51..848H. doi:10.1103/PhysRevE.51.R848.
^ Gorenflo, Rudolf; Mainardi, Francesco; Vivoli, Alessandro (2005). "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons & Fractals. 34 (1): 87–103. arXiv:cond-mat/0701126. Bibcode:2007CSF....34...87G. doi:10.1016/j.chaos.2007.01.052.

[klages-1] Klages, Rainer; Radons, Guenther; Sokolov, Igor M. (2008-09-08). Anomalous Transport: Foundations and Applications. ISBN 9783527622986.

[PaulBaschnagel2013-2] 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.

[Slanina2013-3] Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014.

[4] Elliott W. Montroll; George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6 (2): 167. Bibcode:1965JMP.....6..167M. 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. Bibcode:1973JSP.....9...45K. doi:10.1007/BF01016796.

[6] Hilfer, R.; Anton, L. (1995). "Fractional master equations and fractal time random walks". Phys. Rev. E. 51 (2): R848–R851. Bibcode:1995PhRvE..51..848H. doi:10.1103/PhysRevE.51.R848.

[7] Gorenflo, Rudolf; Mainardi, Francesco; Vivoli, Alessandro (2005). "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons & Fractals. 34 (1): 87–103. arXiv:cond-mat/0701126. Bibcode:2007CSF....34...87G. doi:10.1016/j.chaos.2007.01.052.

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