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. 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.

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, the noise 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

1. $\vartheta _{0}=\mathrm {id} _{\Omega }:\Omega \to \Omega$, the identity function on $\Omega$;
2. 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 $(\Omega ,{\mathcal {F}},\mathbb {P} )$. For one-sided random dynamical systems, one would consider only positive indices $s$; for discrete-time random dynamical systems, one would consider only integer-valued $s$; in these cases, the maps $\vartheta _{s}$ 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 $(\Omega ,{\mathcal {F}},\mathbb {P} ,\vartheta )$ 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

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

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 $\omega$ of the noise.

Reference

• Crauel, H., Debussche, A., & Flandoli, F. (1997) Random attractors. Journal of Dynamics and Differential Equations. 9(2) 307—341.