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Continuous-time random walk

CTRW was introduced by Montroll and Weiss^[4] 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.^[5] A connection between CTRWs and diffusion equations with fractional time derivatives has been established.^[6] Similarly, time-space fractional diffusion equations can be considered as CTRWs with continuously distributed jumps or continuum approximations of CTRWs on lattices.^[7]

A simple formulation of a CTRW is to consider the stochastic process ${\displaystyle X($t)} defined by

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

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

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

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

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

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

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

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

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

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

The above is called the Montroll–Weiss formula.