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. More generally it can be seen to be a special case of a Markov renewal process.

Motivation

CTRW was introduced by Montroll and Weiss 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. A connection between CTRWs and diffusion equations with fractional time derivatives has been established. Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.

Formulation

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 $\Delta X_{i}$ 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 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 $t$.

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

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

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

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

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.