In mathematics, 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 from S into itself that can be thought of as the set of all possible equations of motion, and a probability distribution Q 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

-dimensional vector field to the stochastic differential equation

-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 by

(whenever the right hand side is well-defined) 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

.

• .

• matrix representation, called deterministic transition matrix. It is a binary matrix but it has exactly one entry 1 in each row and 0s otherwise.

• .

The discrete random dynamical system comes as follows,

1. in step 1.

2. .

3. The procedure repeats.

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:

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.

be a probability space, the noise space. Define the base flow be a measure-preserving measurable function:

1. ;

Suppose also that

1. , the identity function;

2. .

, forms a group 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 is ergodic.

be a complete separable metric space, the phase space-measurable function such that

1. ;

2. is continuous;

3. satisfies the (crude) cocycle property: for almost all,

would be given by

1. .

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. It is necessary to "rewind time", as in the definition of a pullback attractor, so that the noise close to the final time remains consistent. of the noise.

See also

• Chaos theory

• Diffusion process

• Stochastic control

References

Further reading

• Stochastic dynamical systems on Scholarpedia