Random dynamical system

In mathematics, a random dynamical system is a measure-theoretic formulation of a dynamical system with an element of "randomness", such as the dynamics of solutions to a stochastic differential equation. It consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space.

Motivation: solutions to a stochastic differential equation

Let $f:\mathbb {R} ^{d}\to \mathbb {R} ^{d}$ be a $d$ -dimensional vector field, and let $\varepsilon >0$ . Suppose that the solution $X(t,\omega ;x_{0})$ to the stochastic differential equation

\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 $\omega \in \Omega$ , where $W:\mathbb {R} \times \Omega \to \mathbb {R} ^{d}$ denotes a $d$ -dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space

(\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) $\varphi :\mathbb {R} \times \Omega \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}$ by

\varphi (t,\omega ,x_{0}):=X(t,\omega ;x_{0})

(whenever the right hand side is well-defined). Then $\varphi$ (or, more precisely, the pair $(\mathbb {R} ^{d},\varphi )$ ) is a (local, left-sided) random dynamical system.

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 $(\Omega ,{\mathcal {F}},\mathbb {P} )$ be a probability space. Define the base flow $\vartheta :\mathbb {R} \times \Omega \to \Omega$ as follows: for each "time" $s\in \mathbb {R}$ , let $\vartheta _{s}:\Omega \to \Omega$ be a measure-preserving measurable function:

\mathbb {P} (E)=\mathbb {P} (\vartheta _{s}^{-1}(E))

for all

E\in {\mathcal {F}}

and

s\in \mathbb {R}

;

Suppose also that

$\vartheta _{0}=\mathrm {id} _{\Omega }:\Omega \to \Omega$ , the identity function on $\Omega$ ;
for all $s,t\in \mathbb {R}$ , $\vartheta _{s}\circ \vartheta _{t}=\vartheta _{s+t}$ .

That is, $\vartheta _{s}$ , $s\in \mathbb {R}$ , forms a group of measure-preserving transformation of the noise $(\Omega ,{\mathcal {F}},\mathbb {P} )$ . (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 $(\Omega ,{\mathcal {F}},\mathbb {P} ,\vartheta )$ is ergodic.)

Now let $(X,d)$ be a complete separable metric space, the phase space. Let $\varphi :\mathbb {R} \times \Omega \times X\to X$ be a $({\mathcal {B}}(\mathbb {R} )\otimes {\mathcal {F}}\otimes {\mathcal {B}}(X),{\mathcal {B}}(X))$ -measurable function such that

for all $\omega \in \Omega$ , $\varphi (0,\omega )=\mathrm {id} _{X}:X\to X$ , the identity function on $X$ ;
for (almost) all $\omega \in \Omega$ , $(t,\omega ,x)\mapsto \varphi (t,\omega )x$ is continuous in both $t$ and $x$ ;
$\varphi$ satisfies the (crude) cocycle property: for almost all $\omega \in \Omega$ ,

\varphi (t,\vartheta _{s}(\omega ))\circ \varphi (s,\omega )=\varphi (t+s,\omega ).

In the case of random dynamical systems driven by a Wiener process $W:\mathbb {R} \times \Omega \to X$ , the base flow $\vartheta _{s}:\Omega \to \Omega$ would be given by

W(t,\vartheta _{s}(\omega ))=W(t+s,\omega )-W(s,\omega )

.

This can be read as saying that $\vartheta _{s}$ "starts the noise at time $s$ instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition $x_{0}$ with some noise $\omega$ for $s$ seconds and then through $t$ seconds with the same noise (as started from the $s$ seconds mark) gives the same result as evolving $x_{0}$ through $(t+s)$ seconds with that same noise.

Attractors for random dynamical systems

Intuitively, it seems obvious that should an attractor exist for a random dynamical system, it should depend upon the realisation $\omega$ of the noise.

The random global attractor ${\mathcal {A}}(\omega )$ for a random dynamical system $\varphi$ is a $\mathbb {P}$ -almost surely unique random set such that

${\mathcal {A}}(\omega )$ is a random compact set: ${\mathcal {A}}(\omega )\subseteq X$ is almost surely compact and $\omega \mapsto \mathrm {dist} (x,{\mathcal {A}}(\omega ))$ is a measurable function for every $x\in X$ ;
${\mathcal {A}}(\omega )$ is invariant: for all $\varphi (t,\omega )({\mathcal {A}}(\omega ))={\mathcal {A}}(\vartheta _{t}(\omega ))$ almost surely;
${\mathcal {A}}(\omega )$ is attractive: for any bounded set $B\subsetneq X$ ,