Continuous-time random walk
CTRW was introduced by Montroll and Weiss  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.  A connection between CTRWs and diffusion equations with fractional time derivatives has been established.  Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices. 
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 .
Denoting the waiting time distribution in between two jumps of by , its Laplace transform is defined by
Similarly, for the jump distribution of the increments, the Fourier transform is given by
One can show that the Laplace-Fourier transform of the probability is given by
- 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 and 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. (2003). "On fractional diffusion and continuous time random walks". Physica A 329 (1): 35–40. doi:10.1103/PhysRevE.51.R848.
- 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.
|This probability-related article is a stub. You can help Wikipedia by expanding it.|