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.

Definition

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 measurable function

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

(sometimes known as the flow map or solution operator) 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.

Properties of the flow map

The map $\varphi$ has three important properties:

1. $\varphi (0,\omega )=\mathrm {id} :\mathbb {R} ^{d}\to \mathbb {R} ^{d}$, the identity function on $\mathbb {R} ^{d}$, for all $\omega \in \Omega$;
2. $(t,\omega ,x)\mapsto \varphi (t,\omega )x$ is continuous in both $t$ and $x$ for (almost) all $\omega \in \Omega$;
3. $\varphi$ 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 above, $\vartheta _{s}:\Omega \to \Omega$ denotes the shift map defined by

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

This can be read as saying that $\vartheta _{s}$ "starts time 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

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

$\mathrm {dist} \left(\varphi (t,\vartheta _{-t}(\omega ))(B),{\mathcal {A}}(\omega )\right)\to 0$ as $t\to +\infty$ almost surely.

There is a slight abuse of notation in the above: the first use of "dist" refers to the Hausdorff semi-distance from a point to a set,

$\mathrm {dist} (x,A):=\inf _{a\in A}\|x-a\|,$

whereas the second use of "dist" refers to the full, symmetric, Hausdorff distance between two sets,

$\mathrm {dist} (A,B):=\max \left\{\sup _{a\in A}\inf _{b\in B}\|a-b\|,\sup _{b\in B}\inf _{a\in A}\|a-b\|\right\}.$