continuous-time random walk

CTRW was introduced by Montroll and Weiss{{cite journal

|author1=Elliott W. Montroll |author2=George H. Weiss | title = Random Walks on Lattices. II

| journal = J. Math. Phys.

| volume = 6

| issue =2

| pages = 167

| date = 1965

| doi = 10.1063/1.1704269

| bibcode =1965JMP.....6..167M}} 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 journal

|author1=. M. Kenkre |author2=E. W. Montroll |author3=M. F. Shlesinger | title = Generalized master equations for continuous-time random walks

| journal = Journal of Statistical Physics

| volume = 9

| issue = 1

| pages = 45–50

| date = 1973

| doi = 10.1007/BF01016796

| bibcode =1973JSP.....9...45K}} A connection between CTRWs and diffusion equations with fractional time derivatives has been established.{{cite journal

|author1=Hilfer, R. |author2=Anton, L.

| title = Fractional master equations and fractal time random walks

| journal = Phys. Rev. E

| volume = 51

| issue = 2

| pages = R848–R851

| date = 1995

| doi = 10.1103/PhysRevE.51.R848

| bibcode =1995PhRvE..51..848H}} Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.{{cite journal

| last1 = Gorenflo | first1 = Rudolf | author1-link = Rudolf Gorenflo

| last2 =Mainardi | first2 = Francesco

| last3 = Vivoli | first3 = Alessandro

| title = Continuous-time random walk and parametric subordination in fractional diffusion

| journal = Chaos, Solitons & Fractals

| volume = 34

| issue = 1

| pages = 87–103

| date = 2005

| doi = 10.1016/j.chaos.2007.01.052

| arxiv =cond-mat/0701126| bibcode =2007CSF....34...87G}}

A simple formulation of a CTRW is to consider the stochastic process $X(t)$ defined by

:$$

X(t) = X_0 + \sum_{i=1}^{N(t)} \Delta X_i,

whose increments $\backslash Delta\; X\_i$ are iid random variables taking values in a domain $\backslash Omega$ 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 by

:$$

P(X,t) = \sum_{n=0}^\infty 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 $t$.

We denote by $\backslash tau$ the waiting time in between two jumps of $N(t)$ and by $\backslash psi(\backslash tau)$ its distribution. The Laplace transform of $\backslash psi(\backslash tau)$ is defined by

:$$

\tilde{\psi}(s)=\int_0^{\infty} d\tau \, e^{-\tau s} \psi(\tau).

Similarly, the characteristic function of the jump distribution $f(\backslash Delta\; X)$ is given by its Fourier transform:

:$$

\hat{f}(k)=\int_\Omega d(\Delta X) \, e^{i k\Delta X} 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) = \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\tilde{\psi}(s)\hat{f}(k)}.

The above is called the Montroll–Weiss formula.

{{reflist}}

{{Stochastic processes}}

Category:Variants of random walks