In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
dynamical systems, a random dynamical system is a dynamical system in which the
equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
have an element of randomness to them. Random dynamical systems are characterized by a
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the to ...
''S'', a
set of
map
A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
s
from ''S'' into itself that can be thought of as the set of all possible equations of motion, and a
probability distribution ''Q'' on the set
that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state
evolving according to a succession of maps randomly chosen according to the distribution ''Q''.
An example of a random dynamical system is a
stochastic differential equation
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
; in this case the distribution Q is typically determined by ''noise terms''. It consists of a
base flow, the "noise", and a
cocycle dynamical system on the "physical"
phase space. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.
Motivation 1: Solutions to a stochastic differential equation
Let
be a
-dimensional
vector field, and let
. Suppose that the solution
to the stochastic differential equation
:
exists for all positive time and some (small) interval of negative time dependent upon
, where
denotes a
-dimensional
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
(
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). Implicitly, this statement uses the
classical Wiener probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
:
In this context, the Wiener process is the coordinate process.
Now define a flow map or (solution operator)
by
:
(whenever the right hand side is
well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
). Then
(or, more precisely, the pair
) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems.
Motivation 2: Connection to Markov Chain
An i.i.d random dynamical system in the discrete space is described by a triplet
.
*
is the state space,
.
*
is a family of maps of
. Each such map has a
matrix representation, called ''deterministic transition matrix''. It is a binary matrix but it has exactly one entry 1 in each row and 0s otherwise.
*
is the probability measure of the
-field of
.
The discrete random dynamical system comes as follows,
# The system is in some state
in
, a map
in
is chosen according to the probability measure
and the system moves to the state
in step 1.
# Independently of previous maps, another map
is chosen according to the probability measure
and the system moves to the state
.
# The procedure repeats.
The random variable
is constructed by means of composition of independent random maps,
. Clearly,
is a
Markov Chain.
Reversely, can, and how, a given MC be represented by the compositions of i.i.d. random transformations? Yes, it can, but not unique. The proof for existence is similar with Birkhoff–von Neumann theorem for
doubly stochastic matrix In mathematics, especially in probability and combinatorics, a doubly stochastic matrix
(also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e.,
:\sum_i x_=\sum_j x_=1 ...
.
Here is an example that illustrates the existence and non-uniqueness.
Example: If the state space
and the set of the transformations
expressed in terms of deterministic transition matrices. Then a Markov transition matrix
can be represented by the following decomposition by the min-max algorithm,
In the meantime, another decomposition could be
Formal definition
Formally, a random dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.
Let
be a
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, the noise space. Define the base flow
as follows: for each "time"
, let
be a measure-preserving
measurable function:
:
for all
and
;
Suppose also that
#
, the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on
;
# for all
,
.
That is,
,
, forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
of measure-preserving transformation of the noise
. For one-sided random dynamical systems, one would consider only positive indices
; for discrete-time random dynamical systems, one would consider only integer-valued
; in these cases, the maps
would only form a
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
instead of a group.
While true in most applications, it is not usually part of the formal definition of a random dynamical system to require that the
measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
is
ergodic.
Now let
be a
complete separable metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, the phase space. Let
be a
-measurable function such that
# for all
,
, the identity function on
;
# for (almost) all
,
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
;
#
satisfies the (crude) cocycle property: for
almost all ,
::
In the case of random dynamical systems driven by a Wiener process
, the base flow
would be given by
:
.
This can be read as saying that
"starts the noise at time
instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition
with some noise
for
seconds and then through
seconds with the same noise (as started from the
seconds mark) gives the same result as evolving
through
seconds with that same noise.
Attractors for random dynamical systems
The notion of an
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain ...
for a random dynamical system is not as straightforward to define as in the deterministic case. For technical reasons, it is necessary to "rewind time", as in the definition of a
pullback attractor
In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatmen ...
.
Moreover, the attractor is dependent upon the realisation
of the noise.
See also
*
Chaos theory
*
Diffusion process
In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diff ...
*
Stochastic control
References
{{Stochastic processes
*
Stochastic differential equations
Stochastic processes