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 in the case of discrete jump lengths and fractional differential equations for the case where the process varies continuously.

Formulation

A simple formulation of the process can be given by considering 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 $V$ 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

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

where $P_{n}(X)$ of 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.