Random dynamical system

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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 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.[1]

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[edit]

Let f : \mathbb{R}^{d} \to \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\{ \begin{matrix} \mathrm{d} X = f(X) \, \mathrm{d} t + \varepsilon \, \mathrm{d} W (t); \\ X (0) = x_{0}; \end{matrix} \right.

exists for all positive time and some (small) interval of negative time dependent upon \omega \in \Omega, where W : \mathbb{R} \times \Omega \to \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 \right).

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

Now define a flow map or (solution operator) \varphi : \mathbb{R} \times \Omega \times \mathbb{R}^{d} \to \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[edit]

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} \times \Omega \to \Omega as follows: for each "time" s \in \mathbb{R}, let \vartheta_{s} : \Omega \to \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

  1. \vartheta_{0} = \mathrm{id}_{\Omega} : \Omega \to \Omega, the identity function on \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 (\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} \times \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 \Omega, \varphi (0, \omega) = \mathrm{id}_{X} : X \to X, the identity function on X;
  2. for (almost) all \omega \in \Omega, (t, \omega, x) \mapsto \varphi (t, \omega,x) is continuous in both t and x;
  3. \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} \times \Omega \to X, the base flow \vartheta_{s} : \Omega \to \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[edit]

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[edit]

  1. ^ Bhattacharya, Rabi, and Mukul Majumdar. "Random dynamical systems: a review." Economic Theory 23, no. 1 (2003): 13-38., Rabi; Mukul Majumdar (2003). "Random dynamical systems: a review". Economic Theory 23 (1): 13–38. doi:10.1007/s00199-003-0357-4. Retrieved 26 June 2013. 
  • Crauel, H., Debussche, A., & Flandoli, F. (1997) Random attractors. Journal of Dynamics and Differential Equations. 9(2) 307—341.