Continuous-time random walk

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 ${\displaystyle X(t)}$ defined by

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

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

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

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

Montroll–Weiss formula

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

${\displaystyle {\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).}$

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

${\displaystyle {\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\Delta X}f(\Delta X).}$

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

${\displaystyle {\hat {\tilde {P}}}(k,s)={\frac {1-{\tilde {\psi }}(s)}{s}}{\frac {1}{1-{\tilde {\psi }}(s){\hat {f}}(k)}}.}$

The above is called the Montroll–Weiss formula.

Examples

References