Random dynamical system

In the mathematical field of dynamical systems, 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 T 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 T that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state $X \in S$ 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.

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.

See also

• Stochastic control

References

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