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 Γ {\displaystyle \Gamma } 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 Γ {\displaystyle \Gamma } that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state X ∈ S {\displaystyle X\in S} 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 f : R d → R d {\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} ^{d}} be a d {\displaystyle d} -dimensional vector field, and let ε > 0 {\displaystyle \varepsilon >0} . Suppose that the solution X ( t , ω ; x 0 ) {\displaystyle X(t,\omega ;x_{0})} to the stochastic differential equation
{ d X = f ( X ) d t + ε d W ( t ) ; X ( 0 ) = x 0 ; {\displaystyle \left\{{\begin{matrix}\mathrm {d} X=f(X)\,\mathrm {d} t+\varepsilon \,\mathrm {d} W(t);\\X(0)=x_{0};\end{matrix}}\right.}
exists for all positive time and some (small) interval of negative time dependent upon ω ∈ Ω {\displaystyle \omega \in \Omega } , where W : R × Ω → R d {\displaystyle W:\mathbb {R} \times \Omega \to \mathbb {R} ^{d}} denotes a d {\displaystyle d} -dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space
( Ω , F , P ) := ( C 0 ( R ; R d ) , B ( C 0 ( R ; R d ) ) , γ ) . {\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ):=\left(C_{0}(\mathbb {R} ;\mathbb {R} ^{d}),{\mathcal {B}}(C_{0}(\mathbb {R} ;\mathbb {R} ^{d})),\gamma \right).}
In this context, the Wiener process is the coordinate process. Now define a flow map or (solution operator) φ : R × Ω × R d → R d {\displaystyle \varphi :\mathbb {R} \times \Omega \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}} by
φ ( t , ω , x 0 ) := X ( t , ω ; x 0 ) {\displaystyle \varphi (t,\omega ,x_{0}):=X(t,\omega ;x_{0})}
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