On potential kernels associated with random dynamical systems

Abstract

Let $(\theta,\varphi)$ be a continuous random dynamical system defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and taking values on a locally compact Hausdorff space $E$. The associated potential kernel $V$ is given by $Vf(\omega ,x)= \int\limits_{0}^{\infty} f(\theta_{t}\omega,\varphi(t,\omega)x)dt, \quad \omega \in \Omega, x\in E.$ In this paper, we prove the equivalence of the following statements: 1. The potential kernel of $(\theta,\varphi)$ is proper, i.e. $Vf$ is $x$-continuous for each bounded, $x$-continuous function $f$ with uniformly random compact support. 2. $(\theta,\varphi)$ has a global Lyapunov function, i.e. a function $L:\Omega\times E \rightarrow (0,\infty)$ which is $x$-continuous and $L(\theta_t\omega, \varphi(t,\omega)x)\downarrow 0$ as $t\uparrow \infty$. In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems.