Random dynamical system

In mathematics, a random dynamical system is a measure-theoretic formulation of a dynamical system with an element of "randomness", such as the dynamics of solutions to a stochastic differential equation. 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 $f\; :\; mathbb\{R\}^\{d\}\; o\; mathbb\{R\}^\{d\}$ be a $d$-dimensional vector field, and let $varepsilon\; >\; 0$. Suppose that the solution $X(t,\; omega;\; x\_\{0\})$ to the stochastic differential equation

:

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

:$(Omega,\; mathcal\{F\},\; mathbb\{P\})\; :=\; left(\; C\_\{0\}\; (mathbb\{R\};\; mathbb\{R\}^\{d\}),\; mathcal\{B\}\; (C\_\{0\}\; (mathbb\{R\};\; mathbb\{R\}^\{d\})),\; gamma\; ight).$

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

Now define a flow map or (solution operator) $varphi\; :\; mathbb\{R\}\; imes\; Omega\; imes\; mathbb\{R\}^\{d\}\; o\; mathbb\{R\}^\{d\}$ 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 $vartheta\; :\; mathbb\{R\}\; imes\; Omega\; o\; Omega$ as follows: for each "time" $s\; in\; mathbb\{R\}$, let $vartheta\_\{s\}\; :\; Omega\; o\; Omega$ 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
# $vartheta\_\{0\}\; =\; mathrm\{id\}\_\{Omega\}\; :\; Omega\; o\; Omega$, the identity function on $Omega$;
# 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 $varphi\; :\; mathbb\{R\}\; imes\; Omega\; imes\; X\; o\; X$ be a $(mathcal\{B\}\; (mathbb\{R\})\; otimes\; mathcal\{F\}\; otimes\; mathcal\{B\}\; (X),\; mathcal\{B\}\; (X))$-measurable function such that

# for all $omega\; in\; Omega$, $varphi\; (0,\; omega)\; =\; mathrm\{id\}\_\{X\}\; :\; X\; o\; X$, the identity function on $X$;
# for (almost) all $omega\; in\; Omega$, $(t,\; omega,\; x)\; mapsto\; varphi\; (t,\; omega,x)$ is continuous in both $t$ and $x$;
# $varphi$ 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 $W\; :\; mathbb\{R\}\; imes\; Omega\; o\; X$, the base flow $vartheta\_\{s\}\; :\; Omega\; o\; Omega$ 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

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