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

:$$left{ egin{matrix} mathrm{d} X = f(X) , mathrm{d} t + varepsilon , mathrm{d} W (t); \ X (0) = x_{0}; end{matrix} ight.

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.