Random dynamical systems are characterized by a state space S, a set of maps
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.
[1] An example of a random dynamical system is a stochastic differential equation; 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.
to the stochastic differential equation exists for all positive time and some (small) interval of negative time dependent upon
-dimensional Wiener process (Brownian motion).
Implicitly, this statement uses the classical Wiener probability space In this context, the Wiener process is the coordinate process.
Now define a flow map or (solution operator)
) 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.
An i.i.d random dynamical system in the discrete space is described by a triplet
is constructed by means of composition of independent random maps,
Reversely, can, and how, a given MC be represented by the compositions of i.i.d.
The proof for existence is similar with Birkhoff–von Neumann theorem for doubly stochastic matrix.
expressed in terms of deterministic transition matrices.
can be represented by the following decomposition by the min-max algorithm,
Formally,[3] a random dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space.
, forms a group 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
would only form a commutative 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
-measurable function such that In the case of random dynamical systems driven by a Wiener process
Thus, the cocycle property can be read as saying that evolving the initial condition
seconds mark) gives the same result as evolving
The notion of an attractor 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.