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 with time R+. Let and denote the set of all -invariant measures on Ω×M and the set of all ergodic -invariant measures whose marginal on Ω coincide with respectively. A function is sub-additive with respect to if F(t+s,ω,x) F(t,ω,x)+F(s,σ(t)ω, (t,ω,x)). We define the maximal growth rate of F to be for a.e. ω. It is shown that it is equal to , where and there exists 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.