Table 2.

Summary of the random dynamical system formulation.

Variable	Description
$X=R\times S\in {\mathbb{R}}^d$	Physical state space a random dynamical system
$\vartheta :R\times \Omega \to \Omega$	Base flow of a random dynamical system
φ : R × Ω×X → X	Flow or mapping to physical states
$\omega \left(t\right)≔{\vartheta }_t\left(\omega \right)\in \Omega$	Fluctuations generated by the base flow
R ⊂ X	Internal state space
S ⊂ X	External state space
S ⊂ X	Active states
φS : R × Ω × A×S → S	Mapping to external states
φR : R×R×S → R	Mapping to internal states