Continuous-time random walk

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.^{[1] [2] [3]} More generally it can be seen to be a special case of a Markov renewal process.

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 ${\displaystyle X(t)}${\displaystyle X(t)} defined by

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

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

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

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

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

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

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

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

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

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

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

• Variants of random walks

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