# 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 defined by

whose increments are iid random variables taking values in a domain and is the number of jumps in the interval . The probability for the process taking the value at time is then given by

Here is the probability for the process taking the value after jumps, and is the probability of having jumps after time .

## Montroll-Weiss formula

We denote by the waiting time in between two jumps of and by its distribution. The Laplace transform of is defined by

Similarly, the characteristic function of the jump distribution is given by its Fourier transform:

One can show that the Laplace-Fourier transform of the probability is given by

The above is called Montroll-Weiss formula.

## Examples

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

- ↑ Klages, Rainer; Radons, Guenther; Sokolov, Igor M.
*Anomalous Transport: Foundations and Applications*. - ↑ 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**: 167. 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. doi:10.1007/BF01016796. - ↑ Hilfer, R.; Anton, L. (1995). "Fractional master equations and fractal time random walks".
*Phys. Rev. E*.**51**(2): R848––R851. 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*. Elsevier.**34**(1): 87–103. doi:10.1016/j.chaos.2007.01.052.