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. 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 defined by
whose increments are iid random variables taking values in a domain and is the number of jumps in the interval . The probability for the process taking the value at time is then given by
Here is the probability for the process taking the value after jumps, and is the probability of having jumps after time .
We denote by the waiting time in between two jumps of and by its distribution. The Laplace transform of is defined by
One can show that the Laplace–Fourier transform of the probability is given by
The homogeneous Poisson point process is a continuous time random walk with exponential holding times and with each increment deterministically equal to 1.
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