Abstract. Let (θ, ϕ) be a continuous random dynamical system defined on a probability space (Ω, F, P) and taking values on a locally compact Hausdorff space E. The associated potential kernel V is given byIn this paper, we prove the equivalence of the following statements:1. The potential kernel of (θ, ϕ) is proper, i.e. V f is x-continuous for each bounded,x-continuous function f with uniformly random compact support. 2. (θ, ϕ) has a global Lyapunov function, i.e. a function L : Ω × E → (0, ∞) which is x-continuous and L(θtω, ϕ(t, ω)x) ↓ 0 as t ↑ ∞.In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.
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