Continuous-time random walk

From HandWiki
Short description: Random walk with random time between jumps

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 by

X(t)=X0+i=1N(t)ΔXi,

whose increments ΔXi are iid random variables taking values in a domain Ω 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 by

P(X,t)=n=0P(n,t)Pn(X).

Here Pn(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 τ the waiting time in between two jumps of N(t) and by ψ(τ) its distribution. The Laplace transform of ψ(τ) is defined by

ψ~(s)=0dτeτsψ(τ).

Similarly, the characteristic function of the jump distribution f(ΔX) is given by its Fourier transform:

f^(k)=Ωd(ΔX)eikΔXf(ΔX).

One can show that the Laplace–Fourier transform of the probability P(X,t) is given by

P~^(k,s)=1ψ~(s)s11ψ~(s)f^(k).

The above is called the Montroll–Weiss formula.

Examples

References

  1. Klages, Rainer; Radons, Guenther; Sokolov, Igor M. (2008-09-08). Anomalous Transport: Foundations and Applications. ISBN 9783527622986. https://books.google.com/books?id=N1xD7ay06Z4C. 
  2. Paul, Wolfgang; Baschnagel, Jörg (2013-07-11). Stochastic Processes: From Physics to Finance. Springer Science & Business Media. pp. 72–. ISBN 9783319003276. https://books.google.com/books?id=OWANAAAAQBAJ&pg=PA72. Retrieved 25 July 2014. 
  3. Slanina, Frantisek (2013-12-05). Essentials of Econophysics Modelling. OUP Oxford. pp. 89–. ISBN 9780191009075. https://books.google.com/books?id=3CJoAgAAQBAJ&pg=PA89. Retrieved 25 July 2014. 
  4. Elliott W. Montroll; George H. Weiss (1965). "Random Walks on Lattices. II". J. Math. Phys. 6 (2): 167. doi:10.1063/1.1704269. Bibcode1965JMP.....6..167M. 
  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. doi:10.1007/BF01016796. Bibcode1973JSP.....9...45K. 
  6. 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. Bibcode1995PhRvE..51..848H. 
  7. "Continuous-time random walk and parametric subordination in fractional diffusion". Chaos, Solitons & Fractals 34 (1): 87–103. 2005. doi:10.1016/j.chaos.2007.01.052. Bibcode2007CSF....34...87G.