Per Enflo

My name is Jestine (34 years old) and my hobbies are Origami and Microscopy.

Here is my web site; http://Www.hostgator1centcoupon.info/ (support.file1.com) 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 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 $\epsilon >0$. Suppose that the solution $X(t,\omega ;x_0)$ to the stochastic differential equation

$\{\begin{array}{c}\mathrm{d}X=f(X)\mathrm{d}t+\epsilon \mathrm{d}W(t);\\ X(0)=x_0;\end{array}$

exists for all positive time and some (small) interval of negative time dependent upon $\omega \in \mathrm{\Omega }$, where $W:\mathbb{ℝ}\times \mathrm{\Omega }\to {\mathbb{ℝ}}^d$ denotes a $d$-dimensional Wiener process (Brownian motion). Implicitly, this statement uses the classical Wiener probability space

$(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}}):=(C_0(\mathbb{ℝ};{\mathbb{ℝ}}^d),\mathcal{ℬ}(C_0(\mathbb{ℝ};{\mathbb{ℝ}}^d)),\gamma ).$

In this context, the Wiener process is the coordinate process.

Now define a flow map or (solution operator) $\phi :\mathbb{ℝ}\times \mathrm{\Omega }\times {\mathbb{ℝ}}^d\to {\mathbb{ℝ}}^d$ by

$\phi (t,\omega ,x_0):=X(t,\omega ;x_0)$

(whenever the right hand side is well-defined). Then $\phi$ (or, more precisely, the pair $({\mathbb{ℝ}}^d,\phi )$) 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 $(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}})$ be a probability space, the noise space. Define the base flow $\vartheta :\mathbb{ℝ}\times \mathrm{\Omega }\to \mathrm{\Omega }$ as follows: for each "time" $s\in \mathbb{ℝ}$, let ${\vartheta }_s:\mathrm{\Omega }\to \mathrm{\Omega }$ be a measure-preserving measurable function:

$\mathbb{\mathbb{P}}(E)=\mathbb{\mathbb{P}}({\vartheta }_s^{-1}(E))$ for all $E\in \mathcal{ℱ}$ and $s\in \mathbb{ℝ}$;

Suppose also that

1. ${\vartheta }_0={\mathrm{i}\mathrm{d}}_{\mathrm{\Omega }}:\mathrm{\Omega }\to \mathrm{\Omega }$, the identity function on $\mathrm{\Omega }$;
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 $(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}})$. 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 $(\mathrm{\Omega },\mathcal{ℱ},\mathbb{\mathbb{P}},\vartheta )$ is ergodic.

Now let $(X,d)$ be a complete separable metric space, the phase space. Let $\phi :\mathbb{ℝ}\times \mathrm{\Omega }\times X\to X$ be a $(\mathcal{ℬ}(\mathbb{ℝ})\otimes \mathcal{ℱ}\otimes \mathcal{ℬ}(X),\mathcal{ℬ}(X))$-measurable function such that

1. for all $\omega \in \mathrm{\Omega }$, $\phi (0,\omega )={\mathrm{i}\mathrm{d}}_X:X\to X$, the identity function on $X$;
2. for (almost) all $\omega \in \mathrm{\Omega }$, $(t,\omega ,x)\mapsto \phi (t,\omega ,x)$ is continuous in both $t$ and $x$;
3. $\phi$ satisfies the (crude) cocycle property: for almost all $\omega \in \mathrm{\Omega }$,

$\phi (t,{\vartheta }_s(\omega ))\circ \phi (s,\omega )=\phi (t+s,\omega ).$

In the case of random dynamical systems driven by a Wiener process $W:\mathbb{ℝ}\times \mathrm{\Omega }\to X$, the base flow ${\vartheta }_s:\mathrm{\Omega }\to \mathrm{\Omega }$ 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.

References

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