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 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.

CTRW was introduced by Montroll and Weiss as a generalization of physical diffusion process 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.

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

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

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