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.

Motivation
CTRW was introduced by Montroll and Weiss 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. 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.

Formulation
A simple formulation of a CTRW is to consider the stochastic process $$X(t)$$ defined by



X(t) = X_0 + \sum_{i=1}^{N(t)} \Delta X_i, $$

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



P(X,t) = \sum_{n=0}^\infty P(n,t) P_n(X). $$

Here $$P_n(X)$$ is the probability for the process taking the value $$X$$ after $$n$$ jumps, and $$P(n,t)$$ is the probability of having $$n$$ jumps after time $$t$$.

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



\tilde{\psi}(s)=\int_0^{\infty} d\tau \, e^{-\tau s} \psi(\tau). $$

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



\hat{f}(k)=\int_\Omega d(\Delta X) \, e^{i k\Delta X} f(\Delta X). $$

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



\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.