Continuous-time random walk

In mathematics, a continuous-time random walk (CTRW) is a generalization of a stochastic jump process with arbitrary distributions of jump lengths and waiting times.

Motivation

CTRW was introduced by Montroll and Weiss as a generalization of physical diffusion process 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

Denoting the waiting time distribution in between two jumps of $N(t)$ by $\psi (\tau )$, its Laplace transform is defined by

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

Similarly, for the jump distribution $f(\Delta X)$ of the increments, the Fourier transform is given by

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

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

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

The above is called Montroll-Weiss formula.

Examples

The Wiener process is the standard example of a continuous time random walk in which the waiting times are exponential and the jumps are continuous and normally distributed.