In mathematics, a continuous-time random walk (CTRW) is a generalization of a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb ...
where the wandering particle waits for a random time between jumps. It is a
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
jump process
A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement, typically modelled as a simple or compound Poisson process.
In finance, various stochastic ...
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
Weiss or Weiß may refer to:
People
* Weiss (surname), including spelling Weiß
* Weiss Ferdl (1883-1949), German actor
Places
* Mount Weiss, Jasper National Park, Alberta, Canada
* Weiss Lake, Alabama
* Weiß (Sieg), a river in North Rhine-West ...
as a generalization of physical diffusion process to effectively describe
anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), \langle r^(\tau )\rangle , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process descri ...
, i.e., the super- and sub-diffusive cases. An equivalent formulation of the CTRW is given by generalized
master equation
In physics, chemistry and related fields, master equations are used to describe the time evolution of a system that can be modelled as being in a probabilistic combination of states at any given time and the switching between states is determine ...
s. 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
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
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
of
is defined by
:
Similarly, the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
of the jump distribution
is given by its
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
:
:
One can show that the Laplace–Fourier transform of the probability
is given by
:
The above is called
Montroll–
Weiss
Weiss or Weiß may refer to:
People
* Weiss (surname), including spelling Weiß
* Weiss Ferdl (1883-1949), German actor
Places
* Mount Weiss, Jasper National Park, Alberta, Canada
* Weiss Lake, Alabama
* Weiß (Sieg), a river in North Rhine-West ...
formula.
Examples
References
{{Stochastic processes
Variants of random walks