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Inmathematics, arandom dynamical system is adynamical system in which theequations of motion have an element of randomness to them. Random dynamical systems are characterized by astate spaceS, aset ofmaps${\displaystyle \mathrm{\Gamma }}$ fromS into itself that can be thought of as the set of all possible equations of motion, and aprobability distributionQ on the set${\displaystyle \mathrm{\Gamma }}$ that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state${\displaystyle X\in S}$ evolving according to a succession of maps randomly chosen according to the distributionQ.^[1]

An example of a random dynamical system is astochastic differential equation; in this case the distribution Q is typically determined bynoise terms. It consists of abase flow, the "noise", and acocycle dynamical system on the "physical"phase space. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.^[2]

Let${\displaystyle f:{\mathbb{R}}^d\to {\mathbb{R}}^d}$ be a${\displaystyle d}$-dimensionalvector field, and let${\displaystyle \epsilon >0}$. Suppose that the solution${\displaystyle X(t,\omega ;x_0)}$ to the stochastic differential equation

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

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

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

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

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

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

(whenever the right hand side iswell-defined). Then${\displaystyle \phi }$ (or, more precisely, the pair${\displaystyle ({\mathbb{R}}^d,\phi )}$) 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.

An i.i.d random dynamical system in the discrete space is described by a triplet${\displaystyle (S,\mathrm{\Gamma },Q)}$.

• ${\displaystyle S}$ is the state space,${\displaystyle \{s_1,s_2,\cdots ,s_n\}}$.
• ${\displaystyle \mathrm{\Gamma }}$ is a family of maps of${\displaystyle S\to S}$. Each such map has a${\displaystyle n\times n}$ matrix representation, calleddeterministic transition matrix. It is a binary matrix but it has exactly one entry 1 in each row and 0s otherwise.
• ${\displaystyle Q}$ is the probability measure of the${\displaystyle \sigma }$-field of${\displaystyle \mathrm{\Gamma }}$.

The discrete random dynamical system comes as follows,

1. The system is in some state${\displaystyle x_0}$ in${\displaystyle S}$, a map${\displaystyle {\alpha }_1}$ in${\displaystyle \mathrm{\Gamma }}$ is chosen according to the probability measure${\displaystyle Q}$ and the system moves to the state${\displaystyle x_1={\alpha }_1(x_0)}$ in step 1.
2. Independently of previous maps, another map${\displaystyle {\alpha }_2}$ is chosen according to the probability measure${\displaystyle Q}$ and the system moves to the state${\displaystyle x_2={\alpha }_2(x_1)}$.
3. The procedure repeats.

The random variable${\displaystyle X_n}$ is constructed by means of composition of independent random maps,${\displaystyle X_n={\alpha }_n\circ {\alpha }_{n-1}\circ \cdots \circ {\alpha }_1(X_0)}$. Clearly,${\displaystyle X_n}$ is aMarkov Chain.

Reversely, can, and how, a given MC be represented by the compositions of i.i.d. random transformations? Yes, it can, but not unique. The proof for existence is similar with Birkhoff–von Neumann theorem fordoubly stochastic matrix.

Here is an example that illustrates the existence and non-uniqueness.

Example: If the state space${\displaystyle S=\{1,2\}}$ and the set of the transformations${\displaystyle \mathrm{\Gamma }}$ expressed in terms of deterministic transition matrices. Then a Markov transition matrix can be represented by the following decomposition by the min-max algorithm,

In the meantime, another decomposition could be

Formally,^[3] arandom dynamical system consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.

Let${\displaystyle (\mathrm{\Omega },\mathcal{F},\mathbb{P})}$ be aprobability space, thenoise space. Define thebase flow${\displaystyle \vartheta :\mathbb{R}\times \mathrm{\Omega }\to \mathrm{\Omega }}$ as follows: for each "time"${\displaystyle s\in \mathbb{R}}$, let${\displaystyle {\vartheta }_s:\mathrm{\Omega }\to \mathrm{\Omega }}$ be a measure-preservingmeasurable function:

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

Suppose also that

1. ${\displaystyle {\vartheta }_0={\mathrm{i}\mathrm{d}}_{\mathrm{\Omega }}:\mathrm{\Omega }\to \mathrm{\Omega }}$, theidentity function on${\displaystyle \mathrm{\Omega }}$;
2. for all${\displaystyle s,t\in \mathbb{R}}$,${\displaystyle {\vartheta }_s\circ {\vartheta }_t={\vartheta }_{s+t}}$.

That is,${\displaystyle {\vartheta }_s}$,${\displaystyle s\in \mathbb{R}}$, forms agroup of measure-preserving transformation of the noise${\displaystyle (\mathrm{\Omega },\mathcal{F},\mathbb{P})}$. For one-sided random dynamical systems, one would consider only positive indices${\displaystyle s}$; for discrete-time random dynamical systems, one would consider only integer-valued${\displaystyle s}$; in these cases, the maps${\displaystyle {\vartheta }_s}$ would only form acommutativemonoid 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 themeasure-preserving dynamical system${\displaystyle (\mathrm{\Omega },\mathcal{F},\mathbb{P},\vartheta )}$ isergodic.

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

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

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

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

${\displaystyle W(t,{\vartheta }_s(\omega ))=W(t+s,\omega )-W(s,\omega )}$.

This can be read as saying that${\displaystyle {\vartheta }_s}$ "starts the noise at time${\displaystyle s}$ instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition${\displaystyle x_0}$ with some noise${\displaystyle \omega }$ for${\displaystyle s}$ seconds and then through${\displaystyle t}$ seconds with the same noise (as started from the${\displaystyle s}$ seconds mark) gives the same result as evolving${\displaystyle x_0}$ through${\displaystyle (t+s)}$ seconds with that same noise.

The notion of anattractor 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 apullback attractor.^[4] Moreover, the attractor is dependent upon the realisation${\displaystyle \omega }$ of the noise.

• Chaos theory
• Diffusion process
• Stochastic control

• Random dynamical systems
• Stochastic differential equations
• Stochastic processes

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