Abstract
Let (theta; phi) be a continuous random dynamical system defined on a probability space (Omega, F, P) and taking values on a locally compact Hausdorff space E. The associated potential kernel V is given by
V f(omega, x) = integral(infinity)(0) (theta(t)omega, phi(t, omega)x)dt, omega is an element of Omega, x is an element of E.
In this paper, we prove the equivalence of the following statements:
1. The potential kernel of (theta, phi) is proper, i.e. V f is x-continuous for each bounded, x-continuous function f with uniformly random compact support.
2. (theta, phi) has a global Lyapunov function, i.e. a function L : Omega x E -> (0, infinity) which is x-continuous and L(theta(t)omega, phi(t, omega)x) down arrow 0 as t up arrow infinity.
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.