Continuous-time random walk explained

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 byX(t) = X _{0} + \sum ^{N(t)} _{i =1} \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 byP(X,t) = \sum ^{inf ty} _{n =0} 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 ^{inf ty} _{0} d\tau e ^{-\tau} \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 ^{i} 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) = (1- \tilde{\psi)/((s)}{s}) (1)/(1- \tilde{\psi) (s)\hat{f}(k)}.

The above is called the Montroll–Weiss formula.

Notes and References

1. Book: Klages. Rainer. Radons. Guenther. Igor M.. Sokolov. Anomalous Transport: Foundations and Applications. 9783527622986. 2008-09-08.
2. Book: Paul. Wolfgang. Baschnagel. Jörg. Stochastic Processes: From Physics to Finance. 25 July 2014. 2013-07-11. Springer Science & Business Media. 9783319003276. 72–.
3. Book: Slanina, Frantisek. Essentials of Econophysics Modelling. 25 July 2014. 2013-12-05. OUP Oxford. 9780191009075. 89–.
4. Elliott W. Montroll . George H. Weiss . Random Walks on Lattices. II . J. Math. Phys. . 6 . 2 . 167 . 1965 . 10.1063/1.1704269 . 1965JMP.....6..167M.
5. . M. Kenkre . E. W. Montroll . M. F. Shlesinger . Generalized master equations for continuous-time random walks . Journal of Statistical Physics . 9 . 1 . 45–50 . 1973 . 10.1007/BF01016796 . 1973JSP.....9...45K.
6. Hilfer, R. . Anton, L. . Fractional master equations and fractal time random walks . Phys. Rev. E . 51 . 2 . R848–R851 . 1995 . 10.1103/PhysRevE.51.R848 . 1995PhRvE..51..848H.
7. Gorenflo . Rudolf . Rudolf Gorenflo . Mainardi . Francesco . Vivoli . Alessandro . Continuous-time random walk and parametric subordination in fractional diffusion . Chaos, Solitons & Fractals . 34 . 1 . 87–103 . 2005 . 10.1016/j.chaos.2007.01.052 . cond-mat/0701126. 2007CSF....34...87G.