# Random dynamical system

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{{Short description|Mathematical concept}}
{{Technical|date=January 2022}}
In [mathematics](/source/mathematics), a '''random dynamical system''' is a [dynamical system](/source/dynamical_system) in which the [equations of motion](/source/equations_of_motion) have an element of randomness to them. Random dynamical systems are characterized by a [state space](/source/state_space) ''S'', a [set](/source/set_(mathematics)) of [map](/source/map_(mathematics))s <math>\Gamma</math> from ''S'' into itself that can be thought of as the set of all possible equations of motion, and a [probability distribution](/source/probability_distribution) ''Q'' on the set <math>\Gamma</math> that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state <math>X \in S</math> evolving according to a succession of maps randomly chosen according to the distribution ''Q''.<ref name=Bhattacharya2003>{{cite journal|last1=Bhattacharya |first1=Rabi |first2=Mukul |last2=Majumdar |title=Random dynamical systems: a review|journal=[Economic Theory](/source/Economic_Theory_(journal))|year=2003|volume=23|issue=1|pages=13–38|doi=10.1007/s00199-003-0357-4|s2cid=15055697 }}</ref>

An example of a random dynamical system is a [stochastic differential equation](/source/stochastic_differential_equation); in this case the distribution Q is typically determined by ''noise terms''. It consists of a [base flow](/source/base_flow_(random_dynamical_systems)), the "noise", and a [cocycle](/source/Oseledec_theorem) dynamical system on the "physical" [phase space](/source/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.<ref>{{Cite journal|last1=Ye|first1=Felix X.-F.|last2=Wang|first2=Yue|last3=Qian|first3=Hong|title=Stochastic dynamics: Markov chains and random transformations|journal=Discrete and Continuous Dynamical Systems - Series B|volume=21|issue=7|pages=2337–2361|doi=10.3934/dcdsb.2016050|date=August 2016|doi-access=free}}</ref>

==Motivation 1: Solutions to a stochastic differential equation==

Let <math>f : \mathbb{R}^{d} \to \mathbb{R}^{d}</math> be a <math>d</math>-dimensional [vector field](/source/vector_field), and let <math>\varepsilon > 0</math>. Suppose that the solution <math>X(t, \omega; x_{0})</math> to the stochastic differential equation

:<math>\left\{ \begin{matrix} \mathrm{d} X = f(X) \, \mathrm{d} t + \varepsilon \, \mathrm{d} W (t); \\ X (0) = x_{0}; \end{matrix} \right.</math>

exists for all positive time and some (small) interval of negative time dependent upon <math>\omega \in \Omega</math>, where <math>W : \mathbb{R} \times \Omega \to \mathbb{R}^{d}</math> denotes a <math>d</math>-dimensional [Wiener process](/source/Wiener_process) ([Brownian motion](/source/Brownian_motion)). Implicitly, this statement uses the [classical Wiener](/source/classical_Wiener_space) [probability space](/source/probability_space)

:<math>(\Omega, \mathcal{F}, \mathbb{P}) := \left( C_{0} (\mathbb{R}; \mathbb{R}^{d}), \mathcal{B} (C_{0} (\mathbb{R}; \mathbb{R}^{d})), \gamma \right).</math>

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

Now define a '''flow map''' or ('''solution operator''') <math>\varphi : \mathbb{R} \times \Omega \times \mathbb{R}^{d} \to \mathbb{R}^{d}</math> by

:<math>\varphi (t, \omega, x_{0}) := X(t, \omega; x_{0})</math>

(whenever the right hand side is [well-defined](/source/well-defined)). Then <math>\varphi</math> (or, more precisely, the pair <math>(\mathbb{R}^{d}, \varphi)</math>) 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.

== Motivation 2: Connection to Markov Chain ==
An i.i.d random dynamical system in the discrete space is described by a triplet <math>(S, \Gamma, Q)</math>.
* <math>S</math> is the state space, <math>\{s_1, s_2,\cdots, s_n\}</math>.
* <math>\Gamma</math> is a family of maps of <math>S\rightarrow S</math>. Each such map has a <math>n\times n</math> matrix representation, called ''deterministic transition matrix''. It is a binary matrix but it has exactly one entry 1 in each row and 0s otherwise.
* <math>Q</math> is the probability measure of the <math>\sigma</math>-field of <math>\Gamma</math>.
The discrete random dynamical system comes as follows,
# The system is in some state <math>x_0</math> in <math>S</math>, a map <math>\alpha_1</math> in <math>\Gamma</math> is chosen according to the probability measure <math>Q</math> and the system moves to the state <math>x_1=\alpha_1(x_0)</math> in step 1. 
# Independently of previous maps, another map <math>\alpha_2</math> is chosen according to the probability measure <math>Q</math>  and the system moves to the state <math>x_2=\alpha_2(x_1)</math>.
# The procedure repeats.
The random variable <math>X_n</math> is constructed by means of composition of independent random maps, <math>X_n=\alpha_n\circ \alpha_{n-1}\circ \dots \circ \alpha_1(X_0)</math>.  Clearly, <math>X_n</math> is a [Markov Chain](/source/Markov_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 for [doubly stochastic matrix](/source/doubly_stochastic_matrix).

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

'''Example:''' If the state space <math>S=\{1, 2\}</math> and the set of the transformations <math>\Gamma</math> expressed in terms of deterministic transition matrices. Then a Markov transition matrix<math> M =\left(\begin{array}{cc}
          0.4  &  0.6 \\   0.7 & 0.3 
          \end{array}\right)</math> can be represented by the following decomposition by the min-max algorithm, <math>   M =0.6\left(\begin{array}{cc}
          0 &  1 \\   1 &  0
          \end{array}\right)+0.3 \left(\begin{array}{cc}
          1  &  0 \\   0 & 1 
          \end{array}\right)+ 0.1\left(\begin{array}{cc}
          1  &  0 \\   1 & 0 
          \end{array}\right).</math>

In the meantime, another decomposition could be <math>          M = 0.18 \left(\begin{array}{cc}
              0 &  1 \\   0 & 1 
          \end{array}\right)+ 0.28\left(\begin{array}{cc}
          1  &  0 \\   1 & 0 
          \end{array}\right)
          +0.42\left(\begin{array}{cc}
       0  &  1 \\   1 & 0 
          \end{array}\right)+0.12\left(\begin{array}{cc}
       1 & 0  \\   0 & 1
          \end{array}\right).</math>

==Formal definition==

Formally,<ref>{{cite book |url=https://books.google.com/books?id=W5AY5A3S2kQC|title = Random Dynamical Systems|isbn = 9783540637585|last1 = Arnold|first1 = Ludwig|author-link1=:de:Ludwig Arnold (Mathematiker)|year = 1998}}</ref> a '''random dynamical system''' consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.

Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [probability space](/source/probability_space), the '''noise''' space. Define the '''base flow''' <math>\vartheta : \mathbb{R} \times \Omega \to \Omega</math> as follows: for each "time" <math>s \in \mathbb{R}</math>, let <math>\vartheta_{s} : \Omega \to \Omega</math> be a measure-preserving [measurable function](/source/measurable_function):

:<math>\mathbb{P} (E) = \mathbb{P} (\vartheta_{s}^{-1} (E))</math> for all <math>E \in \mathcal{F}</math> and <math>s \in \mathbb{R}</math>;

Suppose also that
# <math>\vartheta_{0} = \mathrm{id}_{\Omega} : \Omega \to \Omega</math>, the [identity function](/source/identity_function) on <math>\Omega</math>;
# for all <math>s, t \in \mathbb{R}</math>, <math>\vartheta_{s} \circ \vartheta_{t} = \vartheta_{s + t}</math>.

That is, <math>\vartheta_{s}</math>, <math>s \in \mathbb{R}</math>, forms a [group](/source/group_(mathematics)) of measure-preserving transformation of the noise <math>(\Omega, \mathcal{F}, \mathbb{P})</math>. For one-sided random dynamical systems, one would consider only positive indices <math>s</math>; for discrete-time random dynamical systems, one would consider only integer-valued <math>s</math>; in these cases, the maps <math>\vartheta_{s}</math> would only form a [commutative](/source/commutative) [monoid](/source/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](/source/measure-preserving_dynamical_system) <math>(\Omega, \mathcal{F}, \mathbb{P}, \vartheta)</math> is [ergodic](/source/ergodic).

Now let <math>(X, d)</math> be a [complete](/source/complete_space) [separable](/source/separable_space) [metric space](/source/metric_space), the '''phase space'''. Let <math>\varphi : \mathbb{R} \times \Omega \times X \to X</math> be a <math>(\mathcal{B} (\mathbb{R}) \otimes \mathcal{F} \otimes \mathcal{B} (X), \mathcal{B} (X))</math>-measurable function such that

# for all <math>\omega \in \Omega</math>, <math>\varphi (0, \omega) = \mathrm{id}_{X} : X \to X</math>, the identity function on <math>X</math>;
# for (almost) all <math>\omega \in \Omega</math>, <math>(t,x) \mapsto \varphi (t, \omega,x) </math> is [continuous](/source/continuous_function);
# <math>\varphi</math> satisfies the (crude) '''cocycle property''': for [almost all](/source/almost_all) <math>\omega \in \Omega</math>,
::<math>\varphi (t, \vartheta_{s} (\omega)) \circ \varphi (s, \omega) = \varphi (t + s, \omega).</math>

In the case of random dynamical systems driven by a Wiener process <math>W : \mathbb{R} \times \Omega \to X</math>, the base flow <math>\vartheta_{s} : \Omega \to \Omega</math> would be given by

:<math>W (t, \vartheta_{s} (\omega)) = W (t + s, \omega) - W(s, \omega)</math>.

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

==Attractors for random dynamical systems==
The notion of an [attractor](/source/attractor) for a random dynamical system is not as straightforward to define as in the deterministic case. It is necessary to "rewind time", as in the definition of a [pullback attractor](/source/pullback_attractor), so that the noise close to the final time remains consistent.<ref>{{cite journal |doi=10.1007/BF02219225|title=Random attractors|journal=Journal of Dynamics and Differential Equations|volume=9|issue=2|pages=307–341|year=1997|last1=Crauel|first1=Hans|last2=Debussche|first2=Arnaud|last3=Flandoli|first3=Franco|bibcode=1997JDDE....9..307C|s2cid=192603977}}</ref> Moreover, the attractor is dependent upon the realisation <math>\omega</math> of the noise.

==See also==
*[Chaos theory](/source/Chaos_theory)
*[Diffusion process](/source/Diffusion_process)
*[Stochastic control](/source/Stochastic_control)

==References==
{{reflist}}

==Further reading==
*[http://www.scholarpedia.org/article/Stochastic_dynamical_systems Stochastic dynamical systems] on [Scholarpedia](/source/Scholarpedia)

{{Stochastic processes}}

Category:Random dynamical systems
Category:Stochastic differential equations
Category:Stochastic processes

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Adapted from the Wikipedia article [Random dynamical system](https://en.wikipedia.org/wiki/Random_dynamical_system) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Random_dynamical_system?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.