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 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; 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 interval of negative time dependent upon, where denotes a -dimensional Wiener process. 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 by
. Then is a 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.
The discrete random dynamical system comes as follows,
  1. 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.
  2. Independently of previous maps, another map is chosen according to the probability measure and the system moves to the state.
  3. 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.
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, the noise space. Define the base flow as follows: for each "time", let be a measure-preserving measurable function:
Suppose also that
  1. , the identity function on ;
  2. for all,.
That is,,, 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 ; in these cases, the maps 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.
Now let be a complete separable metric space, the phase space. Let be a -measurable function such that
  1. for all,, the identity function on ;
  2. for all, is continuous in both and ;
  3. satisfies the 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 gives the same result as evolving through 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 of the noise.