Random walk with random time between jumps
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.
Motivation
CTRW was introduced by
Formulation
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 .
Montroll–Weiss formula
We denote by the waiting time in between two jumps of and by its distribution. The Laplace transform of is defined by
Similarly, the characteristic function of the jump distribution is given by its Fourier transform:
One can show that the Laplace–Fourier transform of the probability is given by
The above is called the Montroll–Weiss formula.
Examples
References
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