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En matemáticas, un sistema dinámico aleatorio es una formulación de la teoría de la medida de un sistema dinámico con un elemento de "aleatoriedad". Esto consiste en un flujo base, el "ruido", y un cociclo de un sistema dinámico en el espacio "físico" de fase.

Motivación: solución a ecuaciones diferenciales estocásticas
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.