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 ) {\displaystyle X(t)} defined by

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

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

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

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

Montroll–Weiss formula

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

ψ ~ ( s ) = ∫ 0 ∞ d τ e − τ s ψ ( τ ) . {\displaystyle {\tilde {\psi }}(s)=\int _{0}^{\infty }d\tau \,e^{-\tau s}\psi (\tau ).}

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

f ^ ( k ) = ∫ Ω d ( Δ X ) e i k Δ X f ( Δ X ) . {\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 P ( X , t ) {\displaystyle P(X,t)} is given by

P ~ ^ ( k , s ) = 1 − ψ ~ ( s ) s 1 1 − ψ ~ ( s ) f ^ ( k ) . {\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

The homogeneous Poisson point process is a continuous time random walk with exponential holding times and with each increment deterministically equal to 1.