To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time. 4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds # 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.

• 1/5
Views:
982
93 211
59 158
830
9 610
• ✪ Derivation of PDE for Random Walk
• ✪ What is a Random Walk? | Infinite Series
• ✪ (ML 14.2) Markov chains (discrete-time) (part 1)
• ✪ Brian Berkowitz: Breakthroughs in Groundwater Oct.28 2015
• ✪ 5. Random Walks

## Motivation

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.

## 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:

${\hat {f}}(k)=\int _{\Omega }d(\Delta X)\,e^{ik\Delta X}f(\Delta X).$ 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 Montroll-Weiss formula.

## Examples

The homogeneous Poisson point process is a continuous time random walk with exponential holding times and with each increment deterministically equal to 1.

Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.