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 Z ...
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 themselv ...
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 mod ...
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
In probability and statistics, a Markov renewal process (MRP) is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chains, Poisson processes and renewal processes can be derived as special ...
.
Motivation
CTRW was introduced by
Montroll and
Weiss 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 descr ...
, 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
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independence (probability theory), ...
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
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
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 points ...
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 formula.
Examples
References
{{Stochastic processes
Variants of random walks