Random dynamical system

In the mathematical field of dynamical systems, a random dynamical system is a dynamical system in which the equations of motion have an element of randomness to them. Random dynamical systems are characterized by a state space "S", a set of maps formula_1 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 formula_1 that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state formula_3 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; 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 formula_4 be a formula_5-dimensional vector field, and let formula_6. Suppose that the solution formula_7 to the stochastic differential equation
exists for all positive time and some (small) interval of negative time dependent upon formula_9, where formula_10 denotes a formula_5-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) formula_13 by
(whenever the right hand side is well-defined). Then formula_15 (or, more precisely, the pair formula_16) 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 formula_17.
The discrete random dynamical system comes as follows,
The random variable formula_35 is constructed by means of composition of independent random maps, formula_36. Clearly, formula_35 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.
Here is an example that illustrates the existence and non-uniqueness.
Example: If the state space formula_38 and the set of the transformations formula_1 expressed in terms of deterministic transition matrices. Then a Markov transition matrixformula_40 can be represented by the following decomposition by the min-max algorithm, formula_41
In the meantime, another decomposition could be formula_42
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 formula_43 be a probability space, the noise space. Define the base flow formula_44 as follows: for each "time" formula_45, let formula_46 be a measure-preserving measurable function:
Suppose also that
That is, formula_54, formula_45, forms a group of measure-preserving transformation of the noise formula_43. For one-sided random dynamical systems, one would consider only positive indices formula_57; for discrete-time random dynamical systems, one would consider only integer-valued formula_57; in these cases, the maps formula_54 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 formula_60 is ergodic.
Now let formula_61 be a complete separable metric space, the phase space. Let formula_62 be a formula_63-measurable function such that
In the case of random dynamical systems driven by a Wiener process formula_74, the base flow formula_46 would be given by
This can be read as saying that formula_54 "starts the noise at time formula_57 instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition formula_79 with some noise formula_80 for formula_57 seconds and then through formula_69 seconds with the same noise (as started from the formula_57 seconds mark) gives the same result as evolving formula_79 through formula_85 seconds with that same noise.
Attractors for random dynamical systems.
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. Moreover, the attractor is dependent upon the realisation formula_86 of the noise.