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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.cite-ref-klages-1-0[1]cite-ref-paulbaschnagel2013-2-0[2]cite-ref-slanina2013-3-0[3] More generally it can be seen to be a special case of a Markov renewal process.

• Motivation

• Formulation

• Montroll–Weiss formula

• Examples

• References

CTRW was introduced by Montroll and Weisscite-ref-4[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.cite-ref-5[5] A connection between CTRWs and diffusion equations with fractional time derivatives has been established.cite-ref-6[6] Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.cite-ref-7[7]

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

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

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

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

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

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

ψ ψ ~ ~ ( s ) = ∫ ∫ 0 ∞ ∞ d τ τ e − − τ τ s ψ ψ ( τ τ ) . {\displaystyle {\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).}

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

f ^ ^ ( k ) = ∫ ∫ Ω Ω d ( Δ Δ X ) e i k Δ Δ X f ( Δ Δ X ) . {\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 P ( X , t ) {\displaystyle P(X,t)} is given by

P ~ ~ ^ ^ ( k , s ) = 1 − − ψ ψ ~ ~ ( s ) s 1 1 − − ψ ψ ~ ~ ( s ) f ^ ^ ( k ) . {\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 the Montroll–Weiss formula.

cite-note-klages-11. ↑ citerefklagesradonssokolov2008Klages, Rainer; Radons, Guenther; Sokolov, Igor M. (2008-09-08). Anomalous Transport: Foundations and Applications. ISBN 9783527622986.

cite-note-paulbaschnagel2013-22. ↑ citerefpaulbaschnagel2013Paul, Wolfgang; Baschnagel, Jörg (2013-07-11). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. pp. 72–. ISBN 9783319003276. Retrieved 25 July 2014.

cite-note-slanina2013-33. ↑ citerefslanina2013Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. Retrieved 25 July 2014.

cite-note-44. ↑ citerefelliott-w-montrollgeorge-h-weiss1965Elliott 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.

cite-note-55. ↑ citeref-m-kenkree-w-montrollm-f-shlesinger1973. 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.

cite-note-66. ↑ citerefhilfer-r-anton-l-1995Hilfer, 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.

cite-note-77. ↑ citerefgorenflomainardivivoli2005Gorenflo, 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.