J r me rousseau hitting time statistics for random dynamical systems
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
Contents
- J r me rousseau hitting time statistics for random dynamical systems
- Motivation 1 solutions to a stochastic differential equation
- Motivation 2 Connection to Markov Chain
- Formal definition
- Attractors for random dynamical systems
- References
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
exists for all positive time and some (small) interval of negative time dependent upon
In this context, the Wiener process is the coordinate process.
Now define a flow map or (solution operator)
(whenever the right hand side is well-defined). Then
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,
- The system is in some state
x 0 S , a mapα 1 Γ is chosen according to the probability measureQ and the system moves to the statex 1 = α 1 ( x 0 ) in step 1. - Independently of previous maps, another map
α 2 Q and the system moves to the statex 2 = α 2 ( x 1 ) . - The procedure repeats.
The random variable
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
In the mean time, 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
Suppose also that
-
ϑ 0 = i d Ω : Ω → Ω , the identity function onΩ ; - for all
s , t ∈ R ,ϑ s ∘ ϑ t = ϑ s + t
That is,
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
Now let
- for all
ω ∈ Ω ,φ ( 0 , ω ) = i d X : X → X , the identity function onX ; - for (almost) all
ω ∈ Ω ,( t , ω , x ) ↦ φ ( t , ω , x ) is continuous in botht andx ; -
φ satisfies the (crude) cocycle property: for almost allω ∈ Ω ,
In the case of random dynamical systems driven by a Wiener process
This can be read as saying that
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