Random dynamical system

From Wikipedia, the free encyclopedia

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.

1 Motivation: solutions to a stochastic differential equation
2 Formal definition
3 Attractors for random dynamical systems
4 Reference

[edit] 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 $X (t,ω; x 0)$ 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.

[edit] 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

$\vartheta_{0} = \mathrm{id}_{\Omega} : \Omega \to \Omega$ , the identity function on $Ω$ ;
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

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 $ω$ 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.

[edit] 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 $ω$ of the noise.