Английская Википедия:Continuous-time random walk

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Шаблон:Short description 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 <math>X(t)</math> defined by

<math>

X(t) = X_0 + \sum_{i=1}^{N(t)} \Delta X_i, </math>

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

<math>

P(X,t) = \sum_{n=0}^\infty P(n,t) P_n(X). </math>

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

Montroll–Weiss formula

We denote by <math>\tau</math> the waiting time in between two jumps of <math>N(t)</math> and by <math>\psi(\tau)</math> its distribution. The Laplace transform of <math>\psi(\tau)</math> is defined by

<math>

\tilde{\psi}(s)=\int_0^{\infty} d\tau \, e^{-\tau s} \psi(\tau). </math>

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

<math>

\hat{f}(k)=\int_\Omega d(\Delta X) \, e^{i k\Delta X} f(\Delta X). </math>

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

<math>

\hat{\tilde{P}}(k,s) = \frac{1-\tilde{\psi}(s)}{s} \frac{1}{1-\tilde{\psi}(s)\hat{f}(k)}. </math>

The above is called the Montroll–Weiss formula.