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 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

A simple formulation of a CTRW is to consider the stochastic process ${\displaystyle X(t)}$ defined by

${\displaystyle X(t)=X_{0}+\sum _{i=1}^{N(t)}\Delta X_{i},}$

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

${\displaystyle P(X,t)=\sum _{n=0}^{\infty }P(n,t)P_{n}(X).}$

Here ${\displaystyle P_{n}(X)}$ is the probability for the process taking the value ${\displaystyle X}$ after ${\displaystyle n}$ jumps, and ${\displaystyle P(n,t)}$ is the probability of having ${\displaystyle n}$ jumps after time ${\displaystyle t}$.

Montroll-Weiss formula

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

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

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

${\displaystyle {\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\Delta X}f(\Delta X).}$

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

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

The above is called Montroll-Weiss formula.

Examples

The homogeneous Poisson point process is a continuous time random walk with exponential holding times and with each increment deterministically equal to 1.

References

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