Continuous-time random walk

Short description: Random walk with random time between jumps

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 [math]\displaystyle{ X(t) }[/math] defined by

[math]\displaystyle{ X(t) = X_0 + \sum_{i=1}^{N(t)} \Delta X_i, }[/math]

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

[math]\displaystyle{ P(X,t) = \sum_{n=0}^\infty P(n,t) P_n(X). }[/math]

Here [math]\displaystyle{ P_n(X) }[/math] is the probability for the process taking the value [math]\displaystyle{ X }[/math] after [math]\displaystyle{ n }[/math] jumps, and [math]\displaystyle{ P(n,t) }[/math] is the probability of having [math]\displaystyle{ n }[/math] jumps after time [math]\displaystyle{ t }[/math].

Montroll–Weiss formula

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

[math]\displaystyle{ \tilde{\psi}(s)=\int_0^{\infty} d\tau \, e^{-\tau s} \psi(\tau). }[/math]

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

[math]\displaystyle{ \hat{f}(k)=\int_\Omega d(\Delta X) \, e^{i k\Delta X} f(\Delta X). }[/math]

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

[math]\displaystyle{ \hat{\tilde{P}}(k,s) = \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\tilde{\psi}(s)\hat{f}(k)}. }[/math]

The above is called the Montroll–Weiss formula.