On growth rates of sub-additive functions for semi-flows: Determined and random cases

Let be a measurable random dynamical systems on the compact metric space M over (Ω,F,P,(σ(t))t∈R+) with time R+. Let MP(ϕ) and EP(ϕ) denote the set of all ϕ-invariant measures on Ω×M and the set of all ergodic ϕ-invariant measures whose marginal on Ω coincide with P respectively. A function is sub-additive with respect to ϕ if F(t+s,ω,x)⩽F(t,ω,x)+F(s,σ(t)ω,ϕ(t,ω,x)). We define the maximal growth rate of F to be for P a.e. ω. It is shown that it is equal to , where and there exists ν∈EP(ϕ) such that . The result may have some applications in the study of the dynamical spectrum of infinite dimension random dynamical systems and robust permanence for differential equations.