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
are iid random variables taking values in a domain
The probability for the process taking the value
is the probability for the process taking the value
the waiting time in between two jumps of
is defined by Similarly, the characteristic function of the jump distribution